Abstract:
A process for producing a compressed image of a color picture. The color picture is optically scanned to obtain an array of optical parameters representing the basic colors at each coordinate point of the picture which are then converted into digital values. From such digital values, a selection is made of sets of data values representing a series of elements of the picture having discontinuities therein. A model is formulated for each of the elements having a first component containing all of the discontinuities and a second component representing the differentiable content of the elements. The second component is represented by a Taylor polynomial having a selected polynomial degree of at least 2. A maximum allowable error is then selected. The coefficients of the polynomials of the second component are calculated at selected points of a grid with a grid pitch defined by a specified formula. A compressed color picture is derived from the data values constituting the compressed image.

Description:
This application is a continuation of application Ser. No. 538,976, filed Jun. 15, 1990, now abandoned. 
    
    
     BACKGROUND OF THE INVENTION 
     This invention relates to a method for producing an image of an object, storing, transmitting and processing the same. 
     Image means any entity that represents an object exactly, or more or less approximately. The image is the reproduction of a picture. &#34;Intermediate image&#34; means an image that is produced for the purpose of transforming it later into a different image of the same object. When such a transformation occurs, the image finally produced will be called hereinafter &#34;the final image&#34;. An image which is to be processed in any way elaborated to produce another image of the same nature, will be called a &#34;temporary image&#34;, which, if the processing is a correction or adjustment, is an &#34;unadjusted image&#34;. 
     In a great many technical processes, an image of an object must be produced, and quite often must be stored, transmitted or processed. For instance, it is a common occurrence that two-dimensional figures or pictures be represented by digital data which are stored, processed and transmitted, according to needs. This occurs in word processing by computers, message transmission by telefax, etc. 
     It is obviously desirable to reduce as much as possible the amount of data defining the image which represents a given object, without distorting the image to the extent that it might cease to represent the corresponding object with an acceptable degree of accuracy. Such a reduction of the required data, or &#34;data compression&#34; or &#34;image compression&#34;, as it is sometimes called, serves to simplify, reduce and render more economical the equipment required for the storage of an image, its processing and transmission. For instance, it is well known that in modern technology, transmission lines, including frequency bands available for radio transmission, are increasingly overcrowded, and every effort is being made to exploit them as fully as possible, one of the means for so exploiting them being to reduce the amount of data that are sent through a given transmission line in order to convey a given amount of information. 
     It is a general purpose of this invention to provide a method for producing the image of an object, storing it, processing and transmitting it, while minimizing the amount of data that are required for carrying out the said operations. 
     More specific objects of the invention and specific applications thereof, will become apparent as the description proceeds. 
     BRIEF SUMMARY OF THE INVENTION 
     The following considerations are preliminary to an understanding of the process according to the invention. The process according to the invention will be described with a reference to an object which may be broken up into a number of plane lines, corresponding to functions of one variable. 
     In one of its simplest forms, the object, an image of which is to be constructed, may be a plane line. The object lines, as any other object, may be defined in many different ways, but, for the purposes of illustration only, it will be treated as defined by a graph or by a corresponding function, it being evident that the information conveyed by a graph can be conveyed in other suitable ways. In any case, in order to carry out the process according to the invention, the object line must be translated into digital values or into a computer program or subroutine or an analogical process or into the structure of a special purpose digital or analogical computer, which can be entered and memorized in an elaborator, and which define couples of values x, y or each point of the line. The object line may be considered in its entirety, or, more frequently, it will be divided into segments, to each of which the process of the invention will be separately applied. Therefore, if the line has been so divided, the expression &#34;object line&#34;, when used hereinafter, must be construed as meaning the particular segment under consideration at the moment. 
     The process, then, comprises, in a restrictive definition, the following steps: 
     (1) Approximating a line by a model which includes at least one differentiable component. 
     2) Establishing the maximum allowable error ε and the degree k of the Taylor polynomials by which the differentiable component(s) of the model are to be approximated. 
     (3) Establishing at least a pitch grid h and constructing a grid each region of which has one of said pitches h. 
     (4) Computing the coefficients of the Taylor polynomials of the aforesaid differentiable component or components at selected points of said grid. 
     Two or more of the aforesaid steps may be carried out concurrently, in whole or in part, or divided into successive stages, which may be intercalated to a greater or a lesser extent. 
     Further operations, hereinafter described, may be carried out and are often desirable to minimize the effect of inaccuracies in the said coefficients, for rounding them off, for taking into account different scales which may be present in the data, and for obtaining, if desired, an image which has the same nature as the object. &#34;Non-differentiable component&#34; means herein a component comprising one or more points at which it is not differentiable, or, a component that is not differentiable at all its points. 
     The process according to the invention can be extended to objects that are more complex than plane lines by simple generalizations, as will be explained hereinafter. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The invention will be better understood from the following description of preferred embodiments, with reference to the appended drawings, where: 
     FIGS. 1a and 1b illustrate an example of an object line and its image, respectively; 
     FIGS. 2a and 2b illustrate a temporary image line the segments of which do not match at meeting points, and a corresponding adjusted image line, respectively; 
     FIGS. 3a, 3b, 3c, and 3d illustrate respectively an object line and the corresponding model line, final image and non-differentiable component of the model, with reference to Example 1; 
     FIGS. 4a and 4b represent a picture and its image, respectively, with reference to Example 2; 
     FIG. 5 represents a processed image of the picture of FIG. 4a, with reference to Example 3; and 
     FIGS. 6a and 6b represent the negative of the picture of FIG. 4a and its image, respectively, with reference to Example 4. 
    
    
     DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
     The process steps hereinbefore defined will now be more fully explained. 
     Step (1)--The object line, the data defining which have been physically stored e.g. in an electronic memory, is approximated by a model, preferably defined in the same way as the object line, which model preferably consists of at least a first component embodying the characteristics of the object, if any, which render it non-differentiable at some points or regions--it being of course possible to omit said first component if there are no significant characteristics of non-differentiability of the object--and at least a second component which embodies all the differentiable content of the object. 
     Typical cases of models are the following: 
     Case a) The first component is a base line, which is a simple--desirably, the simplest--line having qualitatively the same discontinuities as the object line, and the second components is a curve which represents the deviations therefrom of the object line, and which will be differentiable and can be called interpolating function. The base line may be constructed in each individual instance, or, more conveniently, may be chosen, according to the actual discontinuities of the object line, from a number of normal forms, which are the simplest functions having the required discontinuities. The following standard form of model can be used in this case: 
     
         Φ(x)=Hx.sub.o,a,b,c,d(x)+φ(x)                      (1) 
    
     wherein H is a normal form defined by H(x)=a(x-x o )+b, if x≧x o  or H(x)=c(x-x o )+d, if x is less than x o . The values of the parameters x o ,a,b,c,d are determined, in a preferred embodiment of the invention, by minimizing a quantity representing an error, e.g. the quadratical error, as hereinafter set forth. The base line can be predetermined, or chose, in general according to predetermined criteria, from a list prepared in advance, or it can be chosen in each case by the operator. This case is illustrated at FIG. 1a, 1b showing respectively an object line and its model. 
     Case b) The model is a differentiable function of another function which embodies the non-differentiable characteristics, viz the discontinuities, of the object line. It can be epressed as: 
     
         Φ(x)=Φ&#39;[φ(x)],                                 (2) 
    
     wherein φ is the first component, which will be called the base curve, and Φ&#39; is the second component. φ(x) can be looked at as defining a change of coordinates: in the differentiable component Φ&#39;, the ordinates are referred to abscissae which are not x, but φ(x). 
     Case c) This case will be mentioned here, though it is not applicable to a line, but only to surfaces in a space having three or more dimensions. In the case of three dimensions, a coordinate (say, the elevation) z of a surface, is a function z 1  in a certain region of the plane x-y of the two remaining coordinates and is another function z 2  in another region thereof, the two regions being separated by a border line defined e.g. by a relationship y=φ(x). Then the model φ(z,y) consists of the function z 1  if y is greater than φ(x), and z 2  if y is smaller than φ(x), one or the other of the z 1  and z 2  applying when y=φ(x). 
     Case d) The object line is differentiable at all points, and the model consists only of a differentiable component. 
     In a form of the invention, all the parameters of the model the values of which have to be chosen, are determined by minimizing a quantity representing an error--e.g. The quadratical error, viz. Σ[f(x i )-Φ(x i )] 2  --the minimization being carried out by means of a predetermined subroutine with respect to all the parameters of the model Φ, for the function f(x) representing the object, the values of f(x) for each x being determined by known subroutines. Programs for this purpose are available, e.g. from the ILSM library. 
     Step (2) 
     a)--The maximum allowable error ε, which is to be tolerated in approximating the object line, viz. which expresses the desired precision of the image, is established. 
     b)--The degree k of the Taylor polynomials, which will be used to approximate the differentiable component or interpolating curve, is established. 
     Step (3)--The grid need not be cartesian and its coordinate lines may be curved, although for simplicity&#39;s sake a cartesian grid will always be illustrated herein. The grid may be divided into different regions having different grid pitches or even different types of coordinate lines. The grid pitch h (viz., the distance between adjacent coordinate lines which define the grid cells) is selected according to the precision desired of the image, and may be different in different parts of the regions, although a regular grid will often be preferred. 
     In an embodiment of the invention, h is calculated, by a suitable subroutine, from the formula 
     
         CMh.sup.k+1 ≦ε                              (3) 
    
     wherein C=1/(k+1)! and M is the maximum, at each grid point, of the absolute value of the (partial, in the case of an object which is a function of more than one variable) derivatives of degree k+1 of the differentiable component or components, in the segment or zone of the object under consideration, M being determined by using a known subroutine which computes the derivatives of order k+1, produced e.g. by a package such as MAXIMA OR MATHEMATICA. 
     Step (4)--The nodes of the grid are taken as base points, and a (known e.g. a MAXIMA) subroutine is applied at each base point to compute the Taylor polynomials of degree k of the interpolating curve. 
     At this stage, the following data have been obtained: 
     A) The coefficients of the Taylor polynomials of the differentiable component or components of the model; 
     B) The number or other identification or analytical definition of the non-differentiable component(s), if any, of the model, such as the base line or the base curve; 
     C) The values of the parameters of the said non-differentiable component(s), if any; 
     and these define an image, which will usually be an intermediate image, but could be a final one, according to cases. Hereinafter it will be assumed that it is an intermediate image, from which the final image, in the same form as the original object, is to be constructed; however this is done merely for the sake of simplicity and involves no limitation. 
     In many cases, as will be explained below, the image thus obtained may require further elaboration without changing its nature, viz. while remaining a set of data of the same kind, and it will be only a temporary, in particular an unadjusted image. Then some or all of the steps from (5) on will be carried out. 
     Step (5)--In the case of the presence of so-called noise or inaccuracies in said temporary image line, or if the numerical noise, viz. the inaccuracies of the computations, which are large in comparison with the accuracy required, the Taylor polynomials which make up the temporary image line or its differentiable component may disagree at their meeting points by more than allowed by the required accuracy, as represented, by way of example, in FIG. 2a. 
     In this case, an adjusted image line is constructed by applying to each differentiable component a subroutine, hereinafter referred to as &#34;Whitney subroutine&#34;, which computes W, wherein W is a quantity representing the discrepancies of the Taylor polynomials. In particular, W can be given by a formula: 
     
         W=Σ.sub.ij ∥p.sub.i -(p.sub.j).sub.i ∥.sup.2 (3) 
    
     Here the sum is taken over all the adjacent grid points i, j (possibly belonging to different segments of the image). p i , p j  denote the Taylor polynomials, obtained in steps (1)-(4) at the grid points i, j, and (p j ) i  denotes the polynomial P j , expressed in coordinates, centered at the i-th grid point. ∥p-q∥ 2  denotes, for any two polynomials p and q of the same degree and number of variables, the sum of squares of the differences of corresponding coefficients. 
     For any values of the coefficients of p i , W is computed by using known subroutines, produced e.g. by a package such as MATHEMATICA. W is then minimized (e.g. by standard gradient methods), using, as starting values of the coefficient of the Taylor polynomials, those obtained by the previous steps, and under such constraints that the result of the minimization do not deviate from the initial data by more than the allowed error, e.g. under the condition that the zero degree coefficients of said polynomials remain unchanged. An adjusted image line, corresponding to the unadjusted image line of FIG. 2a, is illustrated by way of example in FIG. 2b. 
     Step (6)--If the accuracy of the adjusted coefficients of the Taylor polynomials obtained from step (5) is excessive with respect to that desired in the final image, they are rounded off to a maximum allowable error ε&#39; by any suitable method (not necessarily the same for coefficients of different degrees). The data thus obtained represent the adjusted image line. 
     Step (7)--Sometimes the data of the object to be represented may require the use of different grid resolutions, or such use may be desirable. An example which clarifies this case is the following. 
     Let us assume that the object represents a periodic phenomenon, e.g. an oscillatory phenomenon such as an oscillating electrical impulse or an electromagnetic wave. Such a phenomenon can be analyzed and is usually represented by the combination of two or more superimposed components, specifically, a relatively low frequency carrier wave and a higher frequency modulating wave. The modulation can be sometimes considered as resulting from a first, intermediate frequency modulation, and one or more high frequency modulation or modulations, and in this case the object will have three or more components. The image can be conveniently constructed from images of the various components, e.g. of the carrier wave and of the modulating wave or waves, and obviously the lower frequencies will require lower resolutions and larger grid pitches will be suitable for them. Likewise, the frequency of an oscillatory phenomenon may vary at different times or in different spatial regions and its components will not be superimposed, but separated in space. Similar situations may occur in various cases. Generally, many kinds of objects may comprise superimposed or separated components which have details of different fineness, which require different degrees of resolution. Sine oscillatory phenomena are a typical case of objects requiring different grid resolutions, the word &#34;frequency&#34; will be used to indicate the fineness of the required grid, but this is not to be taken as a limitation, since the same procedure can be applied to non-oscillatory phenomena. 
     In such cases, the following procedure is preferably followed: 
     a) Steps 1 to 6 (or such among them which are necessary in the specific case) are carried out and a first temporary image is obtained. 
     b) A new maximum error ε 2 , bigger than ε (or ε&#39;, as the case may be) is chosen. 
     c) A grid which is sparser than the one used for carrying out the steps under a), and the pitch of which is determined by the resolution required by the lowest frequency of the components existing in the object (e.g. that of a carrier wave) is established. 
     d) Steps 1 to 6 are repeated using ε 2  and the sparser grid and a second temporary image is obtained. 
     e) The second temporary image thus obtained is substracted from the first and a first residual image is obtained, which contains only data relating to higher frequency components of the object. 
     f) The same procedure--steps b) to e)--is repeated for successively higher frequencies of components, correspondingly obtaining successive residual images increasingly restricted to higher frequency components. 
     As a result, coefficients of Taylor polynomials are obtained on several grids having increasingly higher resolutions, viz. smaller pitches, separately corresponding to the object components required increasingly higher resolutions. 
     The data obtained after steps (1) to (4) and those among (5), (6), (7), which it has been found necessary to perform, constitute an intermediate image or sometimes a final one. Usually these are the compressed data which can be stored, transmitted and processed. 
     If a further compression is desirable, one of the standard methods of encoding coefficients (e.g. Hoffman coding) can be applied. If necessary, the resulting string of data can be further compressed by one of the standard methods of unstructured data compression (e.g. entropy compression). However, this last step reduces the possibility of a compressed data processing. 
     If a final image, which has the same nature as the object, is to be constructed, the following procedure is followed: 
     Step (8) 
     a) The Taylor polynomial coefficients obtained after completion of steps (1) to (4) and of those among steps (5) and (6) which it has been found necessary to perform, are treated as if they represented an unadjusted temporary image, which is affected by noise, and are subjected once more to step (5), using them as starting data. 
     b) The domain in which the temporary image has been defined is divided into regions by means of a grid, each region being a portion of the grid around a grid node or base point. These regions may overlap. 
     c) A curve or curves representing the Taylor polynomials of degree k in the above regions are constructed from the coefficients defining the temporary image--e.g. obtained as in step (8) a)--at each node of the grid or of that grid having the highest resolution (smallest pitch), if there is more than one grid (particularly if step (7) has been carried out), using a known subroutine. 
     Said curve or curves constitute the final image of the object line. The Aforementioned curves may diverge at the meeting points of the regions mentioned above under b) (or on their overlapping parts.) If this disagreement does not exceed the allowable error ε, any of the overlapping curves can be used at the meeting points on the overlapping parts of the above regions. 
     If as the result of the noise of the data or the computational noise, the above discrepancies are large in comparison with the accuracy required, average values can be used on the overlapping parts. This is done by averaging the values of the overlapping curves with the appropriate weights. 
     Actually, other polynomials or functions could be used for approximation purposes, such as Tchebicheff polynomials, trigonometric exponential functions, etc., without departing from the invention, but Taylor polynomials are preferred. 
     The above described process applies, with obvious generalization, to a wide range of objects. Some examples follow. 
     I--A surface in a three-dimensional space corresponds to a function of two variables. If the surface is defined in a space that has more than three, say, n+1 dimensions, the independent variables will be more than two, say, n (x 1 ,x 2 , . . . x n ), but the operations to be carried out will be essentially the same, and the necessary generalizations will be obvious to skilled persons. In any case, any surface can be translated, as well as a line, to digital values, which can be entered and stored. The model will be constructed in the same way as for a line. Case c) of model construction, already described, applied to surfaces in any space. Analogously to case a), a model may consists of a simple base surface, which presents the discontinuities of the object, surface, and by a differentiable or interpolating surface, which represents the deviations of the object from the base surface. One can also operate analogously to case b), by using functions of more than one variable. The minimization of the quadratical error is effected in the same way as in the case of an object line, using values of Φ ij  . . . n and f(x i ,x j , . . . , x n ) which depend on n variables. The remaining steps are likewise adapted to the existence of n variables. All derivatives, of course, will be partial derivatives. The construction of the final image from the temporary image--step (8)--can likewise be carried out with obvious generalizations in the case of images defined in a space having any number of dimensions. 
     II--A surface can be considered as a family of lines, which are obtained by the intersection of the surface with a family of planes, e.g. vertical planes the orientation of which is taken as that of the x-axis, identified by a parameter, e.g. their y coordinate. A family of curves in a plane, depending on one parameter, as may result from the representation of any number of phenomena, is obviously equivalent to that of a surface and may be treated as such, or vice versa. 
     III--A particular case of an object which is a surface is, e.g., a terrain, wherein the surface is defined by the elevation as a function of two plane (cartesian or polar) coordinates. 
     IV--A building can be represented in the same way, if it is very simple. If its shape is complex, however, it must be broken up into a number of component parts. However, if it is desired to represent it as it is seen from the outside, say by an observer which can place himself at any vantage point within a certain distance from the building, the observer&#39;s position can be identified by three coordinates, x, y and z (or polar coordinates), or by two, if it assumed that the observer&#39;s eye is at a given level. From each position of the observer point it is possible, if the configuration of the terrain is known, to determine the distance D on each line of sight from the observer&#39;s eye to the building surface, and this will determine how the building is seen. Each line of sight can be identified by two coordinates: e.g. its inclination (the angle thereof with the vertical in a vertical plane which contains it), and its azimuth (the angle of said vertical plane with a reference vertical plane, e.g. one that contains the geographic or magnetic north). The way in which the building appears to be observer, is therefore defined by a function D of five variables, viz. by a surface in a six-dimensional space. 
     V--A family of curves in a plane, depending on more than one parameter, is obviously equivalent to a surface in a space having more than three dimensions. 
     VI--If in example IV above the coordinates of the observer are known as a function of a single variable, say, when he approaches the building along a given line, in which case the variable is the distance covered from a starting point, or in motion, as in a vehicle, along a given line, in which case the variable is time. In this case the variables of the surface become three (e.g. distance or time and inclination and azimuth) and the space is only four-dimensional, but the four-dimensional surface is subject to the constraint represented by the definition of the observer&#39;s motion. In general, in many cases, the degree of the space in which the surface is defined may be reduced by the introduction of suitable constraints. 
     VII--The final image of a colour picture is another colour picture, that is not identical, but sufficiently similar to the object picture. The object picture can be scanned by known apparatus (scanners), by means of white light, and for each point the intensity of the three basic colours (magenta, cyan and yellow) may be measured and registered. The object is thus reduced to three partial or component objects, each consisting of the distribution of one basic colour over the picture and having a physical reality, as it is equivalent to the colour picture that would be contained by exposing the original through three filters, having colours complementary to the three basic colours, or, in practice, to an array of digital data representing such one-coloured picture. Each of said partial objects can be subjected to the process of the invention, to produce a reduced or compressed array of data, constituting a partial image, and the partial images can be transformed into a combined final image approximating the original object, by processes known to those skilled in the art. If the partial images must be stored and/or transmitted, the process of the invention will facilitate doing this and render it more economical. In the same way a dynamic coloured picture, such as a movie or a TV broadcast, can be reduced to a final dynamic image. 
     A particular advantage and a preferred aspect of the invention consists in the possibility of processing the compressed intermediate image obtained as set forth hereinbefore and producing from it a processed final image, which does not represent the object but represents what would have been the result of processing the object. The processed intermediate image can be stored and transmitted with the already mentioned savings and advantages inherent in the reduction of the number of data, but said reduction is even more advantageous in the processing, for it is obviously more convenient to process a reduced instead of a larger amount of data. Said processing in a compressed form, as it may be called, is made possible by the following property: 
     Let F be an operator which is analytic in nature, viz. can be defined by mathematical relationships. Let O be an object of any nature, but which can be represented by Taylor polynomials p i . Then by applying operator F to the p i  &#39;s, one obtains polynomials which represent the object that would be obtained by applying the operator F to the object O. If one uses the symbol≃to indicate that an array of polynomials represents an object, one can write: 
     
         if p.sub.i ≃O, then F(p.sub.i)≃F(O). 
    
     Elementary examples of analytic operators are algebraic operations, rotations of geometrical figures, changes of coordinates in general, etc. These operators are represented by mathematical operations. If F(O) is to be constructed, such operations must be carried out on all the data, e.g. digital data, which define the object. But of a compressed image has been obtained as set forth above, and an array of Taylor polynomial coefficients has been obtained, which are in a much smaller number than the said digital data, said mathematical operations can be carried out on said coefficients, and a processed intermediate image will be obtained, which represents F(O) and from which F(O) can be constructed as set forth in step (9) above. 
     The following examples illustrate a number of embodiments of the invention. 
     EXAMPLE 1 
     An object line f in the plane (x, y) is given by an array A=(y 0 , y 1 , . . . , y 100 ), where y i  =f(x i ), x i  =i/100, i=0,1, . . . , 100. In this specific example the array (array 1) is the following: 
     
         __________________________________________________________________________0.1152    0.1155   0.1191       0.1131           0.1174               0.1133                   0.1108                       0.1149                           0.11050.1182    0.1167   0.1206       0.1238           0.1196               0.1264                   0.1282                       0.1313                           0.13150.1299    0.1330   0.1366       0.1409           0.1402               0.1462                   0.1569                       0.1608                           0.16310.1604    0.1693   0.1779       0.1797           0.1826               0.1826                   0.1888                       0.1963                           0.20110.2034    0.2084   0.2170       0.2244           0.2265               0.2327                   0.2429                       0.2468                           0.24720.2523    0.2661   0.2673       0.2702           0.2796               0.2811                   0.2845                       0.2949                           0.30220.3078    0.3049   0.3121       0.3157           0.3256               0.3270                   0.3346                       0.3413                           0.34050.3428    0.3503   0.3513       0.3530           0.3571               0.3675                   0.3616                       0.4648                           0.46650.4659    0.4607   0.4600       0.4536           0.4473               0.4441                   0.4427                       0.4330                           0.43290.4268    0.4243   0.4185       0.4135           0.4107               0.3961                   0.3925                       0.3877                           0.37740.3698    0.3671   0.3583       0.3449           0.3397               0.3338                   0.3271                       0.3091                           0.30310.2929__________________________________________________________________________ 
    
     An object line itself is shown in FIG. 3a. The required accuracy of representing this line is 0.035. The compressed image of this line is produced as follows. 
     Firstly it is subdivided into three segments lying over the segments [0.0, 0.6], [0.6, 0.8], [0.8, 1.0] in the x-axis. The following model is chosen on the segments [0.0, 0.6] and [0.8, 1.0]: 
     y=Q(x)=c 1  sin (ω 1  x+φ 1 )+c 2  cos (ω 2  x+φ 2 )+c 3  x 2  +c 4  x+c 5  with c 1 , c 2 , ω 1 , ω 2 , φ 1 , φ 2 , c 3 , c 4 , c 5  -the parameters. 
     On the segment [0.6, 0.8] the following model is chosen: 
     y=Q(x)+Hx 0 , a, b, c, d (x), where Q(x) is as above, and the normal form H is defined by H(x)=a(x-x 0 )+b, if x≧x 0  or H(x)=c(x-x 0 )+d, if x is less than x 0 . Said normal form is illustrated in FIG. 3d. Approximation on each segment is carried out by minimization, with respect to the corresponding parameters, of the quadratic error: 
     Σ(y i  -Q(x i )) 2  (Σ(y i  -Q(x i )-H(x i )) 2  on [0.6, 0.8]). 
     The values of the parameters found are given in the following array 2. ##EQU1## 
     The corresponding model curve is shown in FIG. 3b. 
     The error of the approximation of the object line by the model found turns out to be 0.005. Respectively, on the step 2, ε is chosen to be 0.03. k is chosen to be 2 on each segment. 
     M, equal to the maximum absolute value of the third derivative of the smooth component in the above mode, as computed by the standard subroutine, is 8. The maximal possible pitch h of the grid to be constructed, is defined by (1/6) M (h/2) 3  =ε, or h≃0.24. In order to provide a uniform grid, a smaller value h=0.2 is chosen on each segment. The corresponding grid points are the following: 0.1, 0.3, 0.5 on [0.0, 0.6], the only grid point 0.7 on [0.6, 0.8] and the only grid point 0.9 on [0.8, 1.0]. Taylor polynomials at these points, as computed by the standard &#34;MATHEMATICA&#34; subroutine, are given in the following array 3. 
     
         ______________________________________Zi   a0           a1            a2______________________________________0.1  0.121582031250             0.105957031250                           0.9936523437500.3  0.180175781250             0.454345703125                           0.6320800781250.5  0.285644531250             0.542724609375                           -0.2360839843750.9  0.381103515625             -0.727050781250                           -1.3940429687500.7  0.272460937500             0.125244140625                           -1.083496093750______________________________________a =   0.142822265625               b =       0.099853515625c =   -0.333251953125               d =       0.199951171875______________________________________ 
    
     Now the coefficients of order 0 are rounded off up to 3 digits, the coefficients of order 1 are rounded off up to 2 digits and the coefficients of order 2 up to 1 digit. The parameters of the normal form H are rounded off up to three digits. These data, listed in the following array 4 represent the intermediate compressed image. 
     
         ______________________________________Zi     a0             a1      a2______________________________________0.1    0.121          0.10    0.90.3    0.180          0.45    0.60.5    0.285          0.54    -0.20.9    0.381          -0.72   -1.30.7    0.272          0.12    -1.0______________________________________a =        0.142   b =          0.100c =        -0.333  d =          0.200______________________________________ 
    
     The compression ratio is 4*100 digits/37 digits≃10.8. 
     The final image is obtained by computing the values of the Taylor polynomials (and the normal form H on [0.6, 0.8]) at the initial points x, i=0, . . . , 100. Each polynomial is used for x, belonging to the corresponding cell of the grid z i . The result is shown in the following array 5. 
     
         __________________________________________________________________________0.1196    0.1190   0.1185       0.1183           0.1182               0.1183                   0.1186                       0.1190                           0.11970.1205    0.1215   0.1227       0.1240           0.1256               0.1273                   0.1292                       0.1313                           0.13350.1360    0.1386   0.1426       0.1460           0.1496               0.1532                   0.1570                       0.1609                           0.16490.1691    0.1733   0.1777       0.1822           0.1868               0.1916                   0.1964                       0.2014                           0.20650.2117    0.2171   0.2225       0.2281           0.2318               0.2376                   0.2433                       0.2490                           0.25460.2602    0.2658   0.2713       0.2768           0.2822               0.2876                   0.2930                       0.2983                           0.30360.3088    0.3140   0.3192       0.3243           0.3294               0.3344                   0.3380                       0.3424                           0.34660.3506    0.3545   0.3581       0.3615           0.3648               0.3678                   0.3706                       0.4709                           0.46850.4660    0.4633   0.4603       0.4572           0.4539               0.4503                   0.4466                       0.4427                           0.43760.4328    0.4276   0.4223       0.4166           0.4107               0.4046                   0.3981                       0.3915                           0.38450.3773    0.3699   0.3621       0.3542           0.3459               0.3374                   0.3287                       0.3196                           0.31040.3008__________________________________________________________________________ 
    
     The corresponding final curve is shown in FIG. 3c. The maximal error in representing the object curve by the final one is 0.033. 
     EXAMPLE 2 
     The object (black and white, continuous tone) picture is the standard test picture, called &#34;Lena&#34; (see FIG. 4a). It is represented by a 512×512 array, each pixel containing 8 bits, representing one of the gray levels between 0 to 255. The file representing this picture is available in test collections in the field of imaging. A part of this array, representing the piece S, marked on FIG. 4a, is the following. 
     
         __________________________________________________________________________97 97 97 97 97 97 100                98 96 94 92 90 89 86 84 84 8689 93 96 101       109          120             133                152                   171                      188                         202                            213                               222                                  214                                     200                                        185                                           167148   127 138    136       133          130             128                125                   115                      114                         113                            111                               110                                  10997 97 97 97 97 97 99 97 95 93 91 89 89 85 83 83 8589 92 95 100       108          119             132                150                   169                      186                         200                            211                               220                                  217                                     203                                        187                                           169149   127 138    135       132          130             127                124                   115                      114                         113                            111                               110                                  10997 97 97 97 97 97 98 96 94 92 90 88 88 84 83 83 8488 92 94 100       108          118             132                148                   167                      184                         198                            209                               218                                  221                                     206                                        189                                           171150   128 137    134       132          129             126                124                   115                      114                         113                            111                               110                                  10997 97 97 97 97 97 98 96 94 92 90 88 87 84 82 82 8488 92 94 99 107          118             131                146                   165                      182                         196                            207                               216                                  224                                     208                                        191                                           172152   129 136    134       131          128             126                123                   115                      114                         123                            111                               110                                  10997 97 97 97 97 97 97 95 93 91 89 87 87 83 81 81 8387 90 93 98 106          117             130                144                   163                      180                         194                            205                               214                                  227                                     211                                        193                                           174153   130 136    133       130          128             125                122                   115                      114                         113                            111                               110                                  10997 97 97 97 97 97 96 94 92 90 88 86 86 82 81 81 8286 90 92 98 106          116             130                142                   161                      178                         192                            203                               212                                  230                                     214                                        196                                           176154   131 135    132       130          127             124                122                   115                      114                         113                            111                               110                                  10999 96 94 93 93 94 93 94 93 92 90 87 85 85 85 85 8585 86 90 96 104          114             125                136                   145                      158                         175                            195                               219                                  220                                     222                                        213                                           193161   117 134    132       129          127             125                122                   121                      118                         115                            111                               108                                  10595 93 91 91 91 92 95 95 95 93 91 88 84 84 84 84 8484 87 91 97 104          113             124                134                   144                      157                         174                            194                               218                                  220                                     223                                        215                                           195163   120 133    131       129          127             125                123                   121                      118                         115                            111                               108                                  10592 90 89 88 89 91 96 96 96 95 92 89 84 84 84 84 8484 88 92 97 104          112             123                133                   143                      156                         172                            193                               216                                  220                                     224                                        216                                           196165   122 131    129       128          126             125                123                   121                      118                         118                            111                               108                                  10588 86 86 86 87 89 97 98 97 96 94 91 84 84 84 84 8484 89 92 97 103          112             122                132                   141                      154                         171                            191                               215                                  220                                     224                                        217                                           198167   125 128    127       125          124             123                122                   121                      118                         115                            111                               108                                  10584 83 83 83 85 87 99 99 99 97 95 92 86 86 86 86 8686 91 93 97 103          111             121                130                   140                      153                         110                            190                               214                                  221                                     225                                        218                                           199169   127 124    123       122          122             121                120                   121                      118                         115                            111                               108                                  10580 80 80 81 83 86 100                100                   100                      99 96 93 88 88 88 88 8888 92 94 97 103          110             119                129                   139                      152                         168                            189                               212                                  221                                     226                                        219                                           201171   130 119    119       118          118             118                118                   121                      118                         115                            111                               108                                  10565 71 78 85 91 98 100                100                   100                      101                         101                            102                               102                                  100                                     97 94 8883 91 91 93 97 105             115                128                   143                      157                         172                            187                               201                                  223                                     229                                        223                                           205174   130 118    118       118          118             118                118                   118                      114                         111                            107                               103                                  9965 72 79 85 92 99 104                104                   104                      104                         104                            104                               107                                  105                                     103                                        99 9590 89 88 90 95 102             112                128                   143                      158                         172                            187                               202                                  219                                     226                                        220                                           202172   129 118    118       118          118             118                118                   115                      112                         109                            105                               102                                  9966 73 79 86 93 99 106                105                   105                      104                         104                            104                               108                                  107                                     105                                        102                                           9894 86 85 87 92 99 109                127                   141                      156                         171                            185                               200                                  214                                     221                                        216                                           198168   126 118    118       118          118             118                118                   113                      110                         108                            105                               102                                  9967 73 80 87 93 100             105                104                   103                      102                         101                            100                               107                                  106                                     105                                        102                                           9995 83 83 85 89 97 107                123                   138                      153                         167                            182                               197                                  207                                     214                                        210                                           193163   121 118    118       118          118             118                118                   112                      110                         107                            105                               102                                  10067 74 81 87 94 101             101                100                   99 97 96 95 103                                  103                                     102                                        100                                           9793 81 80 82 87 94 104                118                   133                      148                         162                            177                               192                                  198                                     206                                        202                                           185156   114 118    118       118          118             118                118                   112                      110                         108                            106                               104                                  10268 75 81 88 95 101             95 93 92 90 88 86 97 97 96 95 9289 78 77 79 84 91 101                111                   126                      141                         155                            170                               185                                  187                                     196                                        192                                           176147   106 118    118       118          118             118                118                   113                      111                         110                            108                               106                                  10579 83 88 93 97 102             94 69 48 30 15 4  40 51 59 65 6868 80 80 82 85 88 93 94 110                      128                         146                            164                               184                                  192                                     176                                        160                                           146132   120 116    115       115          115             115                114                   103                      106                         104                            102                               98 9480 83 86 90 93 96 89 68 50 36 26 19 44 53 59 62 6361 71 72 14 76 80 84 86 102                      119                         137                            156                               176                                  185                                     171                                        157                                           145133   122 117    116       115          115             114                113                   109                      108                         106                            104                               100                                  9681 83 85 87 89 91 86 69 56 46 39 36 53 59 63 64 6358 66 67 69 72 75 80 81 97 114                         132                            151                               171                                  179                                     166                                        155                                           144134   125 118    117       116          114             113                112                   110                      109                         107                            104                               101                                  9684 85 85 86 87 87 86 73 64 58 55 56 66 70 71 70 6760 65 66 68 70 74 78 80 96 113                         131                            150                               170                                  173                                     162                                        152                                           143134   127 119    117       116          114             113                111                   110                      109                         107                            104                               101                                  9688 87 86 86 85 84 89 80 74 72 74 79 83 85 84 81 7566 67 68 70 72 76 80 82 98 115                         133                            152                               172                                  166                                     157                                        149                                           141135   129 120    118       116          114             112                110                   109                      108                         106                            104                               100                                  9692 90 88 86 84 82 94 89 88 90 95 104                               105                                  105                                     102                                        96 8877 73 74 76 78 82 86 88 104                      121                         139                            158                               178                                  160                                     152                                        146                                           140136   132 121    119       116          114             111                109                   107                      106                         104                            102                               98 94107   104 101    98 94 91 86 84 83 83 84 86 91 97 98 93 8368 62 71 82 96 113             132                158                   162                      166                         168                            170                               171                                  154                                     148                                        141                                           134128   121 124    192       119          116             114                111                   105                      104                         102                            101                               100                                  98113   110 107    105       102          99 91 89 88 88 89 91 94 103                                     106                                        104                                           9784 87 96 107       121          138             157                169                   172                      174                         176                            176                               176                                  154                                     147                                        140                                           134127   120 124    121       118          116             113                110                   104                      103                         102                            100                               99 98117   114 112    110       108          106             95 94 93 93 94 95 99 111                                     117                                        117                                           113103   105 114    125       139          156             175                178                   179                      180                         180                            179                               178                                  153                                     146                                        140                                           133126   120 123    120       118          115             112                110                   104                      102                         101                            100                               98 97119   117 115    114       112          110             100                98 97 97 98 100                               106                                  120                                     129                                        132                                           130123   116 124    136       150          166             186                184                   184                      184                         182                            180                               177                                  152                                     146                                        139                                           132126   119 122    120       117          114             112                109                   103                      102                         100                            99 98 96119   118 117    115       114          113             105                103                   102                      102                         103                            105                               114                                  131                                     142                                        148                                           149144   119 128    139       153          170             189                187                   186                      184                         182                            178                               174                                  152                                     145                                        138                                           132125   118 122    119       116          114             111                108                   102                      101                         100                            98 97 96118   117 116    115       114          114             109                108                   107                      107                         108                            109                               125                                  144                                     158                                        167                                           170168   116 124    136       150          166             186                188                   185                      182                         178                            173                               168                                  151                                     144                                        138                                           131124   118 121    118       116          113             110                108                   102                      100                         99 98 96 95118   118 118    118       118          118             111                110                   111                      114                         118                            125                               131                                  156                                     133                                        183                                           184177   152 145    145       153          169             194                185                   184                      182                         180                            179                               177                                  154                                     145                                        136                                           129123   117 112    114       114          112             108                102                   96 108                         111                            107                               95 75118   118 118    118       118          118             114                113                   114                      117                         121                            127                               136                                  161                                     178                                        186                                           187180   157 148    147       155          170             193                188                   186                      184                         182                            180                               178                                  153                                     144                                        136                                           128122   116 112    114       114          112             108                102                   98 107                         108                            100                               85 62118   118 118    118       118          118             117                116                   117                      119                         124                            130                               141                                  166                                     182                                        190                                           190183   161 151    150       156          171             193                190                   188                      185                         183                            180                               178                                  152                                     143                                        135                                           128121   116 112    114       114          112             108                102                   101                      106                         104                            93 75 49118   118 118    118       118          118             119                118                   119                      122                         126                            133                               147                                  170                                     186                                        194                                           194185   165 155    152       158          171             193                191                   188                      185                         182                            180                               177                                  152                                     143                                        134                                           127121   115 112    114       114          112             108                102                   103                      106                         100                            87 65 35118   118 118    118       118          118             122                121                   122                      125                         129                            135                               152                                  175                                     190                                        198                                           197188   169 158    155       159          172             193                191                   188                      185                         181                            178                               775                                  151                                     142                                        134                                           126120   114 112    114       114          112             108                102                   106                      105                         96 80 55 22118   118 118    118       118          118             125                124                   125                      127                         132                            138                               157                                  180                                     195                                        201                                           200191   174 161    157       161          173             192                191                   187                      183                         179                            176                               172                                  150                                     141                                        133                                           126119   114 112    114       114          112             108                102                   108                      104                         93 73 45 9114   117 119    122       125          127             120                123                   126                      130                         136                            142                               160                                  176                                     188                                        196                                           198196   158 152    151       154          162             174                198                   191                      184                         178                            173                               169                                  160                                     140                                        125                                           116112   113 112    111       110          109             108                107                   121                      90 64 44 29 19112   115 118    120       123          126             124                126                   128                      132                         136                            141                               168                                  184                                     195                                        202                                           204202   175 168    166       168          174             185                200                   194                      189                         185                            182                               180                                  161                                     140                                        125                                           115110   111 111    110       109          109             108                107                   111                      82 58 39 26 18111   114 113    119       122          125             128                128                   130                      132                         136                            140                               168                                  183                                     194                                        200                                           202199   186 178    174       174          180             189                193                   189                      186                         184                            183                               182                                  160                                     139                                        123                                           113108   108 109    109       108          108             108                108                   102                      74 52 35 23 17111   114 117    119       122          125             130                130                   130                      132                         134                            138                               158                                  193                                     183                                        189                                           190187   190 181    176       175          179             187                179                   177                      175                         175                            175                               177                                  158                                     137                                        121                                           110104   104 107    107       108          108             108                108                   92 66 46 30 21 16112   115 118    120       123          126             131                130                   130                      130                         132                            134                               140                                  154                                     164                                        169                                           170166   189 178    172       170          172             179                156                   156                      156                         158                            160                               163                                  156                                     134                                        117                                           106100   99 105    106       107          107             108                109                   82 58 39 26 18 15114   117 119    122       125          127             132                130                   128                      128                         129                            130                               112                                  126                                     136                                        140                                           141137   181 169    161       158          159             164                126                   127                      130                         133                            137                               142                                  153                                     130                                        113                                           10195 94 104    105       106          107             108                109                   72 50 33 22 15 14117   115 113    112       113          114             109                105                   102                      101                         100                            100                               97 102                                     105                                        108                                           110111   134 134    131       127          121             113                110                   113                      116                         119                            122                               125                                  111                                     100                                        93 9193 100 91 110       120          121             114                97 29 27 25 24 22 20100   98 96 96 96 97 95 92 90 88 88 88 87 91 95 97 99100   117 116    113       108          102             94 81 85 89 93 97 101                                  91 82 78 7882 91 100    114       120          116             104                83 25 24 23 21 20 1986 84 82 82 82 83 82 80 78 77 77 78 77 82 85 88 9091 103 102    98 93 86 78 61 66 71 76 81 86 78 72 69 7278 89 107    117       118          110             94 68 23 22 21 20 19 1875 73 71 70 71 72 71 68 67 67 67 68 70 75 78 81 8383 93 91 87 82 74 65 50 56 62 68 73 79 73 68 68 7382 95 113    119       116          103             82 52 21 20 20 20 19 1966 64 62 62 62 63 60 58 57 57 58 60 65 69 73 75 7778 86 84 80 74 66 56 48 55 62 68 75 82 74 72 74 8192 108 119    120       112          96 70 36 20 20 20 20 20 2060 58 56 56 56 57 50 48 48 48 50 52 61 65 69 71 7374 83 80 76 69 61 51 55 63 70 78 85 93 83 83 87 96110   127 124    120       108          67 57 18 20 20 21 21 22 22__________________________________________________________________________ 
    
     The compressed image is produced as follows: first, the picture is subdivided into square segments, containing 6×6=36 pixels each one. See FIG. 4a and the array S above, where one of such segments is marked. The step 1 consists in approximating the picture on each segment by the model, which is chosen to be the quadratic polynomial 
     
         z=a.sub.0 +a.sub.1 x+a.sub.2 y+a.sub.11 x.sup.2 +2a.sub.12 xy+a.sub.22 y.sup.2 
    
     where z represents the gray level, and x and y are the coordinates on the picture plane centered at the center of the corresponding segment. The values of the coefficient &#34;a&#34; are found by the standard subroutine, minimizing the quadratic error of the approximation of the gray level on each segment by the model chosen. 
     The array of 8×8=64 polynomials, obtained on the segments, covering the piece S of the picture, is given in the following array 7. 
     
         __________________________________________________________________________0.37500000 -0.0058594        -0.0055246              0.001883                     -0.007146                           0.0109860.35937500 -0.0104353        -0.0276228              0.007010                     0.013632                           0.0092080.31640625 -0.0081473        0.0078125              0.010149                     -0.001263                           0.0330640.39843750 -0.0127790        0.0998326              0.003348                     0.014263                           0.0623600.74218750 -0.0247767        0.1643973              -0.015172                     0.000459                           -0.0497000.70703125 0.0304130        -0.2262835              -0.009312                     -0.027809                           -0.0381910.50781250 -0.0092076        -0.0348772              -0.015067                     -0.003071                           0.0078470.43359375 0.0021763        -0.0233817              0.007010                     0.004908                           -0.0049180.33593750 -0.0344308        -0.0031808              0.003348                     0.027637                           0.0262630.37109375 0.0164063        -0.0194196              -0.005022                     0.008954                           -0.0226000.32421875 0.0092634        -0.0077009              0.016532                     -0.015383                           -0.0038710.38671875 -0.0045201        0.0907478              0.006069                     -0.026834                           0.0355750.63671875 -0.0199777        0.1973214              0.009626                     -0.011422                           0.0661270.81250000 0.0224331        -0.2329241              0.003558                     0.016875                           -0.2058110.48828125 -0.0261161        -0.0174107              -0.025321                     0.024337                           0.0060690.43750000 0.0035156        -0.0440290              -0.002511                     -0.010188                           -0.0075330.32031250 0.0101563        0.0811384              0.006278                     0.006256                           0.0025110.40234375 -0.0292969        -0.0090960              -0.058594                     -0.026260                           0.0151720.40234375 -0.0013951        -0.0376674              -0.050642                     0.022643                           -0.0223910.33984375 -0.0376116        0.0617188              0.004290                     -0.008265                           0.0469800.63281250 -0.0425223        0.1786830              -0.036516                     0.004305                           0.0061730.80468750 -0.0712053        -0.2079241              -0.040388                     0.029043                           -0.2343750.45703125 - 0.0006696        0.0003348              -0.000314                     -0.009184                           -0.0113000.41075625 0.0016741        -0.0319196              0.020299                     0.019056                           -0.0007320.33203125 -0.0089286        0.0193080              0.020194                     -0.052519                           0.0145440.21093750 0.1237165        -0.0967076              0.060059                     0.156036                           0.0770090.25781250 0.0914063        0.0062500              0.087995                     -0.093673                           -0.0554550.26562500 -0.0221540        0.0343192              0.074916                     -0.002899                           0.0181010.47265625 -0.0229911        0.2156250              0.078055                     0.008839                           0.0159040.57812500 -0.0305245        -0.1201451              -0.003557                     0.072006                           0.0212400.44531950 0.0000558        -0.0220424              0.014230                     -0.028096                           0.0016740.41015625 -0.0073103        -0.0318638              -0.019148                     0.008409                           -0.0169500.43750000 0.0410714        -0.0239955              -0.033378                     0.020663                           -0.0007320.36718750 0.0566406        0.0056362              0.014962                     -0.005309                           0.0162180.48046875 0.1614397        0.0278460              0.038295                     0.096687                           -0.1092360.53906250 0.1313058        0.1659040              -0.128697                     0.012.025                           0.0546180.71093750 0.0371094        -0.0090960              -0.059329                     -0.061617                           -0.0310760.53125000 -0.0079799        -0.0807478              0.012347                     -0.010418                           0.0154860.44921875 -0.0116630        -0.0317522              -0.008789                     0.007950                           0.0000000.38671875 -0.0111607        -0.0160714              -0.011300                     0.000402                           0.0000000.45703125 0.0071429        -0.0061384              0.010568                     0.007691                           -0.0023020.46093750 0.0342634        0.0351562              0.001988                     0.005568                           0.0352610.73828125 0.0527344        0.0965960              -0.012660                     -0.02.8109                           -0.1507740.59765625 0.0304687        0.0774553              -0.012556                     -0.041212                           0.1465890.71875000 0.0060268        -0.0338170              -0.019880                     -0.024452                           -0.0114050.51171875 -0.0079799        -0.0932478              -0.001569                     -0.002612                           0.0156950.43750000 -0.0044085        -0.0244978              0.009312                     -0.004334                           -0.0330640.37500000 -0.0677456        -0.1410714              0.003557                     -0.121397                           -0.1430320.45703125 -0.0024554        0.0328125              0.024484                     0.014522                           -0.0056500.51171875 -0.0028460        0.0307478              -0.024170                     -0.036993                           0.0266810.75390625 -0.1251674        0.0778460              -0.168248                     -0.019142                           -0.0885180.68359375 0.0162947        0.0053571              -0.122001                     -0.059694                           0.0782650.70703125 -0.1199218        -0.0175781              -0.155797                     0.070226                           0.0243790.44921875 -0.0365513        -0.1284040              -0.030866                     -0.027522                           0.1015970.41796875 -0.0125000        0.0001116              0.000628                     0.019687                           -0.0081610.15234375 -0.0695313        -0.1944197              0.006906                     0.062679                           0.1086080.29296875 -0.1353237        -0.0140067              0.060582                     0.000373                           0.0156950.27734375 -0.1326451        -0.0152344              0.019357                     0.018683                           0.0256350.32031250 -0.0872768        0.0389509              0.044155                     -0.000804                           -0.0249020.34765625 -0.1336496        -0.0653460              0.070103                     -0.026145                           -0.0425850.26171875 -0.1049665        0.0693639              0.164062                     0.038772                           -0.0004190.26562500 0.0065848        0.0437500              0.130371                     0.086556                           0.0901930.43750000 -0.0600446        -0.1228795              -0.031076                     -0.157902                           -0.1569480.07421875 -0.0148995        -0.0127790              0.015904                     0.030048                           0.013079__________________________________________________________________________ 
    
     (The coefficients are given after rescaling the x and y variables to the square [-1,1][-1,1], and the gray level z to [0,1]). 
     Step 2 The required accuracy ε is chosen to be 5 gray levels, k is fixed to be 2, and the grid on each segment is chosen to contain the only point--the center of this segment. Thus the Taylor polynomials computed on this step are identical to the approximating polynomials found on the step 1. The 6 digits accuracy with which the coefficients of these polynomials are given in the array P above is excessive, and the coefficients are rounded off up to 8 bits in degree 0, up to 7 bits in degree 1 and up to 6 bits in degree 2. The corresponding binary array is the intermediate compressed image. It is approximately represented by the following digital array P&#39; (corresponding to the same piece S of the picture, as the above array P). 
     
         __________________________________________________________________________0.37500000 0.0000000        0.0000000              0.000000                     0.000000                           0.0000000.35937500 -0.0078125        -0.0234375              0.000000                     0.000000                           0.0000000.31640625 -0.0078125        0.0000000              0.000000                     0.000000                           0.0312500.39843750 -0.0078125        0.0937500              0.000000                     0.000000                           0.0468750.74218750 -0.0234375        0.1640625              0.000000                     0.000000                           -0.0468750.70703125 0.0234375        -0.2187500              0.000000                     -0.015625                           -0.0312500.50781250 -6.0078125        -0.0312500              0.000000                     0.000000                           0.0000000.43359375 0.0000000        -0.0156250              0.000000                     0.000000                           0.0000000.33593750 -0.0312500        0.0000000              0.000000                     0.015625                           0.0156250.37109375 0.0156250        -0.0156250              0.000000                     0.000000                           -0.0256250.32421875 0.0078125        0.0000000              0.015625                     0.000000                           0.0000000.38671875 0.0000000        0.0781250              0.000000                     -0.015625                           0.0312500.63671875 -0.0156250        0.1953125              0.000000                     0.000000                           0.0625000.81250000 0.0156250        -0.2265625              0.000000                     0.015625                           -0.2031250.49218750 -0.0234375        -0.0156250              -0.015625                     0.015625                           0.0000000.43750000 0.0000000        -0.0390625              0.000000                     0.000000                           0.0000000.32031250 0.0078125        0.0781250              0.000000                     0.000000                           0.0000000.40234375 -0.0234375        -0.0078125              -0.046875                     -0.015625                           0.0000000.40234375 0.0000000        -0.0312500              -0.046875                     0.015625                           -0.0155250.33984375 -0.0312500        0.0546875              0.000000                     0.000000                           0.0468750.63281250 -0.0390625        0.1718750              -0.031250                     0.000000                           0.0000000.80468750 -0.0703125        -0.2031250              -0.031250                     0.015625                           -0.2187500.45703125 0.0000000        0.0000000              0.000000                     0.000000                           0.0000000.41015625 0.0000000        -0.0312500              0.015625                     0.015625                           0.0000000.33203125 -0.0078125        0.0156250              0.015625                     -0.046875                           0.0000000.21093750 0.1171875        -0.0937500              0.046875                     0.140625                           0.0625000.25781250 0.0859375        0.0000000              0.078125                     -0.078125                           -0.0468750.26562500 -0.0156250        0.0312500              0.062500                     0.000000                           0.0156250.47656250 -0.0156250        0.2109375              0.062500                     0.000000                           0.0156250.57812500 -0.0234375        -0.1171875              0.000000                     0.062500                           0.0156250.44531250 0.0000000        -0.0156250              0.000000                     -0.015625                           0.0000000.41015625 0.0000000        -0.0312500              -0.015625                     0.000000                           -0.0156250.43750000 0.0390625        -0.0234375              -0.031250                     0.015625                           0.0000000.36718750 0.0546875        0.0000000              0.000000                     0.000000                           0.0156250.48437500 0.1562500        0.0234375              0.031250                     0.093750                           -0.0937500.53906250 0.1250000        0.1640625              -0.125000                     0.000000                           0.0468750.71093750 0.0312500        -0.0078125              -0.046875                     -0.046875                           -0.0156250.53125000 -0.0078125        -0.0781250              0.000000                     0.000000                           0.0000000.44921875 -0.0078125        -0.0312500              0.000000                     0.000000                           0.0000000.38671875 -0.0078125        -0.0156250              0.000000                     0.000000                           0.0000000.45703125 0.0000000        0.0000000              0.000000                     0.000000                           0.0000000.46093750 0.0312500        0.0312500              0.000000                     0.000000                           0.0312500.73828125 0.0468750        0.0937500              0.000000                     -0.015625                           -0.1406250.59765625 0.0234375        0.0703125              0.000000                     -0.031250                           0.1406250.71875000 0.0000000        -0.0312500              -0.015625                     -0.015625                           0.0000000.51171875 -0.0078125        -0.0859375              0.000000                     0.0000g0                           0.0156250.43750000 0.0000000        -0.0234375              0.000000                     0.000000                           -0.0312500.37500000 -0.0625000        -0.1406250              0.000000                     -0.109375                           -0.1406250.45703125 0.0000000        0.0312500              0.015625                     0.000000                           0.0000000.51171875 0.0000000        0.0234375              -0.015625                     -0.031250                           0.0156250.75390625 -0.1250000        0.0703125              -0.156250                     -0.015625                           -0.0781250.68359375 0.0156250        0.0000000              0.109375                     -0.046875                           0.0781250.70703125 -0.1171875        -0.0156250              -0.140625                     0.062500                           0.0156250.44921875 -0.0312500        -0.1250000              -0.015625                     -0.015625                           0.0937500.41796875 -0.0078125        0.0000000              0.000000                     0.015625                           0.0000000.15234375 -0.0625000        -0.1875000              0.000000                     0.062500                           0.0937500.29296875 -0.1328125        -0.0078125              0.046875                     0.000000                           0.0156250.27734375 -0.1250000        -0.0078125              0.015625                     0.015625                           0.0156250.32031250 -0.0859375        0.0312500              0.031250                     0.000000                           -0.0156250.34765625 -0.1328125        -0.0625000              0.062500                     -0.015625                           -0.0312500.26171875 -0.1015625        0.0625000              0.156250                     0.031250                           0.0000000.26562500 0.0000000        0.0390625              0.125000                     0.078125                           0.0781250.43750000 -0.0546875        -0.1171875              -0.015625                     -0.156250                           -0.1562500.07421875 -0.0078125        -0.0078125              0.015625                     0.015625                           0.000000__________________________________________________________________________ 
    
     The compression ratio is 512*512*8 bits/86*86*(8+2*7÷3*6) bits≃6.7. 
     The final image is obtained by computing the values of the Taylor polynomials, representing the intermediate image, at each pixel of the corresponding segment. The part S&#39; of the obtained array, representing the final image (and corresponding to the piece S of the initial picture), is the following array 9. 
     
         __________________________________________________________________________137   137 137    136       138          129             138                134                   140                      136                         135                            134                               130                                  139                                     135                                        129                                           134131   138 139    127       132          129             134                131                   133                      130                         125                            130                               129                                  127                                     131                                        127                                           134128   135 133    144       135          142             143                142                   143                      150                         148                            154                               153                                  149149   149 143    154       145          142             135                133                   118                      120                         127                            110                               98 82 86 74 6365 63 61 55 67 72 66 73 69 73 74 73 73 70 78 80 7675 73 72 80 80 78 77 75 75 76 78 78 75 7878 79 78 78 73 75 73 75 79 79 89 77 81 86 90 86 7889 89 92 95 92 90 92 91 101                      97 92 89 94 104                                     100                                        101                                           96104   98 96 98 100          95 106                108                   102                      103                         100                            100                               96 99104   91 98 95 105          101             98 101                   97 102                         97 97 105                                  104                                     105                                        103                                           95102   97 105    102       102          103             103                99 103                      96 106                            102                               99 102                                     101                                        106                                           100104   103 101    102       100          100             100                104                   108                      103                         104                            99 104                                  99105   106 101    105       105          105             101                109                   106                      104                         102                            105                               104                                  105                                     102                                        104                                           112102   99 110    104       101          100             102                102                   98 100                         103                            103                               99 98 100                                        101                                           97107   105 104    106       101          100             100                101                   99 102                         97 98 99 103105   103 101    103       101          105             104                109                   102                      99 102                            103                               96 96 103                                        108                                           109114   104 102    102       97 99 97 100                   102                      98 101                            102                               102                                  99 98 99 97100   102 93 100       105          105             99 97 96 102                         96 101                               104                                  9495 103 99 95 99 95 102                93 97 99 92 96 95 102                                     95 101                                           9094 93 89 86 93 91 90 89 86 83 85 87 74 72 73 71 8283 88 90 102       106          111             110                116                   118                      125                         128                            126                               127                                  136136   136 141    135       142          141             131                130                   121                      122                         129                            122                               130                                  125                                     123                                        131                                           125131   129 126    131       130          128             128                135                   133                      129                         132                            129                               127                                  124                                     133                                        127                                           126128   127 129    126       127          127             126                124                   124                      136                         129                            129                               128                                  127129   134 129    136       135          134             129                130                   131                      136                         136                            136                               132                                  132                                     132                                        127                                           131134   134 161    181       189          198             201                206                   207                      206                         210                            212                               213                                  211                                     213                                        215                                           210204   197 183    156       131          96 77 77 71 74 76 83 73 8587 89 91 88 90 90 94 93 101                      93 88 88 92 96 92 87 9493 98 101    92 90 94 90 89 92 94 88 93 88 91 90 95 93100   86 93 97 94 97 96 97 93 95 94 94 87 86100   92 97 96 95 101             95 97 93 89 101                            95 97 97 88 94 8291 84 88 89 84 91 93 107                   139                      144                         147                            149                               148                                  130                                     99 137                                           137137   136 138    129       138          134             140                136                   135                      134                         130                            139                               135                                  129134   131 138    139       127          132             129                134                   131                      133                         130                            125                               130                                  129                                     127                                        131                                           127134   128 135    133       144          135             142                143                   142                      143                         150                            148                               154                                  153                                     149                                        149                                           149143   154 145    142       135          133             118                120                   127                      110                         98 82 86 7463 65 63 61 55 67 72 66 73 69 73 74 73 73 70 78 8076 75 73 72 80 80 78 77 75 75 76 78 78 75 78 78 7978 78 73 75 73 75 79 79 89 77 81 86 90 8678 89 89 92 85 92 90 92 91 101                         97 92 89 94 104                                        100                                           10196 104 98 96 98 100             95 106                   108                      102                         103                            100                               100                                  96 99 104                                           9198 95 105    101       98 101             97 102                   97 97 105                            104                               105                                  10395 102 97 105       102          102             103                103                   99 103                         96 106                               102                                  99 102                                        101                                           106100   104 103    101       102          100             100                100                   104                      108                         103                            104                               99 104                                     99 105                                           106101   105 105    105       101          109             106                104                   102                      105                         104                            105                               102                                  104112   102 99 110       104          101             100                102                   102                      98 100                            103                               103                                  99 98 100                                           10197 107 105    104       106          101             100                100                   101                      99 102                            97 98 99 103                                        105                                           103101   103 101    105       104          109             102                99 102                      103                         96 96 103                                  108109   114 104    102       102          97 99 97 100                      102                         98 101                               102                                  102                                     99 98 9997 100 102    93 100          105             105                99 97 96 102                            96 101                                  104                                     94 95 10399 95 99 95 102          93 97 99 92 96 95 102                               95 10190 94 93 89 86 93 91 90 89 86 83 85 87 74 72 73 7182 83 88 90 102          106             111                110                   116                      118                         125                            128                               126                                  127                                     136                                        136                                           136141   135 142    141       131          130             121                122                   129                      122                         130                            125                               123                                  131125   131 129    126       131          130             128                128                   135                      133                         129                            132                               129                                  127                                     124                                        133                                           127126   128 127    129       126          127             127                126                   124                      124                         136                            129                               129                                  128                                     127                                        129                                           134129   136 135    134       129          130             131                136                   136                      136                         132                            132                               132                                  127131   134 134    161       181          189             198                201                   206                      207                         206                            210                               212                                  213                                     211                                        213                                           215210   204 197    183       156          131             96 77 77 71 74 76 83 73 85 87 8991 88 90 90 94 93 101                93 88 88 92 96 92 8794 93 98 101       92 90 94 90 89 92 94 88 93 88 91 90 9593 100 86 93 97 94 97 96 97 93 95 94 94 87 86 100                                           9297 96 95 101       95 97 93 89 101                      95 97 97 88 9482 91 84 88 89 84 91 93 107                      139                         144                            147                               149                                  148                                     130                                        99 137137   137 136    138       129          138             134                140                   136                      135                         134                            130                               139                                  135                                     129                                        134                                           131138   139 127    132       129          134             131                133                   130                      125                         130                            129                               127                                  131127   134 128    135       133          144             135                142                   143                      142                         143                            150                               148                                  154                                     153                                        149                                           149149   143 154    145       142          135             133                118                   120                      127                         110                            98 82 86 74 63 6563 61 55 67 72 66 73 69 73 74 73 73 70 7880 76 75 73 72 80 80 78 77 75 75 76 78 78 75 78 7879 78 78 73 75 73 75 79 79 89 77 81 86 90 86 78 8989 92 95 92 90 92 91 101                   97 92 89 94 104                                  100101   96 104    98 96 98 100                95 106                      108                         102                            103                               100                                  100                                     96 99 10491 98 95 105       101          98 101                97 102                      97 97 105                               104                                  105                                     103                                        95 10297 105 102    102       103          103             99 103                   96 106                         102                            99 102                                  101106   100 104    103       101          102             100                100                   100                      104                         108                            103                               104                                  99 104                                        99 105106   101 105    105       105          101             109                106                   104                      102                         105                            104                               105                                  102                                     104                                        112                                           10299 110 104    101       100          102             102                98 100                      103                         103                            99 98 10010 197 107    105       104          106             101                100                   100                      101                         99 102                               97 98 99 103                                           105103   101 103    101       105          104             109                102                   99 102                         103                            96 96 103                                     108                                        109                                           114104   102 102    97 99 97 100                102                   98 101                         102                            102                               99 9899 97 100    102       93 100             105                105                   99 97 96 102                               96 101                                     104                                        94 95103   99 95 99 95 102             93 97 99 92 96 95 102                                  95 101                                        90 9493 89 86 93 91 90 89 86 83 85 87 74 72 7371 82 83 88 90 102             106                111                   110                      116                         118                            125                               128                                  126                                     127                                        136                                           136136   141 135    142       141          131             130                121                   122                      129                         122                            130                               125                                  123                                     131                                        125                                           131L29   126 131    130       128          128             135                133                   129                      132                         129                            127                               124                                  133127   126 128    127       129          126             127                127                   126                      124                         124                            136                               129                                  129                                     128                                        127                                           129134   129 136    135       134          129             130                131                   136                      136                         136                            132                               132                                  132                                     127                                        131                                           134134   161 181    189       198          201             206                2C7                   206                      210                         212                            213                               211                                  213215   210 204    197       183          156             131                96 77 77 71 74 76 83 73 85 8789 91 88 90 90 94 93 101                   93 88 88 92 96 92 87 94 9398 101 92 90 94 90 89 92 94 88 93 88 91 9095 93 100    86 93 97 94 97 96 97 93 95 94 94 87 86 10092 97 96 95 101          95 97 93 89 101                         95 97 97 88 94 82 9184 88 89 84 91 93 107                139                   144                      147                         149                            148                               130                                  99137   137 137    136       138          129             138                134                   140                      136                         135                            134                               130                                  139                                     135                                        129                                           134131   138 139    127       132          129             134                131                   133                      130                         125                            130                               129                                  127                                     131                                        127                                           134128   135 133    144       135          142             143                142                   143                      150                         148                            154                               153                                  149149   149 143    154       145          142             135                133                   118                      120                         127                            110                               98 82 86 74 6365 63 61 55 67 72 66 73 69 73 74 73 73 70 78 80 7675 73 72 80 80 78 77 75 75 76 78 78 75 7878 79 78 78 73 75 73 75 79 79 89 77 81 86 90 86 7889 89 92 95 92 90 92 91 101                      97 92 89 94 104                                     100                                        101                                           96104   98 96 98 100          95 106                108                   102                      103                         100                            100                               96 99104   91 98 95 105          101             98 101                   97 102                         97 97 105                                  104                                     105                                        103                                           95102   97 105    102       102          103             103                99 103                      96 106                            102                               99 102                                     101                                        106                                           100104   103 101    102       100          100             100                104                   108                      103                         104                            99 104                                  99105   106 101    105       105          105             101                109                   106                      104                         102                            105                               104                                  105                                     102                                        104                                           112102   99 110    104       101          100             102                102                   98 100                         103                            103                               99 98 100                                        101                                           97107   105 104    106       101          100             100                101                   99 102                         97 98 99 103105   103 101    103       101          105             104                109                   102                      99 102                            103                               96 96 103                                        108                                           109114   104 102    102       97 99 97 100                   102                      98 101                            102                               102                                  99 98 99 97100   102 93 100       105          105             99 97 96 102                         96 101                               104                                  9495 103 99 95 99 95 102                93 97 99 92 96 95 102                                     95 101                                           9094 93 89 86 93 91 90 89 86 83 85 87 74 72 73 71 8283 88 90 102       106          111             110                116                   118                      125                         128                            126                               127                                  136136   136 141    135       142          141             131                130                   121                      122                         129                            122                               130                                  125                                     123                                        131                                           125131   129 126    131       130          128             128                135                   133                      129                         132                            129                               197                                  124                                     133                                        127                                           126128   127 129    126       127          127             126                124                   124                      136                         129                            129                               128                                  127129   134 199    136       135          134             129                130                   131                      136                         136                            136                               132                                  132                                     132                                        127                                           131134   134 161    181       189          198             201                206                   207                      206                         210                            212                               213                                  211                                     213                                        215                                           210204   197 183    156       131          96 77 77 71 74 76 83 73 8587 89 91 88 90 90 94 93 101                      93 88 88 92 96 92 87 9493 98 101    92 90 94 90 89 92 94 88 93 88 91 90 95 93100   86 93 97 94 97 96 97 93 95 34 94 87 86100   92 97 96 95 101             95 97 93 89 101                            95 97 97 88 94 8291 84 88 89 84 91 93 107                   139                      144                         147                            149                               148                                  130                                     99 137                                           137137   136 138    129       138          134             140                136                   135                      134                         130                            139                               135                                  129134   131 138    139       127          132             129                134                   131                      133                         130                            125                               130                                  129                                     127                                        131                                           127134   128 135    133       144          135             142                143                   142                      143                         150                            148                               154                                  153                                     149                                        149                                           149143   154 145    142       135          133             118                120                   127                      110                         98 82 86 7463 65 63 61 55 67 72 66 73 69 73 74 73 73 70 78 8076 75 73 72 80 80 78 77 75 75 76 78 78 75 78 78 7978 78 73 75 73 75 79 79 89 77 81 86 90 8678 89 89 92 95 92 90 92 91 101                         97 92 89 94 104                                        100                                           10196 104 98 96 98 100             95 106                   108                      102                         103                            100                               100                                  96 99 104                                           9198 95 105    101       98 101             97 102                   97 97 105                            104                               105                                  10395 102 97 105       102          102             103                103                   99 103                         96 106                               102                                  99 102                                        101                                           106100   104 103    101       102          100             100                100                   104                      108                         103                            104                               99 104                                     99 105                                           106101   105 105    105       101          109             106                104                   102                      105                         104                            105                               102                                  104112   102 99 110       104          101             100                102                   102                      98 100                            103                               103                                  99 98 100                                           10197 107 105    104       106          101             100                100                   101                      99 102                            97 98 99 103                                        105                                           103101   103 101    105       104          109             102                99 102                      103                         96 96 103                                  108__________________________________________________________________________ 
    
     The picture representing the final image is shown in FIG. 4b. 
     EXAMPLE 3 (ROTATION OF A PICTURE) 
     The object picture is the same as in Example 2. The required operation is the rotation by 90° in the counterclockwise direction (FIG. 5a represents the result of a rotation of the object picture). 
     The array of the gray levels of the rotated piece S&#39; of the object picture is the following array 10. 
     
         __________________________________________________________________________60 66 75 86 100          117             114                112                   111                      111                         112                            114                               118                                  118                                     118                                        118                                           118118   118 119    119       117          113             107                92 88 84 81 80 79 68 67 67 6665 65 80 84 88 92 95 99 97 97 97 97 97 9758 64 73 84 98 115             117                115                   114                      114                         115                            117                               118                                  118                                     118                                        118                                           118118   117 118    117       114          110             104                90 67 85 83 83 83 75 74 73 7372 71 80 83 86 90 93 96 97 97 97 97 97 9756 62 71 82 96 113             119                118                   117                      117                         118                            119                               118                                  118                                     118                                        118                                           118118   116 117    115       112          107             101                88 86 85 85 86 88 81 81 80 7979 78 80 83 86 89 91 94 97 97 97 97 97 9756 62 70 82 96 112             122                120                   119                      119                         120                            122                               118                                  118                                     118                                        118                                           118118   115 115    114       110          105             98 86 86 86 87 90 93 88 87 87 8685 85 81 83 86 88 91 93 97 97 97 97 97 9756 62 71 82 96 113             125                123                   122                      122                         123                            125                               118                                  118                                     118                                        118                                           118118   114 114    112       108          102             94 84 85 87 89 93 97 95 94 93 9392 91 83 85 87 89 91 93 97 97 97 97 97 9757 63 72 83 97 114             127                126                   125                      125                         126                            127                               118                                  118                                     118                                        118                                           118118   114 113    110       106          99 91 82 84 87 91 96 102                                  101                                     101                                        100                                           9999 98 86 87 89 91 92 94 97 97 97 97 97 9750 60 31 82 95 109             132                131                   130                      128                         124                            120                               125                                  122                                     119                                        117                                           114111   109 105    100       95 91 86 94 89 86 86 89 94 95 101                                        108                                           106104   100 100    99 97 96 95 93 96 97 98 98 99 10048 58 68 80 92 105             130                130                   130                      128                         126                            123                               124                                  121                                     118                                        116                                           113110   108 103    98 94 89 84 89 80 73 69 68 69 93 100                                        104                                           105104   100 100    99 98 96 95 94 94 95 96 96 97 9848 57 67 78 90 102             128                130                   130                      130                         128                            126                               25 122                                     119                                        117                                           114111   107 102    97 93 88 83 88 74 64 56 50 48 92 99 103                                           105104   100 100    99 97 96 95 93 92 93 94 94 95 9648 57 67 77 88 101             128                130                   132                      132                         132                            130                               127                                  125                                     122                                        119                                           117114   107 102    97 93 88 83 90 72 58 46 36 30 90 97 102                                           104104   101 99 97 96 95 93 92 90 91 92 92 93 9450 58 67 77 88 100             129                132                   134                      136                         136                            136                               132                                  129                                     126                                        124                                           121118   108 103    98 94 89 84 95 74 55 39 26 15 88 96 101                                           104104   101 96 95 94 92 91 90 89 89 90 90 91 9252 60 68 78 88 100             130                134                   138                      140                         141                            142                               138                                  135                                     133                                        130                                           127125   109 105    100       95 91 86 104                   79 56 36 19 4  86 95 100                                           104104   102 93 92 91 89 88 87 86 87 88 83 89 9061 65 70 77 87 97 112                140                   158                      168                         168                            160                               157                                  152                                     147                                        141                                           136131   125 114    106       99 94 91 105                   83 66 53 44 40 97 103                                        107                                           108107   102 88 86 34 84 84 85 86 87 87 88 89 8965 69 75 82 91 102             126                154                   173                      183                         184                            176                               180                                  175                                     170                                        166                                           161156   144 131    120       111          103             97 105                   85 70 59 53 51 97 103                                        106                                           107105   100 88 86 84 84 84 85 82 83 84 84 85 8669 73 78 85 95 105             136                164                   183                      194                         195                            188                               195                                  190                                     186                                        182                                           178173   158 142    129       117          106             98 102                   84 71 63 59 59 96 102                                        105                                           105103   97 88 86 84 84 84 85 81 81 82 83 83 8471 75 81 88 97 108             140                169                   189                      200                         202                            195                               201                                  198                                     194                                        190                                           186183   167 148    132       117          104             93 96 81 70 64 62 65 95 100                                        102                                           10299 94 88 86 84 84 84 85 81 81 82 83 83 8473 77 83 90 99 110             141                170                   190                      202                         204                            198                               200                                  197                                     194                                        190                                           187184   170 149    130       113          97 83 88 75 67 63 63 68 92 97 99 9895 89 88 86 84 84 84 85 82 83 84 84 85 8674 78 83 91 100          111             137                166                   187                      199                         202                            196                               191                                  188                                     185                                        183                                           180177   168 144    123       103          84 68 77 66 60 58 61 68 89 93 95 9490 83 88 86 84 84 84 85 86 87 87 88 89 8983 86 93 103       117          134             181                189                   190                      186                         175                            158                               174                                  169                                     165                                        161                                           157152   116 119    116       105          87 62 73 67 65 66 71 80 78 31 83 8689 91 92 91 89 88 87 86 90 90 91 92 92 9380 84 91 102       116          134             169                178                   181                      178                         168                            152                               161                                  158                                     155                                        151                                           148145   124 128    124       114          96 71 74 68 66 67 72 80 77 80 83 8588 91 94 93 92 92 91 90 92 93 94 94 95 9676 80 87 98 113          131             161                172                   176                      174                         166                            151                               157                                  155                                     152                                        150                                           147145   136 139    136       125          107             82 76 70 68 69 74 82 79 82 85 8790 93 97 97 97 97 97 96 98 98 99 100                               100                                  10169 74 82 93 108          123             158                170                   175                      174                         168                            154                               161                                  159                                     158                                        156                                           155153   150 153    150       139          121             96 78 72 70 72 76 85 84 87 89 9295 97 103    103       103          104             104                104                   106                      106                         107                            108                               108                                  10961 66 74 86 102          121             159                172                   179                      180                         174                            162                               173                                  172                                     171                                        171                                           170169   166 170    166       156          138             113                82 76 74 75 80 88 91 94 97 99102   105 110    111       112          112             113                114                   116                      117                         118                            118                               119                                  12051 56 65 78 94 113             164                179                   187                      189                         185                            174                               192                                  193                                     193                                        193                                           193194   186 189    186       175          157             132                86 80 78 80 84 93 101                                     104                                        107                                           109112   115 119    121       122          123             124                125                   130                      130                         131                            132                               132                                  13355 48 50 61 81 110             126                156                   179                      193                         200                            198                               191                                  191                                     191                                        190                                           188185   188 187    184       178          169             158                88 82 80 81 86 94 111                                     118                                        123                                           127128   128 129    130       132          133             134                136                   142                      144                         146                            148                               150                                  15263 55 56 66 85 113             127                156                   177                      189                         194                            191                               787                                  188                                     188                                        188                                           186184   185 186    184       179          172             162                104                   98 96 97 102                               110                                  126                                     133                                        138                                           141143   143 139    140       141          143             144                145                   161                      163                         165                            167                               169                                  17170 62 62 71 89 116             130                156                   175                      186                         189                            184                               183                                  185                                     185                                        183                                           184182   182 184    184       180          174             166                121                   115                      123                         114                            119                               128                                  141                                     148                                        153                                           156158   157 152    153       154          156             157                158                   178                      180                         182                            184                               186                                  18878 68 68 76 93 119             133                158                   175                      184                         185                            178                               179                                  181                                     182                                        183                                           182180   178 182    184       180          176             168                139                   133                      131                         132                            137                               146                                  155                                     162                                        167                                           171172   172 168    170       171          172             174                175                   192                      194                         196                            198                               200                                  20285 74 73 81 97 122             137                160                   175                      183                         182                            173                               176                                  178                                     180                                        180                                           180179   173 178    180       179          176             170                158                   152                      150                         151                            156                               164                                  170                                     177                                        182                                           185187   187 189    190       191          193             194                195                   203                      205                         207                            209                               211                                  21393 82 79 86 101          125             142                163                   177                      182                         180                            169                               172                                  175                                     177                                        178                                           178177   168 174    177       178          176             171                178                   172                      170                         171                            176                               184                                  185                                     192                                        197                                           200202   201 212    214       215          216             218                219                   212                      214                         216                            218                               220                                  22283 74 73 78 91 111             153                156                   158                      160                         101                            160                               150                                  151                                     152                                        152                                           153154   151 152    152       153          154             154                160                   166                      173                         179                            185                               192                                  187                                     198                                        207                                           214219   223 221    221       220          220             220                220                   230                      227                         224                            221                               217                                  21483 72 68 72 82 100             130                134                   137                      139                         140                            140                               141                                  142                                     143                                        143                                           144145   144 145    146       146          147             148                152                   157                      162                         166                            171                               176                                  196                                     206                                        214                                           221226   229 226    225       224          224             223                222                   214                      211                         208                            206                               203                                  20087 74 68 69 78 93 113                117                   121                      123                         125                            125                               133                                  134                                     134                                        135                                           136136   138 138    139       140          140             141                146                   149                      152                         155                            157                               160                                  192                                     202                                        210                                           216220   223 219    218       217          216             215                213                   196                      193                         191                            189                               187                                  18596 81 73 72 78 91 101                106                   110                      173                         115                            116                               126                                  126                                     127                                        128                                           128129   131 132    132       133          134             134                140                   141                      143                         144                            145                               146                                  176                                     185                                        193                                           198202   205 201    199       198          196             195                193                   176                      174                         172                            171                               169                                  167110   92 82 78 82 93 95 100                   104                      108                         110                            112                               119                                  120                                     121                                        121                                           122123   124 125    126       126          127             128                136                   135                      134                         134                            133                               132                                  147                                     156                                        163                                           168172   174 171    169       167          165             163                161                   154                      153                         152                            150                               149                                  148127   108 95 89 91 100             94 99 104                      108                         111                            1Z3                               114                                  114                                     115                                        116                                           116117   118 118    119       120          120             121                132                   129                      Z27                         125                            122                               120                                  106                                     114                                        121                                           126129   130 130    127       125          122             120                117                   131                      130                         129                            128                               127                                  127124   119 113    107       100          91 104                105                   107                      109                         111                            112                               112                                  112                                     112                                        112                                           112112   121 122    122       123          124             124                121                   120                      119                         118                            117                               116                                  118                                     118                                        118                                           118118   118 119    124       128          131             133                134                   135                      136                         136                            137                               138                                  138120   120 119    117       114          110             105                106                   107                      109                         110                            111                               114                                  114                                     114                                        114                                           114114   118 119    120       120          121             122                119                   118                      117                         117                            116                               115                                  118                                     118                                        118                                           118118   118 119    123       127          129             131                132                   132                      133                         134                            134                               135                                  136108   112 116    118       120          120             106                107                   108                      108                         109                            110                               114                                  114                                     114                                        114                                           114114   116 116    117       118          118             119                116                   116                      116                         116                            115                               115                                  118                                     118                                        118                                           118118   118 118    122       125          128             129                129                   130                      130                         131                            132                               132                                  13387 96 103    110       116          121             107                107                   108                      108                         109                            109                               112                                  112                                     112                                        112                                           112112   113 114    114       115          116             116                114                   114                      114                         114                            115                               115                                  118                                     118                                        118                                           118118   118 118    122       124          126             127                127                   127                      128                         128                            129                               130                                  13057 70 82 94 104          114             108                108                   108                      108                         108                            108                               108                                  108                                     108                                        108                                           108108   110 111    112       112          113             114                111                   112                      113                         113                            114                               115                                  118                                     118                                        118                                           118118   118 118    121       123          125             125                125                   124                      125                         126                            126                               127                                  12818 36 52 68 83 97 109                109                   108                      108                         107                            107                               102                                  102                                     102                                        102                                           102102   108 108    109       110          110             111                109                   110                      111                         112                            113                               114                                  118                                     118                                        118                                           118118   118 118    120       122          123             123                122                   122                      122                         123                            124                               124                                  12520 20 21 23 25 29 72 82 92 102                         111                            121                               108                                  106                                     103                                        101                                           9896 102 102    103       104          104             105                107                   109                      110                         110                            109                               107                                  113                                     112                                        112                                           113115   118 121    121       121          121             121                121                   115                      115                         115                            115                               115                                  11520 20 20 22 24 27 50 58 66 74 82 90 104                                  105                                     106                                        106                                           107108   100 101    102       102          103             104                106                   108                      109                         109                            108                               106                                  111                                     110                                        110                                           110112   114 118    118       118          118             118                118                   114                      114                         114                            114                               114                                  11421 20 20 21 23 25 33 39 46 52 58 64 93 96 100                                        104                                           108111   99 100    100       101          102             102                104                   106                      107                         107                            106                               104                                  110                                     108                                        107                                           108109   111 115    115       115          115             115                115                   113                      113                         113                            113                               113                                  11321 20 20 20 21 24 22 26 30 35 39 44 73 80 87 93 100107   98 98 99 100          100             101                102                   104                      104                         104                            104                               102                                  108                                     106                                        105                                           105105   107 111    111       111          111             111                111                   111                      111                         111                            111                               111                                  11122 20 19 19 20 22 15 18 21 23 26 29 45 55 65 75 8595 96 97 98 98 99 100                98 100                      101                         101                            100                               98 106                                     104                                        102                                           102102   103 108    108       108          108             108                108                   110                      110                         110                            110                               110                                  11022 20 19 18 19 20 14 15 16 17 18 19 9  22 35 49 6275 95 96 96 97 98 98 94 96 96 96 96 94 105                                     102                                        100                                           9999 99 105    105       105          105             105                105                   109                      109                         109                            109                               109                                  109__________________________________________________________________________ 
    
     The above rotation acts on the Taylor polynomials, representing the intermediate image, obtained in the Example 2, as follows: let the 6×6 pixel square segments, into which the original picture has been subdivided, be indexed by two indices i and j, in such a way that the middle segment has indices 0,0. Denote the Taylor polynomial corresponding to the segment i, j by p ij . Then: 
     a. The indices i, j of each p ij  are replaced by -j,i 
     b. x is replaced by y, and y by -x. 
     Using the notation already used in discussing processing, F(p ij  (x,y))=p -j ,i (y, -x). 
     The result of the application of the corresponding subroutine to the Taylor polynomials in the intermediate range, obtained in the Example 2, is the intermediate range of the rotated picture. Its part P&#39; corresponding to the rotated piece S&#39;, is the following array 11. 
     
         __________________________________________________________________________0.29296875 -0.0078125        0.0078125              0.015625                     -0.000000                           0.0156250.45703125 0.0312500        -0.0312500              0.000000                     -0.000000                           0.0000000.45703125 0.0000000        -0.0000000              0.000000                     -0.000000                           0.0000000.43750000 -0.0234375        0.0234375              0.000000                     -0.015625                           0.0000000.33203125 0.0156250        -0.0156250              0.000000                     0.046875                           0.0000000.32031250 0.0781250        -0.0781250              0.000000                     -0.000000                           0.0000000.33593750 0.0000000        -0.0000000              0.015625                     -0.015625                           0.0156250.37500000 0.0000000        -0.0000000              0.000000                     -0.000000                           0.0000000.27734375 -0.0078125        0.0078125              0.015625                     -0.015625                           0.0156250.51171875 0.0234375        -0.0234375              0.015625                     0.031250                           0.0156250.46093750 0.0312500        -0.0312500              0.031250                     -0.000000                           0.0312500.36718750 0.0000000        -0.0000000              0.015625                     -0.000000                           0.0156250.21093750 -0.0937500        0.0937500              0.062500                     -0.140625                           0.0625000.40234375 -0.0078125        0.0078125              0.000000                     0.015625                           0.0000000.37109375 -0.0156250        0.0156250              -0.015625                     -0.000000                           -0.0156250.35937500 -0.0234375        0.0234375              0.000000                     -0.000000                           0.0000000.32031250 0.0312500        -0.0312500              -0.015625                     -0.000000                           -0.0156250.75390625 0.0703125        -0.0703125              -0.078125                     0.015625                           -0.0781250.73828125 0.0937500        -0.0937500              -0.140625                     0.015625                           -0.1406250.48437500 0.0234375        -0.0234375              -0.093750                     -0.093750                           -0.0937500.25781250 0.0000000        -0.0000000              -0.046895                     0.078125                           -0.0468750.40234375 -0.0312500        0.0312500              -0.015625                     -0.015625                           -0.0156250.32421875 0.0000000        -0.0000000              0.000000                     -0.000000                           0.0000000.31640625 0.0000000        -0.0000000              0.031250                     -0.000000                           0.0312500.34765625 -0.0625000        0.0625000              -0.031250                     0.015625                           -0.0312500.68359375 0.0000000        -0.0000000              0.078125                     0.046875                           0.0781250.50765625 0.0703125        -0.0703125              0.140625                     0.031250                           0.1406250.53906250 0.1640625        -0.1640625              0.046875                     -0.000000                           0.0468750.26562500 0.0312500        -0.0312500              0.015625                     -0.000000                           0.0156250.33984375 0.0546875        -0.0546875              0.046875                     -0.000000                           0.0468750.38671875 0.0781250        -0.0781250              0.031250                     0.015625                           0.0312500.39843750 0.0937500        -0.0937500              0.046875                     -0.000000                           0.0468750.26171875 0.0625000        -0.0625000              0.000000                     -0.031250                           0.0000000.70703125 -0.0156250        0.0156250              0.015625                     -0.062500                           0.0156250.71875000 -0.0312500        0.0312500              0.000000                     0.015625                           0.0000000.71093750 -0.0078125        0.0078125              -0.015625                     0.046875                           -0.0156250.47656250 0.2109375        -0.2109375              0.015625                     -0.000000                           0.0156250.63281250 0.1718750        -0.1718750              0.000000                     -0.000000                           0.0000000.63671875 0.1953125        -0.1953125              0.062500                     -0.000000                           0.0625000.74218750 0.1640625        -0.1640625              -0.046875                     -0.000000                           -0.0468750.26562500 0.0390625        -0.0390625              0.078125                     -0.078125                           0.0781250.44921875 -0.1250000        0.1250000              0.093750                     0.015625                           0.0937500.51171875 -0.0859375        0.0859375              0.015625                     -0.000000                           0.0156250.53125000 -0.0781250        0.0781250              0.000000                     -0.000000                           0.0000000.57812500 -0.1171875        0.1171875              0.015625                     -0.062500                           0.0156250.80468750 -0.2031250        0.2031250              -0.218750                     -0.015625                           -0.2187500.81250000 -0.2265625        0.2265625              -0.203125                     -0.015625                           -0.2031250.70703125 -0.2187500        0.2187500              -0.031250                     0.015625                           -0.0312500.43750000 -0.1171875        0.1171875              -0.156250                     0.156250                           -0.1562500.41796875 0.0000000        -0.0000000              0.000000                     -0.015625                           0.0000000.43750000 -0.0234375        0.0234375              -0.031250                     -0.000000                           -0.0312500.44921875 -0.0312500        0.0312500              0.000000                     -0.000000                           0.0000000.44531250 -0.0156250        0.0156250              0.000000                     0.015625                           0.0000000.45703125 0.0000000        -0.0000000              0.000000                     -0.000000                           0.0000000.49218750 -0.0156250        0.0156250              0.000000                     -0.015625                           0.0000000.50781250 -0.0312500        0.0312500              0.000000                     -0.000000                           0.0000000.07421875 -0.0078125        0.0078125              0.000000                     -0.015625                           0.0000000.15234375 -0.1875000        0.1875000              0.093750                     -0.062500                           0.0937500.37500000 -0.1406250        0.1406250              -0.140625                     0.109375                           -0.1406250.38671875 -0.0156250        0.0156250              0.000000                     -0.000000                           0.0000000.41015625 -0.0312500        0.0319500              -0.015625                     -0.000000                           -0.0156250.41015625 -0.0312500        0.0312500              0.000000                     -0.015625                           0.0000000.43750000 -0.0390625        0.0390625              0.000000                     -0.000000                           0.0000000.43359375 -0.0156250        0.0156250              0.000000                     -0.000000                           0.000000__________________________________________________________________________ 
    
     The final image, produced from the data rotated in a compressed form, is shown in FIG. 5b. 
     EXAMPLE 4 (PRODUCING A NEGATIVE PICTURE) 
     The object picture is the same as in the Example 2. It is required to produce a negative of this picture. Under this operation each gray level value z must be replaced by z&#39;=255-z. 
     The negative of the original picture is shown in FIG. 6a. The array S&#34; of the gray levels, corresponding to the negative of the piece S, is the following. 
     
         __________________________________________________________________________158   158 158    158       158          158             155                157                   159                      161                         163                            165                               166                                  169                                     171                                        171                                           169166   162 159    154       146          135             122                103                   84 67 53 42 33 41 55 70 88107   128 117    119       122          125             127                130                   140                      741                         142                            144                               145                                  146158   158 158    158       158          158             156                158                   160                      162                         164                            166                               166                                  170                                     172                                        172                                           170166   163 160    155       147          136             123                105                   86 69 55 44 35 38 52 68 86106   128 117    120       123          125             128                131                   140                      141                         142                            144                               145                                  146258   158 158    158       158          158             157                159                   161                      163                         165                            167                               167                                  171                                     172                                        172                                           171167   163 161    155       147          137             123                107                   88 71 57 46 37 34 49 66 84105   127 118    121       123          126             129                131                   140                      141                         142                            144                               145                                  146158   158 158    158       158          158             157                159                   161                      163                         165                            167                               168                                  171                                     173                                        173                                           171168   164 161    156       148          137             124                109                   90 73 59 48 39 31 47 64 83103   126 119    121       124          127             129                132                   140                      141                         142                            144                               145                                  146158   158 158    158       158          158             158                160                   162                      164                         166                            168                               168                                  172                                     174                                        174                                           172168   165 162    157       149          138             125                111                   92 75 61 50 41 28 44 62 81102   125 119    122       125          127             130                133                   140                      141                         142                            144                               145                                  146158   138 158    158       158          158             159                161                   163                      165                         167                            169                               169                                  173                                     174                                        174                                           173169   165 163    157       149          139             125                113                   94 77 63 52 43 25 41 59 79101   124 120    123       125          128             131                133                   140                      141                         142                            144                               145                                  146156   159 161    162       162          161             162                161                   162                      163                         165                            168                               170                                  170                                     170                                        170                                           170170   169 165    159       151          141             130                119                   110                      97 80 60 36 35 33 42 6294 138 121    123       126          128             130                133                   134                      137                         140                            144                               147                                  150160   162 164    164       164          163             160                160                   160                      162                         164                            107                               171                                  171                                     171                                        171                                           171171   168 164    158       151          142             131                121                   111                      98 81 61 37 35 32 40 6092 135 122    124       126          128             130                132                   134                      137                         140                            144                               147                                  150163   165 166    167       166          164             159                159                   159                      160                         163                            166                               171                                  171                                     171                                        171                                           171171   167 163    158       151          143             132                129                   112                      99 83 62 39 35 31 39 5990 133 124    126       127          129             130                132                   137                      137                         140                            144                               147                                  150167   169 169    169       168          166             158                157                   158                      159                         161                            164                               171                                  171                                     171                                        171                                           171171   166 163    158       152          143             133                173                   114                      101                         84 64 40 35 31 38 5788 130 127    128       130          131             132                133                   134                      137                         140                            144                               147                                  150171   172 172    172       170          168             156                156                   156                      158                         160                            163                               169                                  169                                     169                                        169                                           169169   164 162    158       152          144             134                125                   115                      102                         85 65 41 34 30 37 5686 128 131    132       133          133             134                335                   134                      137                         140                            144                               147                                  150175   175 175    174       172          169             155                155                   155                      156                         159                            162                               167                                  167                                     167                                        167                                           167167   163 161    158       152          145             136                126                   116                      103                         87 66 43 34 29 36 5484 125 136    136       137          137             137                137                   134                      137                         140                            144                               147                                  150190   184 177    170       164          157             155                155                   155                      154                         154                            153                               153                                  155                                     158                                        161                                           166172   164 164    162       158          150             140                127                   112                      98 83 68 54 32 26 32 5081 125 137    137       137          137             137                137                   137                      141                         144                            148                               152                                  156190   183 176    170       163          156             151                151                   151                      151                         151                            151                               148                                  150                                     152                                        156                                           160165   166 167    165       160          153             143                127                   112                      97 83 68 53 36 29 35 5383 126 137    137       137          137             137                137                   140                      143                         146                            150                               153                                  156189   182 176    169       162          156             149                150                   150                      151                         151                            151                               147                                  148                                     150                                        153                                           157161   169 170    168       163          156             146                128                   114                      99 84 70 55 41 34 39 5787 129 137    137       137          137             137                137                   142                      145                         147                            150                               153                                  156188   182 175    168       162          155             150                151                   152                      153                         154                            155                               148                                  149                                     150                                        153                                           156160   172 172    170       166          158             148                132                   117                      102                         88 73 58 48 41 45 6292 134 137    137       137          137             137                137                   143                      145                         148                            150                               153                                  155188   181 174    168       161          154             154                155                   156                      158                         159                            160                               152                                  152                                     153                                        155                                           158162   174 175    173       168          161             151                137                   122                      107                         93 78 63 57 49 53 7099 141 137    137       137          137             137                137                   143                      145                         147                            149                               151                                  153187   180 174    167       160          154             160                162                   163                      165                         167                            169                               158                                  158                                     159                                        160                                           163166   177 178    176       171          164             154                144                   129                      114                         100                            85 70 68 59 63 79108   149 137    137       137          137             137                137                   142                      144                         145                            147                               149                                  150176   172 167    162       158          153             161                186                   207                      225                         240                            251                               215                                  204                                     196                                        190                                           187187   175 175    173       170          167             162                161                   145                      127                         109                            91 71 63 79 95 109123   135 139    140       140          140             140                141                   148                      149                         151                            153                               157                                  161175   172 169    165       162          159             166                187                   205                      219                         229                            236                               211                                  202                                     196                                        193                                           192194   184 183    181       179          175             171                169                   153                      136                         118                            99 79 70 84 98 110122   133 138    139       140          140             141                142                   146                      147                         149                            151                               155                                  159174   172 170    168       166          164             269                186                   199                      209                         216                            219                               202                                  196                                     192                                        191                                           192197   189 188    186       183          180             175                174                   158                      141                         123                            104                               84 76 89 100                                           111121   130 137    138       139          141             142                143                   145                      146                         148                            151                               154                                  159171   170 170    169       168          168             169                182                   191                      197                         200                            199                               189                                  185                                     184                                        185                                           188195   190 189    187       185          181             177                175                   159                      142                         124                            105                               85 82 93 103                                           112121   128 136    138       139          141             142                144                   145                      146                         148                            151                               154                                  159167   168 169    169       170          171             166                175                   181                      183                         181                            176                               172                                  170                                     171                                        174                                           180189   188 187    185       183          179             175                173                   157                      140                         122                            103                               83 89 98 106                                           114120   126 135    137       139          141             143                145                   146                      147                         149                            151                               155                                  159163   165 167    169       171          173             161                166                   167                      165                         160                            151                               150                                  150                                     153                                        159                                           167178   182 181    179       177          173             169                167                   151                      134                         116                            97 77 95 103                                        109                                           115119   123 134    136       139          141             144                146                   148                      149                         151                            153                               157                                  161148   151 154    157       161          164             169                171                   172                      172                         171                            169                               164                                  158                                     157                                        162                                           172187   193 184    173       159          142             123                97 93 89 87 85 84 101                                     107                                        114                                           121127   134 131    133       136          139             141                144                   150                      151                         153                            154                               155                                  157142   145 148    150       153          156             164                166                   167                      167                         166                            164                               161                                  152                                     149                                        151                                           158171   168 159    148       134          117             98 86 83 81 79 79 79 101                                     108                                        115                                           121128   135 131    134       137          139             142                145                   151                      152                         153                            155                               156                                  157138   141 143    145       147          149             160                161                   162                      162                         161                            160                               156                                  144                                     138                                        138                                           142152   150 141    130       116          99 80 77 76 35 75 76 77 102                                     109                                        115                                           122129   135 132    135       137          140             143                145                   151                      153                         154                            155                               157                                  158136   138 140    141       143          145             155                157                   158                      158                         157                            155                               149                                  135                                     126                                        123                                           125132   139 131    119       105          89 69 71 71 71 73 75 78 103                                     109                                        116                                           123129   136 133    135       138          141             143                146                   152                      153                         155                            156                               157                                  159136   137 138    140       141          142             150                152                   153                      153                         152                            150                               141                                  124                                     113                                        107                                           106111   136 127    116       102          85 66 68 69 71 73 77 81 103                                     110                                        117                                           123130   137 133    136       139          141             144                147                   153                      154                         155                            157                               158                                  159137   138 139    140       141          141             1L6                147                   148                      148                         147                            146                               130                                  111                                     97 88 8587 139 131    119       105          89 69 67 70 73 77 82 87 104                                     111                                        117                                           124131   137 134    137       139          142             145                147                   153                      155                         156                            157                               159                                  160137   137 137    137       137          137             144                145                   144                      141                         137                            130                               124                                  99 82 72 7178 103 110    110       102          86 61 70 71 73 75 76 78 101                                     110                                        119                                           126132   138 143    141       141          143             147                153                   159                      147                         144                            148                               160                                  180137   137 137    137       137          137             141                142                   141                      138                         134                            128                               119                                  94 77 69 6875 98 107    108       100          85 62 67 69 71 73 75 77 102                                     111                                        119                                           127133   139 143    141       141          143             147                153                   157                      148                         147                            155                               170                                  193137   137 137    137       137          137             138                139                   138                      136                         131                            125                               114                                  89 73 65 6572 94 104    105       99 84 62 65 67 70 72 75 77 103                                     112                                        120                                           127134   139 143    141       141          143             147                153                   154                      149                         151                            162                               180                                  206137   137 137    137       137          137             136                137                   136                      133                         129                            122                               108                                  85 68 61 6170 90 100    103       97 84 62 64 67 70 73 75 78 103                                     112                                        123                                           128134   140 143    141       141          143             147                153                   152                      149                         155                            168                               190                                  220137   137 137    137       137          137             133                134                   133                      130                         126                            120                               103                                  80 65 57 5867 86 97 100       96 83 62 64 67 70 74 77 80 104                                     113                                        121                                           129135   141 143    141       141          143             147                153                   149                      150                         159                            175                               200                                  233137   137 137    137       137          137             130                131                   130                      128                         123                            117                               98 75 150                                        54 5564 81 94 98 94 82 63 64 68 72 76 79 83 105                                     114                                        122                                           129136   141 143    141       141          143             147                153                   147                      151                         162                            182                               210                                  246141   138 136    133       130          128             135                132                   129                      125                         119                            113                               95 79 67 60 5759 97 103    104       101          93 81 57 64 71 77 82 86 95 115                                        130                                           139143   142 143    144       145          146             147                148                   134                      165                         191                            211                               226                                  236143   140 137    135       132          129             131                129                   127                      123                         119                            114                               87 71 60 53 5153 80 87 89 87 81 70 55 61 66 70 73 75 94 115                                        130                                           140145   144 144    145       146          146             147                148                   144                      173                         197                            216                               229                                  237144   141 138    136       133          130             127                127                   125                      123                         119                            115                               87 72 61 55 5356 69 77 81 81 75 66 62 66 69 71 72 73 95 116                                        132                                           142147   147 146    146       147          147             147                147                   153                      181                         203                            220                               232                                  238144   141 138    136       133          130             125                125                   125                      123                         121                            117                               97 82 72 66 6568 65 74 79 80 76 68 76 78 80 80 80 78 97 118                                        134                                           145151   151 148    148       147          147             141                147                   163                      189                         209                            225                               234                                  239143   140 137    135       132          129             124                125                   125                      125                         123                            121                               115                                  101                                     81 86 8589 66 77 83 85 83 76 99 99 99 97 95 92 99 121                                        138                                           149155   156 150    149       148          148             147                146                   173                      197                         216                            229                               237                                  240141   138 136    133       130          128             123                125                   127                      127                         126                            125                               143                                  129                                     119                                        115                                           114118   74 86 94 97 96 91 129                   128                      125                         122                            118                               113                                  102                                     125                                        142                                           154160   161 151    150       149          148             147                146                   183                      205                         222                            233                               240                                  241138   140 142    143       142          141             146                150                   153                      154                         155                            155                               158                                  153                                     150                                        147                                           145144   121 121    124       128          134             142                145                   142                      139                         136                            133                               130                                  144                                     155                                        162                                           164162   155 164    145       135          134             141                158                   226                      228                         230                            231                               233                                  235155   157 159    159       159          158             160                163                   165                      167                         167                            167                               168                                  164                                     160                                        158                                           156155   138 134    142       147          153             161                174                   170                      166                         162                            158                               154                                  164                                     173                                        177                                           177173   164 155    141       135          139             151                172                   230                      231                         232                            234                               235                                  236169   171 173    173       173          172             173                175                   177                      178                         178                            177                               178                                  173                                     170                                        167                                           165164   152 153    157       162          169             177                194                   189                      184                         179                            174                               169                                  177                                     183                                        186                                           183177   166 148    138       137          145             161                187                   232                      233                         234                            235                               236                                  237180   182 184    185       184          183             184                187                   188                      188                         188                            187                               185                                  180                                     177                                        174                                           172172   162 164    168       173          181             190                205                   199                      193                         187                            182                               176                                  182                                     167                                        187                                           182173   160 142    136       139          152             173                203                   234                      235                         235                            235                               236                                  236189   191 193    193       193          192             195                197                   198                      198                         199                            195                               190                                  186                                     182                                        180                                           178177   169 171    175       181          189             199                207                   200                      193                         187                            180                               173                                  181                                     133                                        181                                           174163   147 136    135       143          159             185                219                   235                      235                         235                            235                               235                                  235195   197 199    199       199          198             205                207                   207                      207                         205                            203                               194                                  190                                     186                                        184                                           182181   172 175    179       186          194             204                200                   192                      185                         177                            170                               162                                  172                                     172                                        168                                           159145   128 131    135       147          168             198                237                   235                      235                         234                            234                               233                                  233__________________________________________________________________________ 
    
     The above operation on Taylor polynomials is the following: 
     
         F(a.sub.0 +a.sub.1 x+a.sub.2 y+a.sub.11 x.sup.2 +2a.sub.12 xy+a.sub.22 y.sup.2)=1-a.sub.0 -a.sub.1 x-a.sub.2 y-a.sub.11 a.sup.2 -2a.sub.12 xy-a.sub.22 y.sup.2 
    
     (in the same rescaling as above). 
     The corresponding subroutine, applied to the Taylor polynomials of the intermediate image obtained in the Example 2, gives the intermediate image of the negative. The part of the polynomials array P&#34;, corresponding to the piece S&#34; of the negative, is the following. 
     
         __________________________________________________________________________0.62500000 -0.0000000        -0.0000000              -0.000000                     -0.000000                           -0.0000000.64062500 0.0078125        0.0234375              -0.000000                     -0.000000                           -0.0000000.68359375 0.0078125        -0.0000000              -0.000000                     -0.000000                           -0.0312500.60156250 0.0078125        -0.0937500              -0.000000                     -0.000000                           -0.0468750.25781250 0.0234375        -0.1640625              -0.000000                     -0.000000                           0.0468750.29296875 -0.0234375        0.2187500              -0.000000                     0.015625                           0.0312500.49218750 0.0078125        0.0312500              -0.000000                     -0.000000                           -0.0000000.56640625 -0.0000000        0.0156250              -0.000000                     -0.000000                           -0.0000000.66406250 0.0312500        -0.0000000              -0.000000                     -0.015625                           -0.0156250.62890625 -0.0156250        0.0156250              -0.000000                     -0.000000                           0.0156250.67578125 -0.0078125        -0.0000000              -0.015625                     -0.000000                           -0.0000000.61328125 -0.0000000        -0.0781250              -0.000000                     0.015625                           -0.0312500.36328125 0.0156250        -0.1953125              -0.000000                     -0.000000                           -0.0625000.18750000 -0.0156250        0.2265625              -0.000000                     -0.015625                           0.2031250.50781250 0.0234375        0.0156250              0.015625                     0.015625                           -0.0000000.56250000 -0.0000000        0.0390625              -0.000000                     -0.000000                           -0.0000000.67968750 -0.0078125        -0.0781250              -0.000000                     -0.000000                           -0.0000000.59765625 0.0234375        0.0078125              0.046875                     0.015625                           -0.0000000159765625 -0.0000000        0.0312500              0.046875                     -0.015625                           0.0156250.66015625 0.0312500        -0.0546875              -0.000000                     -0.000000                           -0.0468750.36718750 0.0390625        -0.1718750              0.031250                     -0.000000                           -0.0000000.19531250 0.0703125        0.2031250              0.031250                     -0.015625                           0.2187500.54296875 -0.0000000        -0.0000000              -0.000000                     -0.000000                           -0.0000000.58984375 -0.0000000        0.0312500              -0.015625                     -0.015625                           -0.0000000.66796875 0.0078125        -0.0156250              -0.015625                     0.046875                           -0.0000000.78906250 -0.1171875        0.0937500              -0.046875                     -0.140625                           -0.0625000.74218150 -0.0859375        -0.0000000              -0.078125                     0.078125                           0.0468750.73437500 0.0156250        -0.0312500              -0.062500                     -0.000000                           -0.0156250.52343750 0.0156250        -0.2109375              -0.062500                     -0.000000                           -0.0156250.42187500 0.0234375        0.1171875              -0.000000                     -0.062500                           -0.0156250.55468750 -0.0000000        0.0156250              -0.000000                     0.015625                           -0.0000000.58984375 -0.0000000        0.0312500              0.015625                     -0.000000                           0.0156250.56250000 -0.0390625        0.0234375              0.031250                     -0.015625                           -0.0000000.63281250 -0.0546875        -0.0000000              -0.000000                     -0.000000                           -0.0156250.51562500 -0.1562500        -0.0234375              -0.031250                     -0.093750                           0.0937500.46093750 -0.1250000        -0.1640625              0.125000                     -0.000000                           -0.0468750.28906250 -0.0312500        0.0078125              0.046875                     0.046875                           0.0156250.46875000 0.0078125        0.0781250              -0.000000                     -0.000000                           -0.0000000.55078125 0.0078125        0.0312500              -0.000000                     -0.000000                           -0.0000000.61328125 0.0078125        0.0156250              -0.000000                     -0.000000                           -0.0000000.54296875 -0.0000000        -0.0000000              -0.000000                     -0.000000                           -0.0000000.53906250 -0.0312500        -0.0312500              -0.000000                     - 0.000000                           -0.0312500.26171875 -0.0468750        -0.0937500              -0.000000                     0.015625                           0.1406250.40234375 -0.0234375        -0.0703125              -0.000000                     0.031250                           -0.1406250.28125000 -0.0000000        0.0312500              0.015625                     0.015625                           -0.0000000.48828125 0.0078125        0.0859375              -0.000000                     -0.000000                           -0.0156250.56250000 -0.0000000        0.0234375              -0.000000                     -0.000000                           0.0312500.62500000 0.0625000        0.1406250              -0.000000                     0.109375                           0.1406250.54296875 -0.0000000        -0.0312500              -0.015625                     -0.000000                           -0.0000000.48828125 -0.0000000        -0.0234375              0.015625                     0.031250                           -0.0156250.24609375 0.1250000        -0.0703125              0.156250                     0.015625                           0.0781250.31640625 -0.0156250        -0.0000000              0.109375                     0.046875                           -0.0781250.29296875 0.1171875        0.0156250              0.140625                     -0.062500                           -0.0156250.55078125 0.0312500        0.1250000              0.015625                     0.015625                           -0.0937500.58203125 0.0078125        -0.0000000              -0.000000                     -0.015625                           -0.0000000.84765625 0.0625000        0.1875000              -0.000000                     -0.062500                           -0.0937500.70703125 0.1328125        0.0078125              -0.046875                     -0.000000                           -0.0156250.72265625 0.1250000        0.0078125              -0.015625                     -0.015625                           -0.0156250.67968750 0.0859375        -0.0312500              -0.031250                     -0.000000                           0.0156250.65234375 0.1326125        0.0625000              -0.062500                     0.015625                           0.0312500.73828125 0.1015625        -0.0625000              -0.156250                     -0.031250                           -0.0000000.73437500 -0.0000000        -0.0390625              -0.125000                     -0.078125                           -0.0781250.56250000 0.0546875        0.1171875              0.015625                     0.156250                           0.1562500.92578125 0.0078125        0.0078125              -0.015625                     -0.015625                           -0.000000__________________________________________________________________________ 
    
     The final image produced from the intermediate negative image, obtained as above, is shown in FIG. 6b. 
     While a number of embodiments of the invention have been discussed and illustrated, it will be understood that the invention may be carried out in a number of ways and with many modifications, adaptations, and variations, by persons skilled in the art, without departing from its spirit and from the scope of the appended claims.