Method and arrangement for computing pixel values of pixels of a digital picture signal that are arranged outside a two-dimensional scanning raster

A method and arrangement for computing values of pixels arranged outside a 2-dimensional scanning raster of pixels having pixel values of a digital picture signal, where at least 3 of 4 reference pixels arranged around the searched pixel are determined for computing a searched pixel value s.sub.i (x.sub.0 +.DELTA.x,y.sub.0 +.DELTA.y). A polynomial p(x,y) in the form c.sub.m-1 x.sup.am-1 y.sup.bm-1 +c.sub.m-2 x.sup.am-2 y.sup.bm-2 + . . . +c.sub.2 x+c.sub.1 y+c.sub.0 is constituted corresponding to their values. Next, k 1st derivatives .delta.s(x,y)/.delta.x in the x-direction and .delta.s(x,y)/.delta.x in the y-direction are formed from at least one of the reference pixel values. Then, a matrix is formed whose elements consist of the values of the basic functions x.sup.am-1 y.sup.bm-1,x.sup.am-2 y.sup.bm-2, . . . x,y,1 of the polynomial at at least 3 of the reference pixels and the k 1st derivatives .delta.p(x,y)/.delta.x in the x-direction and .delta.p(x,y)/.delta.y in the y-direction of the basic functions at at least one of the reference pixel values. The basic functions are allocated to a reference pixel or a 1st derivative of a reference pixel in one matrix row. An inverse or pseudo-inverse matrix is formed from the matrix by means of inversion and multiplied by an observation vector for determining the polynomial coefficients c.sub.m-1 through c.sub.0. The polynomial coefficients are introduced into polynomial p(x,y) and the new pixel value s.sub.i (x.sub.0 +.DELTA.x,y.sub.0 +.DELTA.y) is computed from the function value of the polynomial p(.DELTA.x,.DELTA.y) at the location (.DELTA.x, .DELTA.y).

BACKGROUND OF THE INVENTION 
The invention relates to a method and an arrangement for computing values 
s.sub.i (x.sub.0 +.DELTA.x,y.sub.0 +.DELTA.y) of pixels which are arranged 
outside a two-dimensional scanning raster {x.sub.0 .+-.n, y.sub.0 .+-.m, 
with m,n=0,1,2,3, . . . } of pixels having pixel values s(x.sub.0 
.+-.n,y.sub.0 .+-.m) of a digital picture signal. 
In such methods and arrangements for computing new pixel values, it is 
important to determine the values of pixels which are located outside a 
two-dimensional scanning raster. The known pixels are situated at 
locations (x.sub.0 .+-.n,y.sub.0 .+-.m) in which m and n are integers of 
zero up to a maximal value. The pixels of a digital picture signal which 
are located in this scanning raster have known pixel values. By way of 
non-limitative example, it is herein assumed that the distance between two 
adjacent pixels is 1 in the horizontal and vertical directions. 
For example, for the purpose of television signal conversion, conversion 
between different graphic modes in PCs, video data compression, or for 
medical applications, it is often desirable or necessary to determine 
pixel values of pixels which are not located at the predetermined points 
on the two-dimensional scanning raster but are located between these 
points. Fundamentally, the searched pixel values may have arbitrary 
positions between the points on the scanning raster. This complicates 
their computation. 
In the state of the art, methods are known which are based on bilinear or 
bicubic interpolation for computing such new pixel values in their 
simplest case. In bilinear interpolation, a linear weighting of the four 
pixels located closest to a searched pixel in the scanning raster is 
performed. In the interpolation by means of cubic B-splines, an ideal 
interpolation filter is approximated by means of a sin(x)/x-shaped pulse 
response for the one-dimensional case by means of a 3rd-order polynomial, 
which minimizes the interpolation error, but strongly reduces the 
resolution. Cf. K. Pratt: "Digital Image Processing", second edition, pp. 
114 etc. 
For computing new pixel values, it is further possible to perform a 
combined up and down-sampling. 
EP-A-660 514 discloses a filter which operates as a one-dimensional 
polynomial interpolator and in which a one-dimensional polynomial is set 
up which should approximate the pixel values located outside a 
one-dimensional scanning raster as satisfactorily as possible. 
SUMMARY OF THE INVENTION 
It is an object of the invention to provide a method and an arrangement 
allowing the computation of pixels which are located outside a 
two-dimensional scanning raster. The computation should be as accurate as 
possible and yet performed with a reasonable number of components. 
For a method according to the invention, this object is solved in that for 
computing a searched pixel value s.sub.i (x.sub.0 +.DELTA.x,y.sub.0 
+.DELTA.y), at least three pixel values s(x.sub.0,y.sub.0), 
s(x.sub.0,y.sub.0 +1), s(x.sub.0 +1,y.sub.0) and s(x.sub.0 +1,y.sub.0 +1) 
of four reference pixels arranged around the searched pixel are 
determined, 
a polynomial p(x,y) of the form 
EQU p(x,y)=c.sub.m-1 x.sup.am-1 y.sup.bm-1 +c.sub.m-2 x.sup.am-2 y.sup.bm-2 + . 
. . +c.sub.2 x+c.sub.1 y+c.sub.0 
is constituted which, at the positions of at least three of the four 
reference pixels should correspond to their values, so that 
p(0,0)=s(x.sub.0,y.sub.0), p(1,0)=s(x.sub.0 +1,y.sub.0), 
p(0,1)=s(x.sub.0,y.sub.0 +1) and/or p(1, 1)=s(x.sub.0 +1,y.sub.0 +1), 
the k first derivatives .delta.s(x,y)/.delta.x in the x-direction and 
.delta.s(x,y)/.delta.y in the y-direction are formed from at least one of 
the reference pixel values, 
a matrix is formed, whose elements consist of the values of the basic 
functions 
EQU x.sup.am-1 y.sup.bm-1,x.sup.am-2 y.sup.bm-2, . . . , x,y,1 
of the polynomial at at least three of the reference pixels and the k first 
derivatives .delta.p(x,y)/.delta.x in the x-direction and 
.delta.p(x,y)/.delta.y in the y-direction of the basic functions at least 
one of the reference pixels, the basic functions allocated to a reference 
pixel or to a first derivative of a reference pixel being in one row of 
the matrix, 
an inverse or pseudo-inverse matrix is formed from the matrix by means of 
inversion, 
the inverse or pseudo-inverse matrix is multiplied by an observation vector 
for determining the polynomial coefficients c.sub.m-1, c.sub.m-2, . . . , 
c.sub.2, c.sub.1, c.sub.0, which vector comprises as elements at least 
three of the reference pixel values and the k first derivatives 
.delta.s(x,y)/.delta.x in the x-direction and .delta.s(x,y)/.delta.y in 
the y-direction of at least one of the reference pixels, 
the polynomial coefficients are introduced into the polynomial p(x,y) and 
the new pixel value s.sub.i (x.sub.0 +.DELTA.x,y.sub.0 +.DELTA.y) is 
computed by computing the function value of the polynomial 
p(.DELTA.x,.DELTA.y) at the location (.DELTA.x,.DELTA.y). 
In the method according to the invention, a polynomial is set up which 
maximally approximates the values of four pixels which are located in the 
two-dimensional scanning raster and whose values are known. Moreover, the 
polynomial should maximally approximate the k.sup.th derivatives of these 
pixels. Such a computed polynomial allows a reasonably accurate 
determination of pixel values which are arranged in the square between the 
four known pixels. 
For a searched pixel value s.sub.i (x.sub.0 +.DELTA.x,y.sub.0 +.DELTA.y), 
the values of four reference pixels arranged around this searched pixel 
are first determined. These are the pixels which are closest to the 
searched pixel in the two-dimensional scanning raster and whose values are 
known. A polynomial of the above-mentioned form is set up, which, as a 
target, should maximally approximate the pixel values of these four 
reference pixels. 
For example, for the reference pixel value s(x.sub.0,y.sub.0), the 
polynomial should yield its pixel value when the values 0 for x and 0 for 
y are introduced in the polynomial. The polynomial should yield the 
reference pixel value s(x.sub.0 +1,y.sub.0) when the values 1 for x and 0 
for y are introduced in the polynomial. A corresponding situation is valid 
for the two further reference pixel values. 
The k first derivatives .delta.s(x,y)/.delta.x and .delta.s(x,y)/.delta.y 
are formed from at least one of the reference pixel values. The polynomial 
should maximally approximate also these values of the derivatives so that 
the derivative of the polynomial for each pixel is equal to the 
corresponding derivative of the pixel value itself. 
It is generally sufficient when the first derivatives in the x and 
y-directions are formed for one of the reference pixels and when the 
polynomial for one of these reference pixel values maximally approximates 
its derivative. 
The above polynomial is initially present in a general form; for computing 
the searched pixel value s.sub.i (x.sub.0 +.DELTA.x,y.sub.0 +.DELTA.y), 
particularly the polynomial coefficients c.sub.m-1, c.sub.m-2, . . . , 
c.sub.2, c.sub.1, c.sub.0 are to be determined. To this end, a matrix is 
formed whose values only consist of the basic functions of the polynomial. 
In this matrix, the polynomial coefficients thus do not occur but only the 
other elements of the polynomial occur. The polynomial is then set up for 
at least three of the reference pixel values and for the first derivatives 
in the x and y-directions at at least one of the reference pixel values. 
The basic functions are set up within the matrix in such a way that, 
within a row of the matrix, the basic functions allocated to one reference 
pixel value or to a first derivative of a reference pixel value are 
arranged in the same sequence as in the polynomial. Thus, the basic 
functions of the polynomial of at least three of the reference pixel 
values and of at least the first derivatives in the x and y-directions of 
at least one of the reference pixel values are line-sequentially present 
within the matrix. 
An inverse or pseudo-inverse matrix which is multiplied by means of an 
observation vector is formed from this matrix by means of inversion. The 
observation vector line-sequentially has the corresponding values of the 
reference pixels themselves and the corresponding derivatives of the 
reference pixels. The lines within the vector and within the inverse 
matrix are allocated to the same reference pixels or reference pixel 
derivatives. 
The polynomial coefficients c.sub.m-1, c.sub.m-2, . . . , c.sub.2, c.sub.1, 
c.sub.0 are computed by multiplying the inverse matrix by the observation 
vector. 
The polynomial coefficients can now be introduced into the polynomial 
p(x,y). 
The values .DELTA.x and .DELTA.y are introduced into the polynomial for the 
searched pixel value s.sub.i (x.sub.0 +.DELTA.x,y.sub.0 +.DELTA.y). The 
polynomial is now computed and yields, as a result, the searched pixel 
value s.sub.i (x.sub.0 +.DELTA.x,y.sub.0 +.DELTA.y). 
A very good approximation of the searched pixel value is found by means of 
this two-dimensional polynomial computation of this value. Nevertheless, 
the number of components required for the computation remains reasonable, 
because some of the above-mentioned computation steps do not need to be 
repeated for each pixel value. Only the new values to be introduced for 
the reference pixels and their first derivatives are to be determined each 
time. The polynomial in its basic form, as well as the inverse matrix, 
are, however, maintained for changing (.DELTA.x,.DELTA.y). Moreover, the 
polynomial values p(0,0), p(1,0), p(0,1) and p(1,1) remain identical 
because the searched pixel value is introduced in such a way that it is 
arranged at the location x.sub.0 +.DELTA.x and y.sub.0 +.DELTA.y. 
Independent of the fact where the searched pixel value is arranged in a 
two-dimensional raster, the quadrant of reference pixel values is always 
set up with these values, while only the relative location within this 
quadrant is re-introduced by the values .DELTA.x and .DELTA.y for the 
searched pixel value. 
An embodiment of the invention is characterized in that for forming the 
matrix, the first derivatives .delta.p(x,y)/.delta.x in the x-direction 
and .delta.p(x,y)/.delta.y in the y-direction are formed from one of the 
reference pixel values, preferably the reference pixel value 
s(x.sub.0,y.sub.0), and that the observation vector comprises the four 
reference pixel values s(x.sub.0,y.sub.0), s(x.sub.0,y.sub.0 +1), 
s(x.sub.0 +1,y.sub.0) and s(x.sub.0 +1,y.sub.0 +1) and the first 
derivatives .delta.s(x,y)/.delta.x in the x-direction and 
.delta.s(x,y)/.delta.y in the y-direction of one of the reference pixels, 
preferably the reference pixel value s(x.sub.0,y.sub.0). 
For the computation explained above, it is fundamentally sufficient to take 
the four reference pixel values and the first derivatives in the x and 
y-directions of one of the reference pixel values into account. In this 
case, already very good approximations of the searched pixel value are 
obtained, without using elaborate computations. 
According to the invention, the object mentioned hereinbefore is solved for 
an arrangement in that 
first computing means are provided, which 
constitute a polynomial (p(x,y) of the form 
EQU p(x,y)=c.sub.m-1 x.sup.am-1 y.sup.bm-1 +c.sub.m-2 x.sup.am-2 y.sup.bm-2 +. 
. . +c.sub.2 x+c.sub.1 y+c.sub.0 
which, at the positions of at least three of the four reference pixels 
should correspond to their values, so that p(0,0)=s(x.sub.0,y.sub.0), 
p(1,0)=s(x.sub.0 +1,y.sub.0), p(0,1)=s(x.sub.0,y.sub.0 +1) and/or 
p(1,1)=s(x.sub.0 +1,y.sub.0 +1), 
form a matrix whose elements consist of the values of the basic functions 
EQU x.sup.am-1 y.sup.bm-1,x.sup.am-2 y.sup.bm-2, . . . ,x,y,1 
of the polynomial at at least three of the reference pixels and the k first 
derivatives .delta.p(x,y)/.delta.x in the x-direction and 
.delta.p(x,y)/.delta.y in the y-direction of the basic functions at least 
one of the reference pixels, the basic functions allocated to a reference 
pixel or to a first derivative of a reference pixel being in one row of 
the matrix, 
form an inverse or pseudo-inverse matrix from the matrix by means of 
inversion, 
differentiators are provided which form the k first derivatives 
.delta.s(x,y)/.delta.x in the x-direction and .delta.s(x,y)/.delta.y in 
the y-direction from at least one of the reference pixel values, 
second computing means are provided, which 
for computing a searched pixel value s.sub.i (x.sub.0 +.DELTA.x,y.sub.0 
+.DELTA.y), receive at least three pixel values s(x.sub.0,y.sub.0), 
s(x.sub.0,y.sub.0 +1), s(x.sub.0 +1,y.sub.0) and s(x.sub.0 +1,y.sub.0 +1) 
of four reference pixels arranged around the searched pixel, 
multiply the inverse or pseudo-inverse matrix by an observation vector for 
determining the polynomial coefficients c.sub.m-1, c.sub.m-2, . . . , 
c.sub.2, c.sub.1, c.sub.0, which vector comprises as elements at least 
three of the reference pixel values and the k first derivatives 
.delta.s(x,y)/.delta.x in the x-direction and .delta.s(x,y)/.delta.y in 
the y-direction of at least one of the reference pixels, 
and introduce the polynomial coefficients into the polynomial p(x,y) and 
compute the new pixel value s.sub.i (x.sub.0 +.DELTA.x,y.sub.0 +.DELTA.y) 
by computing the function value of the polynomial p(.DELTA.x,.DELTA.y) at 
the location (.DELTA.x,.DELTA.y). 
This arrangement operates in accordance with the method described 
hereinbefore. For the arrangement, the fact that not all of the 
above-mentioned computations have to be repeated for computing a new pixel 
value, may be utilized to advantage. 
Therefore, first computing means are provided in the arrangement, which set 
up the polynomial, compute the polynomials at the four reference pixel 
values and form the matrix and the inverse matrix. These means may be 
implemented in such a way that they perform these computations only once. 
Furthermore, differentiators are provided which perform the required first 
derivations. 
With the aid of second computing means, a searched pixel value s.sub.i 
(x.sub.0 +.DELTA.x,y.sub.0 +.DELTA.y) is computed in the manner described 
above. The computations performed by the computing means are to be 
repeated each time for one searched pixel value s.sub.i (x.sub.0 
+.DELTA.x,y.sub.0 +.DELTA.y). 
Dependent on their realization, the means for computing the polynomial, its 
values and the matrices may be implemented in such a way that they are 
realized in the computing means. This means that the computing means are 
implemented in such a way that the computations of the polynomial, its 
values and the matrices are implicitly realized in their structure. 
The arrangement according to the invention is further characterized in that 
the differentiators in the frequency range for forming the k.sup.th 
derivative approximatively have the transfer function H=(j.omega.).sup.k, 
in which .omega. is the local frequency in the x-direction or the 
y-direction, dependent on whether the derivative is to be realized in the 
x-direction or the y-direction. A very good approximation of the 
derivative values is achieved by this implementation of the transfer 
function. 
In accordance with a further embodiment of the invention, the 
differentiators comprise successively arranged delay elements, and that 
adders are provided which receive, within the differentiator, each time 
those signals which are multiplied by coefficients of the same value, the 
adders preceding a multiplier performing the multiplication by the 
allocated coefficients, and the output signals from the multipliers are 
added by means of an adder which supplies the output signal of the 
differentiator. 
Since the above-mentioned transfer function can be performed by 
multiplication of pixel values delayed by several values and since pixel 
values delayed by several values are to be multiplied by the same 
coefficients, this differentiator structure limits the number of 
components to a maximal extent. 
A further embodiment of the arrangement according to the invention is 
characterized in that 
the values s(x.sub.0 .+-.n,y.sub.0 .+-.m) are applied to a series 
arrangement of line memories whose output signals supply pixel values 
s(x.sub.0 +3,y.sub.0+ 2), s(x.sub.0 +3,y.sub.0 +1), . . . , s(x.sub.0 
+3,y.sub.0 -3), 
a first delay element is provided which receives the pixel value s(x.sub.0 
+3,y.sub.0 +1) and supplies the pixel value s(x.sub.0 +1,y.sub.0 +1) from 
its output, 
a second delay element is provided which receives the pixel value s(x.sub.0 
+1,y.sub.0 +1) and supplies the pixel value s(x.sub.0,y.sub.0 +1) from its 
output, 
a third delay element is provided which receives the pixel value s(x.sub.0 
+3,y.sub.0) and supplies the pixel value s(x.sub.0 +1,y.sub.0) from its 
output, 
a fourth delay element is provided which receives the pixel value s(x.sub.0 
+1,y.sub.0) and supplies the pixel value s(x.sub.0,y.sub.0) from its 
output, 
a first differentiator is provided which receives the pixel values 
s(x.sub.0 +3,y.sub.0 +3), s(x.sub.0 +3,y.sub.0 +2), . . . , s(x.sub.0 
+3,y.sub.0 -3) and performs a first derivation 
.delta.s(x,y)/.delta.y.vertline.(x.sub.0 +3,y.sub.0) of the pixel value 
s(x.sub.0 +3,y.sub.0) in the y-direction, 
a fifth delay element is provided which receives the first derivative in 
the y-direction at the location (x.sub.0 +3,y.sub.0) and supplies from its 
output the third derivative of the pixel values in the y-direction at the 
location (x.sub.0,y.sub.0), 
a second differentiator is provided which receives the pixel value 
s(x.sub.0 +3,y.sub.0) and performs a first derivation 
.delta.s(x,y)/.delta.x.vertline.(x.sub.0,y.sub.0) of the pixel value in 
the x-direction, 
the computing circuit comprises a superposition stage by means of which the 
pixel values and first derivatives s(x.sub.0,y.sub.0), s(x.sub.0,y.sub. 
+1), s(x.sub.0 +1,y.sub.0), s(x.sub.0 +1,y.sub.0 
+1),.delta.s(x,y)/.delta.x.vertline.(x.sub.0,y.sub.0) and 
.delta.s(x,y)/.delta.y.vertline.(x.sub.0,y.sub.0) are superimposed in 
accordance with the inverse matrix in such a way that the superposition 
stage supplies the coefficients from its output, 
the computing circuit comprises multipliers by means of which the 
coefficients are multiplied by the polynomial basic functions which are 
generated from .DELTA.x and .DELTA.y, 
and the output signals from the multiplier are added by means of an adder 
stage in the computing circuit, which adder stage supplies the value of 
the searched pixel s.sub.i (x.sub.0 +.DELTA.x,y.sub.0 +.DELTA.y) from its 
output. 
This arrangement consisting of the line memories, the delay elements and 
the differentiators supplies the required values to the second computing 
circuit which requires these values for setting up the observation vector. 
The computation of the coefficients is then performed by corresponding 
fixed combinations in the superposition stage. The superposition stage 
supplies the coefficients of the polynomial from its output. The 
coefficients are subsequently multiplied by the polynomial basic functions 
which are generated from .DELTA.x and .DELTA.y of the searched pixel value 
s.sub.i. The combination of these products by means of the adder stage 
then yields the searched pixel value s.sub.i (x.sub.0 +.DELTA.x,y.sub.0 
+.DELTA.y). 
In this arrangement, the first computing means are integrated in the second 
computing means in such a way that the second computing circuit and its 
superposition stage have such a structure that implicitly a computation of 
the values in accordance with the polynomial or its k first derivatives is 
performed. Moreover, the combinations in the superposition stage are 
chosen to be such that the inverse or pseudo-inverse matrix is realized by 
predetermined combinations of the pixel values or their first derivatives. 
In this arrangement, the first computing circuit does not explicitly appear 
but is realized by the structure of the second computing circuit. This is 
possible because the computations of the first computing circuit do not 
need to be individually performed for pixel values but represent general 
computing prescriptions which can be realized by the structure of the 
second computing circuit.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
FIG. 1 shows a two-dimensional scanning raster with pixel values s(x,y). 
The section of the scanning raster of FIG. 1 shows raster points x.sub.0 
-1, x.sub.0, x.sub.0 +1 and x.sub.0 +2 in the x-direction. In the 
y-direction, this scanning raster shows raster values y.sub.0 -1, y.sub.0, 
y.sub.0 +1 and y.sub.0 +2. 
It is assumed that a pixel value s.sub.i (x.sub.0 +.DELTA.x,y.sub.0 
+.DELTA.y) is searched. The searched pixel value s.sub.i is thus arranged 
within a square which has reference pixel values s(x.sub.0,y.sub.0), 
s(x.sub.0,y.sub.0 +1), s(x.sub.0 +1, y.sub.0) and s(x.sub.0 +1,y.sub.0 
+1). 
Independent of the fact that where the searched pixel value s.sub.i is 
actually located within the two-dimensional raster, its location in the 
two-dimensional scanning raster is always computed on the basis of a 
reference pixel s(x.sub.0,y.sub.0). This means that the pixels arranged 
around the searched pixel s.sub.i always have the above-mentioned raster 
values. 
FIG. 2 shows a coarse block diagram of an arrangement for computing the 
searched pixel value s.sub.i. The values of the pixels of the 
two-dimensional scanning raster s(x,y) are applied to differentiators 1, 2 
to 3 arranged one behind the other in the arrangement and performing a 
derivation of the received pixel values in the x-direction. An overall 
number of k such differentiators may be provided. Furthermore, the 
differentiators 4, 5 to 6 are provided which perform derivations of the 
received pixel values in the y-direction of the two-dimensional scanning 
raster. Here again, as many differentiators 4 to 6 are provided as first 
derivations are to be performed. 
The output values of all differentiators 1 to 6 are applied to a circuit 
block 7 which forms an observation vector from the values of the reference 
pixel values themselves as well as from their k first derivatives. 
Furthermore, a polynomial of the form 
EQU p(x,y)=c.sub.m-1 x.sup.am-1 y.sup.bm-1 +c.sub.m-2 x.sup.am-2 y .sup.bm-2 + 
. . . +c.sub.2 x+c.sub.1 y+c.sub.0 
is set up in the circuit block 7. A matrix is also formed, whose elements 
consist of the values of the basic functions 
EQU x.sup.am-1 y.sup.bm-1,x.sup.am-2 y.sup.bm-2, . . . , x,y,1 
of the polynomial at the reference pixels and the k first derivatives. The 
basic functions allocated to the reference pixel or to one of the k first 
derivatives of a reference pixel are arranged within a row of the matrix. 
In the arrangement, an inverse or pseudo-inverse matrix A.sup.31 is 
formed from this matrix. This matrix is multiplied by the observation 
vector and supplies the polynomial coefficients c.sub.m-1, c.sub.m-2, . . 
. , c.sub.2, c.sub.1, c.sub.0 at its output. This signal represents the 
output signal of the circuit block 7 which is applied to a second circuit 
block 8 in which a computation of the above-mentioned polynomial is 
performed for the searched pixel value s.sub.i (x.sub.0 +.DELTA.x,y.sub.0 
+.DELTA.y) in that the coefficients are introduced into the polynomial and 
the function value of the polynomial is determined by introducing the 
values .DELTA.x and .DELTA.y. The result of this polynomial thus computed 
yields the pixel value s.sub.i (x.sub.0 +.DELTA.x,y.sub.0 +.DELTA.y). 
The above-mentioned polynomial of the form 
EQU p(x,y)=c.sub.m-1 x.sup.am-1 y.sup.bm-1 +c.sub.m-2 x.sup.am-2 y.sup.bm-2 + . 
. . +c.sub.2 x+c.sub.1 y+c.sub.0 
should maximally approximate the values of the four reference pixels 
s(x.sub.0,y.sub.0), s(x.sub.0,y.sub.0 +1), s(x.sub.0 +1,y.sub.0) and 
s(x.sub.0 +1,y.sub.0 +1). Moreover, the k first derivatives of this 
polynomial at at least one of the reference pixels should maximally 
approximate the k first derivatives of the values of the corresponding 
reference pixels. 
This will be explained hereinafter, assuming that for the computation of a 
searched pixel s.sub.i (x.sub.0 +.DELTA.x,y.sub.0 +.DELTA.y) the four 
reference pixel values s(x.sub.0,y.sub.0), s(x.sub.0,y.sub.0 +1), 
s(x.sub.0 +1,y.sub.0) and s(x.sub.0 +1,y.sub.0 +1) arranged around this 
searched pixel are used for this computation. Moreover, only the first 
derivative of the reference pixel s(x.sub.0,y.sub.0) in the x and 
y-directions are used. 
A matrix is formed whose elements consist of the values of the basic 
functions 
EQU x.sup.am-1 y.sup.bm-1,x.sup.am-2 y.sup.bm-2, . . . , x,y,1 
of the polynomial and the four reference pixel values and the first 
derivative of the basic functions of the polynomial at the reference pixel 
value s(x.sub.0,y.sub.0). The basic functions are arranged 
line-sequentially for the four reference pixel values and the first 
derivatives in the x and y-directions. 
For the general case in which .delta.p(x,y)/.delta.x and 
.delta.p(x,y)/.delta.y are used for the polynomial p(x,y) and for the 
first derivatives, it should hold that: 
##EQU1## 
This equation may be set up in this form because the basic functions which 
are present in the matrix and are multiplied by the allocated polynomial 
coefficients should yield the polynomials and the derivatives of the 
polynomials again. 
For the special, simplified case in which the above-mentioned four 
reference pixels and the first derivatives of one of the reference pixels 
are used for computing a pixel value s.sub.i, this equation is as follows: 
##EQU2## 
By using the reference pixel values and the first derivatives of one of 
the reference pixel values in the basic functions, the matrix is 
simplified in which there are only zeros and ones left. 
For the special case assumed for this equation in which only the values of 
the four reference pixels and the first derivatives in the x and 
y-directions of one of the reference pixel values are used, the 
above-mentioned polynomial can be simplified to the form 
EQU p(x,y)=c.sub.5 x.sup.2 +c.sub.4 y.sup.2 +c.sub.3 xy+c.sub.2 x+c.sub.1 
y+c.sub.0 (3) 
Since in the equation (2) shown above, the coefficients c.sub.0 to c.sub.5 
are initially unknown, the equation (2) is solved in accordance with the 
vector with the coefficients c.sub.5 to c.sub.0. Then the following 
equation is obtained 
##EQU3## 
Since the polynomial should maximally approximate the values of the 
reference pixels and the first derivatives in the manner described above, 
equation (4) can be written as: 
##EQU4## 
In this equation, the matrix set up with the basic functions is now an 
inverse or pseudo-inverse matrix. The vector comprising the pixel values 
and their first derivatives as elements will hereinafter be referred to as 
observation vector. By multiplication of this observation vector with the 
inverse or pseudo-inverse matrix, the vector can be computed with the 
polynomial coefficients c.sub.5 to c.sub.0. 
When the polynomial coefficients c.sub.5 to c.sub.0 are known, they can be 
introduced into the polynomial 
EQU p(x,y)=c.sub.5 x.sup.2 +c.sub.4 y.sup.2 +c.sub.3 xy+c.sub.2 x+c.sub.1 
y+c.sub.0 
For computing a searched pixel s.sub.i (x.sub.0 +.DELTA.x,y.sub.0 
+.DELTA.y), these values should then be introduced into the polynomial so 
that: 
EQU s.sub.i (x.sub.0 +.DELTA.x,y.sub.0 +.DELTA.y)=c.sub.5 .DELTA.x.sup.2 
+c.sub.4 .DELTA.y.sup.2 +c.sub.3 .DELTA.x.DELTA.y+c.sub.2 .DELTA.x+c.sub.1 
.DELTA.y+c.sub.0. (6) 
In the coefficients c.sub.5 to c.sub.0 which are now known, the searched 
pixel value s.sub.i can then be computed directly. 
FIG. 3 is a block diagram of an arrangement according to the invention 
computing the pixel value s.sub.i in accordance with this example in which 
the four reference pixel values s(x.sub.0,y.sub.0),s(x.sub.0 
+1,y.sub.0),s(x.sub.0,y.sub.0 +1) and s(x.sub.0 +1,y.sub.0 +1) surrounding 
the searched pixel are used for computing the pixel value s.sub.i. 
Moreover, the first derivative in the x and y-directions of the reference 
pixel value s(x.sub.0,y.sub.0) is introduced. 
The pixel values s(x,y) present in the two-dimensional scanning raster of 
FIG. 1 are applied to three line memories 31 to 36 arranged one behind the 
other in the arrangement of FIG. 3. The line memories 31 to 36 perform a 
delay by a period of one picture line so that they supply pixels of the 
same x value at the output, which pixels have an y value which is smaller 
than a given value, i.e. they are each time located one line higher in the 
same horizontal position. 
When the output value of the line memory 33 is used as value s(x.sub.0 
+3,y.sub.0), the following pixel values are provided: 
the pixel value s(x.sub.0 +3,y.sub.0 +3) is present at the input of the 
line memory 31, the value s(x.sub.0 +3,y.sub.0 +2) is present at the 
output of the line memory 31, the value s(x.sub.0 +3,y.sub.0 +1) is 
present at the output of the line memory 32, the value s(x.sub.0 
+3,y.sub.0 -1) is present at the output of the line memory 34, the pixel 
value s(x.sub.0 +3,y.sub.0 -2) is present at the output of the line memory 
35, and the pixel value s(x.sub.0 +3,y.sub.0 -3) is present at the output 
of the line memory 36. 
All of these pixel values are applied to a first differentiator 41 which 
performs a derivation of the pixel values in the y-direction, the 
derivation being performed on the basis of the pixel value s(x.sub.0 
+3,y.sub.0). 
Subsequently, a delay member 43 is provided which delays the derivation of 
the pixel values in the y-direction at the position (x.sub.0 +3,y.sub.0) 
by a period of three pixel values so that the derivation of the pixel 
values in the y-direction is present at the output at the position 
(x.sub.0,y.sub.0). 
Furthermore, two delay elements 51 and 52 are provided, each performing a 
delay by a period of two pixel values, thus generating a delay by two 
values in the x-direction of the two-dimensional scanning raster. 
The first delay element 51 receiving the pixel value s(x.sub.0 +3,y.sub.0 
+1) supplies a pixel value s(x.sub.0 +1,y.sub.0 +1) at the output. 
The second delay element, which receives the pixel value s(x.sub.0 
+3,y.sub.0) at the input, supplies the value s(x.sub.0 +1,y.sub.0) at the 
output. 
Moreover, two further delay elements 53 and 54 are provided, each 
performing a delay by a period of one pixel value, thus generating a delay 
by one value in the x-direction of the two-dimensional scanning raster. 
The third delay element 53, which receives the pixel value s(x.sub.0 
+1,y.sub.0 +1), then supplies a pixel value s(x.sub.0,y.sub.0 +1) at the 
output. The fourth delay element 54, which receives the pixel value 
s(x.sub.0 +1,y.sub.0) at the input, supplies the value s(x.sub.0,y.sub.0) 
at the output. 
The value s(x.sub.0,y.sub.0) supplied by the second delay element 52 is 
applied to a second differentiator 42 which performs a first derivation in 
the x-direction at the location of this pixel. 
The arrangement of FIG. 3 further comprises computing means 61 which are 
provided with a superposition stage 62. 
The superposition stage 62 receives the output signals from the 
differentiator 42, the output signals from the delay elements 51 to 54, as 
well as the output signal from the delay element 43. 
The values s(x.sub.0,y.sub.0), s(x.sub.0 +1,y.sub.0), s(x.sub.0,y.sub.0 
+1), s(x.sub.0 +1,y.sub.0 +1) as well as the first derivative of the pixel 
value s(x.sub.0,y.sub.0) in the x and y-directions are thereby available 
at the input of the superposition stage. 
These are the values of the vector in accordance with equation (4) which, 
multiplied by the inverse matrix of this equation, yield the vector with 
the polynomial coefficients c.sub.5 to c.sub.0. 
A further consideration of this equation (5) shows that, for example, the 
polynomial coefficient c.sub.5 results from the sum -s(x.sub.0,y.sub.0), 
s(x.sub.0 +1,y.sub.0) and -.DELTA.s(x,y)/.delta.x at the location 
(x.sub.0,y.sub.0). In a corresponding manner, the coefficients c.sub.4 to 
c.sub.0 are obtained by corresponding inverse matrix superposed values or 
inverted values of the observation vector in equation (5). 
This coherence between the pixel values of the observation vector and the 
polynomial coefficients is shown in the computing means 61 by a 
corresponding connection of adders 63 to 68 within the superposition stage 
62. 
For example, for the polynomial coefficient c.sub.5 taken as an example 
hereinbefore, the adders 63 and 64 are provided which perform a 
superposition of the input values in accordance with the prescription 
indicated by the inverse matrix in equation (5). The adder 63 receives the 
output signal from the second differentiator 42 at an inverting input, 
which differentiator supplies the signal -s(x,y)/.delta.x at the location 
(x.sub.0,y.sub.0). The signal s(x.sub.0 +1,y.sub.0) is applied to a 
non-inverting input of the adder 63. The output signal of the adder 63 is 
applied to a non-inverting input of the adder 64. The signal 
s(x.sub.0,y.sub.0) is applied to an inverting input of the adder 64. 
The adder 64 thereby supplies the coefficient c.sub.5 at the output, in 
accordance with equation (5). 
In a corresponding manner, the adders 65 and 66 are provided for generating 
the polynomial coefficient c.sub.4, the adders 65, 67 and 68 are provided 
for generating the polynomial coefficient c.sub.3. In accordance with the 
inverse matrix in equation (5), the polynomial coefficient c.sub.2 
directly results from the value -s(x,y)/.delta.x at the location 
(x.sub.0,y.sub.0), i.e. from the value of the second differentiator 42. 
The polynomial coefficient c.sub.1 directly results from the value 
-s(x,y)/.delta.y at the location (x.sub.0,y.sub.0), i.e. from the value of 
the delay member 43. The polynomial coefficient c.sub.0 corresponds to the 
pixel value s(x.sub.0,y.sub.0). 
The superposition stage 62 thereby supplies the polynomial coefficients 
c.sub.5 to c.sub.0 at the output. 
Since the pixel value s.sub.i (x.sub.0 +.DELTA.x,y.sub.0 +.DELTA.y) is 
obtained by introducing the values .DELTA.x and .DELTA.y into the 
polynomial, the equation (6) is now valid, which is computed within the 
computing means 61 by multiplying the polynomial coefficients c.sub.5 to 
c.sub.0 by the allocated basic functions at the locations .DELTA.x and 
.DELTA.y. Thus, the multiplier 71 multiplies the polynomial coefficient 
c.sub.5 by the value .DELTA.x.sup.2, the multiplier 72 multiplies the 
polynomial coefficient c.sub.4 by the basic function .DELTA.y.sup.2, the 
multiplier 73 multiplies the polynomial coefficient c.sub.3 by the basic 
function .DELTA.x.multidot..DELTA.y, the multiplier 74 multiplies the 
polynomial coefficient c.sub.2 by the basic function .DELTA.x and the 
multiplier 75 multiplies the polynomial coefficient c.sub.1 by the basic 
function .DELTA.y. Since c.sub.0 should be multiplied only by 1, a 
multiplier is not necessary in this case. 
The polynomial coefficient c.sub.0 as well as the output values of the 
multipliers 71 to 75 are applied to an adder stage 81 which supplies the 
value s.sub.i (x.sub.0 +.DELTA.x,y.sub.0 +.DELTA.y) of the searched pixel 
at the output, in conformity with equation (6). 
In the arrangement of FIG. 3, the computing instruction in conformity with 
the polynomial and the inverse matrix is contained in the implementation 
of the superposition stage. The searched pixel values s.sub.y can be 
directly and continuously computed thereby from the input signal s(x,y) of 
the arrangement of FIG. 3. This may be done very accurately on the basis 
of the method according to the invention, with the number of components in 
the arrangement of FIG. 3 remaining relatively small. 
The differentiators 41 and 42 shown as a block in FIG. 3 preferably have a 
transfer function H=(j.omega.).sup.k, in which .omega. is the local 
frequency in the x or the y-direction, dependent on whether the derivation 
should be effected in the x or the y-direction. 
This transfer function may be realized, for example, by a filter which has 
the filter coefficients 13/128, -40/128, 115/128, 0/128, -115/128, 40/128 
and -13/128. 
This filter is realized by a corresponding circuit in FIG. 4. 
The filter shown in FIG. 4 has six delay elements 11 to 16 which are 
arranged one behind the other, and in which the first delay element 11 
receives the input signal to be differentiated. In conformity with the 
filter coefficients to be selected, the input signal as well as the output 
signal from the last delay element 16, inverted by means of an inverter 
26, is applied to an adder 21 because both are to be multiplied in a 
non-inverted and an inverted form by the factor 13/128. This is realized 
by means of a multiplier 24 arranged subsequent to the adder 21. 
In conformity with the filter coefficients mentioned above, the output 
signals from the first delay element 11 and the fifth delay element 15 are 
superposed in a non-inverted and an inverted form by means of a summing 
device 19 and multiplied by the filter coefficient -40/128. Similarly, the 
output signals from the second delay element 12 and the fourth delay 
element 14 are applied in a non-inverted form and in a form inverted by 
means of an inverter 18 to an adder 17 whose output signal is multiplied 
by the allocated filter coefficient 115/128 by means of a multiplier. 
The output values of the multipliers 22 to 24 are superposed by means of an 
adder 25 so that the overall arrangement of FIG. 4 has a filter behavior 
which is a good approximation of the above-mentioned transfer function. 
Such differentiators can be used in the circuit of FIG. 3. The delay 
elements within the differentiator are provided for the differentiation in 
the x-direction. The delay elements in the arrangement of FIG. 3 are 
externally built up as line memories 31 to 36 for the delay in the 
y-direction. In this case, the input signal and the output signals of the 
line memories 31 to 36 should be applied in a corresponding manner to the 
adders 17, 19 and 21 in a non-inverted and an inverted form, without 
passing through the delay elements 11 to 16. 
Moreover, the delay time of the vertical differentiator 41 in FIG. 3 is to 
be matched with the delay time of the horizontal differentiator 42 by 
means of a delay member 43 which realizes a horizontal delay of three 
pixels, so that the horizontal derivative of the input signal s(x,y) at 
the point (x.sub.0,y.sub.0) is present at the output of the differentiator 
42, and the vertical derivative of the input signal s(x,y) at the same 
point (x.sub.0,y.sub.0) is present at the output of the delay member 43.