APPARATUS AND METHOD FOR IMAGE PROCESSION

Abstract: A system for and methods of image processing are disclosed. The system comprises image acquisition apparatus for acquiring pixels of a digital image, permutation apparatus for receiving the pixels, for reordering pixels of the digital image into a plurality of abstracted images at a large reduction factor and for combining the abstracted images into many sets of abstracted images combinable into abstracted images at different small reduction factors and storage apparatus for receiving and for storing the abstracted sets of digital images. One method comprises an efficient way of rotating images stored as abstracted images. Other methods enable transformation between different storage formats.

Description: APPARATUS AND METHOD FOR IMAGE PROCESSION FIELD OF THE INVENTION The present invention relates to image storage methods generally and to permutations of pixel storage order in particular. BACKGROUND OF THE INVENTION Digitized images are rectangular arrays of pixel data which are usually acquired serially from a scanning device. Some color scanning devices present a complete pixel at a time, comprising three colors, typically cyan, yellow and magenta; other scanning devices present, at one time, a single row of partial pixels of one of the three colors, and alternate rows among the three colors. It is known in the prior art to store digitized images in sequential format. The images are typi¬ cally stored on magnetic disks, in sequentially written blocks of typically between 512 and 8192 bytes, where one block is the minimum number of bytes which can be read during one disk access. This block storage structure is only optimal when all the data accessed on each block is utilized. For data scattered in parts of many blocks, access becomes expensive in thai many more disk reads are necessary to retrieve the desired information than would be needed if the desired data were clustered in complete blocks. Computer memory is random access; sequential access and random access to computer memory each takes generally the same length of time which is typically much less than disk access time. Unfortunately, computer memory storage is much smaller than disk storage and thus, cannot be used to store and to access digitized images of many Mbytes. Digitized images are usually displayed on Cathode Ray Tube (CRT) devices having a reso¬ lution of 75 - 100 pixels per inch. Since most scanners scan at a resolution of about 300 pixels per inch, displaying a digitized image requires a reduction of the volume of information by a factor typically varying between 9 and 16. SUMMARY OF THE INVENTION It is an object of the present invention to utilize the properties of both disk storage and com¬ puter memoiy to store digitized images. Therefore, there is provided in accordance with the present invention, an image processing system comprising image acquisition apparatus for acquiring pixels of a digital image, permutation apparatus for receiving the pixels subsequent to their acquisition by the image acquisition apparatus and for ordering the pixels, thereby to reorder the digital image and storage apparatus for receiving and for storing the multiplicity of abstracted images. Further, there is provided in accordance with an embodiment of the present invention an image processing system comprising an image acquisition apparatus for acquiring pixels of a digital image, permutation apparatus for receiving the pixels subsequent to their acquisition by the image acquisi¬ tion apparatus and for ordering the pixels thereby to reorder the digital image into a multiplicity of abstracted images at a reduction factor and storage apparatus for receiving and for storing the multiplicity of abstracted- images. Additionally, there is provided in accordance with an embodiment of the present invention an image processing system comprising image acquisition apparatus for acquiring pixels of a digital image, permutation apparatus for receiving the pixels subsequent to their acquisition by the image acquisition apparatus, for ordering the pixels thereby to reorder the digital image into a plurality of abstracted images at a" relatively large reduction factor and for combining the abstracted images into a multiplicity of sets of the abstracted images combinable into a plurality of abstracted images at a plurality of different relatively small reduction factors and storage apparatus for receiv¬ ing and for storing the digital image in the multiplicity of sets. Additionally, in accordance with any of the embodiments of the present invention, the permuta¬ tion apparatus may comprise an apparatus for dividing the digital image into a plurality of bands comprising a predetermined number of rows of the digital image and apparatus for operating on the plurality of bands in a band by band manner. The apparatus for operating operates separately on each band. Further, in accordance with any of the embodiments of the present invention, portions of the bands are written by the apparatus for operating to the storage apparatus when the portions of the band fill at least one file system block. There is provided in accordance with the present invention, an image processing method including the steps of acquiring pixels of a digital image, receiving the pixels subsequent to their acquisition in the step of acquiring and ordering the pixels thereby to reorder the digital image into a multiplicity of abstracted images and storing the multiplicity of abstracted images. There is provided in accordance with an embodiment of the present invention an image pro¬ cessing method including the steps of acquiring pixels of a digital image, receiving the pixels subse¬ quent to their acquisition in the step of acquiring and ordering the pixels thereby to reorder the digital image into a plurality of abstracted images at a relatively large reduction factor and combining said abstracted image into a multiplicity of sets of the abstracted images combinable into a plurality of abstracted images at a plurality of different relatively small reduction factors and storing the digital image in the multiplicity of sets. There is provided in accordance with an embodiment of the present invention, an image pro¬ cessing method including the steps of of sequentially acquiring pixels of a digital image, sequentially receiving the pixels subsequent to their acquisition in the step of sequentially acquiring and ord¬ ering the pixels thereby to reorder the digital image into a multiplicity of abstracted images at a reduction factor and of storing the multiplicity of abstracted images. Moreover, there is provided in accordance with the present invention, an image transformation and rotation method including the steps of receiving an image stored in a permuted format, transforming the image in a permuted format to a tile format via an inverse transformation of the permuted format to a sequential format and via a forward transformation from the sequential for¬ mat to the tile format, of rotating the tile format image by rotating the location of each tile of the tile format irr e, utilizing an amount of computer memory less than the image size and retransform- ing the rotated tile format image to the permuted format via an inverse transformation from the tile format to the sequential format and via a forward transformation from the sequential format to the permuted format. Further, in accordance with the present invention, the step of rotation includes the steps of receiving and buffering a number of tiles of a source image to be rotated, of writing to a storage medium a different number of tiles of a destination image that are fully defined by rotations of the buffered tiles of a source image, and repeating the steps of receiving and writing until the source image no longer contains tiles. There is provided in accordance with an embodiment of the present invention an image rota¬ tion method including the steps of receiving and buffering a number of tiles of a source image to be rotatec and of writing to a storage medium a different number of tiles of a destination image that are fully defined by rotations of the buffered tiles of a source image, and repeating the steps of receiving and writing until the source image no longer contains tiles. There is provided in accordance with the present invention, an image processing system comprising an image acquisition apparatus for acquiring pixels of a digital image, apparatus for dividing the digital image into a plurality of bands each comprising k predetermined number of rows, and permutation apparatus for operating on the plurality of bands in a band by band manner and for reordering the location of the pixels. Further, there is provided in accordance with present invention, an image processing system comprising a storage apparatus for receiving and for storing the bands in a band by band manner. BRIEF DESCRIPTION OF THE DRAWINGS The present invention will be understood and appreciated more fully from the following detailed description taken in conjunction with the drawings in which: Fig. 1 is a block diagram illustration of an image storage system using the image storage method of Fig. 2B; Fig. 2A is a pictorial illustration of a sequentially stored image; Fig. 2B is a pictorial illustration of a permutation based image storage method con¬ structed and operative in accordance with the present invention; Fig. 2C is a pictorial illustration of an alternative view of the permutation based image storage method of Fig.2B. Fig. 3 is. a Venn diagram illustration of the regions of overlap between multiple abstraction levels useful in an alternate embodiment of Fig. 2; Fig. 4 is a pictorial illustration of an order for storing the regions of overlap of Fig. 3; Fig. 5 is a pictorial illustration of an alternative order for storing the regions of overlap of Fig. 3; Fig. 6 is a pictorial illustration of a method for transforming an image stored in one format, or permutation, to an image stored in another format; Fig. 7 is a pictorial illustration of a tile format permutation useful for rotating stored images; Fig. 8 is a pictorial illustration of a method for rotating images 90° using the tile format of Fig. 7; Fig. 9 is a pictorial illustration of a method for rotating images 45° using the tile format of Fig. 7; and Fig. 10 is a pictorial illustration of a band permutation image storage method useful in the image storage method of Figs. 2. DETAILED DESCRIPTION OF A PREFERRED EMBODIMENT , Reference is now made to Fig. 1 which illustrates an image acquisition and storage system using the image storage method of the present invention. The system is based on a variation of standard direct memory address (DMA) techniques and comprises an image acquisition unit 110, typically a scanner or a CCD camera, for acquiring a digitized image and a scanner acquisition unit 112, similar in operation to a DMA controller, for receiving pixels of the digitized image, for per¬ muting the addresses of the pixels within the digitized image according to the method of the present invention, and for transmitting the permuted pixels to temporary storage in a computer memory 114. The system additionally comprises a Central Processing Unit (CPU) 116, such as the 80386 microprocessor from Intel Corp., for directing the scanner acquisition unit 112. Since the computer memory 114 has a limited size, typically less than 10 Megabytes, which is generally significantly smaller than the size of the digitized image, typically in the tens of Megabytes, the memory 114 cannot hold the entirety of the permuted image. Thus, the permuted pixels tem¬ porarily stored in memory 114 are typically periodically transferred to a storage medium 118, via a storage medium DMA controller 119, in parallel with the entering of other permuted pixels into memory 114. The storage medium 118 is operative for generally longer term storage. ' Scanner acquisition unit 112 permutes the addresses of the pixels according to a mapping transformation which typically is chosen to optimize frequent operations on the digitized image, such as reducing the size of the image in order to display it on a display device. The mapping transformation is typically a permutation of the set_ ={0,l,2,. V-l} onto itself. A permutation P is denoted i→p t where i is an element of the set of initial pixel addresses, /»,- is a pixel address to which the z ' th initial pixel is permuted, known as a permuted pixel address, and N is the number of pixels in the image. Equation 1 presents a representation for the permuta¬ tion P. The upper row of the notation typically indicates the initial pixel addresses i and is denoted T(P); the lower row of the notation typically indicates the permuted pixel addresses . The nota¬ tion specifies that any untransformed pixels i (i.e. any pixels for which />.=/) in the permutation P are not included in the representation, but they are present in the mapping. Moreover, a per¬ mutation P operates on the entire digitized image. It will be understood that the permutational transformations are invertible and that they do not perform any compression or expansion of the image. The smallest set of sequential pixel addresses, known as an interval, that contains T(P) is typically called P's band, b(P). For example, let C be the transformation given in equation 2. f20 351 C = 35 20 (2) T(C) accordingly contains the addresses {20,35} and the band b(C) is the interval [20.....35]. A further example of a band b(P) is shown in Fig. 10 which illustrates a digitized image, denoted 120, organized into bands, denoted 122a-e. Each band 122, as mentioned hereinabove, is typically comprised of a predetermined interval of pixel addresses of the image 120. Moreover, the bands 122 do not overlap with each other. Thus, it is possible to define non-overlapping (i.e. having disjoint sets biff) ) transformations P j whose bands, for the example shown in Fig. 10, are 122a-e. It will be appreciated that the number of non-overlapping transformations depends on the number of bands 122 chosen and is typically denoted by n b . A transformation P for the entire image 120 is the product of the non- overlapping transformations P j as defined in equation 3. P = P l P 2 ■ ■ ■ P Λb . (3) The transformations P j are typically related to the transformation P x by the addition of a base value c r - If P is defined by i→ρ u , then P j is defined by In other words, each transformation P is the transformation P x shifted by the base value c } . For example, the permutation 0 1 2 3 4 5 6 7 8 9 10 11 12 13 141 E = 4 2 3 1 0 9 7 8 6 5 14 12 13 11 10 (5) can be written as the product of three non-overlapping transformations, [0 1 2 3 4] 5 6 7 8 9 10 11 12 13 14 E = E-ι 'Eo-E * = 4 2 3 1 0 9 7 8 6 5 14 12 13 11 10 (6) The bands associated with each permutation E j are the intervals b(E x ) = [0..4] b(E 2 ) = [5..9] (7) 6(£ 3 ) = [10..14] The base values c j of each band is just The transformation E can then be seen to obey equation (4) which in this case is written: i + C j → e i + c jt (9) where In accordance with a preferred embodiment of the present invention and as mentioned hereinabove, the scanner acquisition unit 112 receives a pixel value /_ from location i of the digi¬ tized image 120 and stores the pixel value in permuted location Zi=Pi+q b of memory 114, where q b is the starting memory location for band b. The permutation is typically performed according to a transformation P by an address generating routine G{fj), described in more detail hereinbelow. Upon termination of the permutation of the entirety of one band 122, the portion of the permuted band and any previously unwritten portions of previous bands still stored in memory 114 which comprise an integral multiple of blocks of data is written to storage medium 118. The remaining portion of the permuted band is typically written to storage medium 118 only upon termination of the permutation of the next band 122. It is a feature of the present invention that the system of Fig. 1 writes a block of data to the storage medium 118 typically only when the block of data is full. It is an additional feature of the present invention that the permutation operation does not significantly slow down the writing to the storage medium versus writing in a unpermuted manner. The acquisition and permutation of a second band 122 of pixel values can be performed in parallel with the write operation for a first already permuted band. This parallel reading and writing is known as double buffering. For a double buffering scheme which neglects block size, of storage medium 118, memory 114 is typically divided into two buffers. Each buffer is minimally the size of one permutation P j . The size of a permutation is defined as the number of elements in a band. For jstorage medium 118 organized in blocks, typically of a predetermined size B, the minimum size of memory 114 for a double buffering scheme is typically twice the size of the permutations P j plus twice the size B of one block. The addition of 2B is necessary because the bands may not be block aligned. It will be appreciated that the maximum data buffer size is principally dependent on the size of the band 122 and secondarily dependent on both the block size, B, and the anticipated location of the band on storage medium 118. It is not dependent on the complexity of the transformation. As mentioned hereinabove, an address generating routine G(P) is defined which generates the permuted location p, in memory 114 for each pixel as ft arrives sequentially. The address generating routine G(P) is less general than the permutation in that the address generating rou¬ tine G(P) is a routine which produces the permuted addresses sequentially, on successive routine executions, whereas the permutation P is a transformation which will produce any permuted address given its address as an argument. According to a preferred embodiment of the present invention, the definition of the permu¬ tation P, and accordingly of permutation P t is typically chosen to reduce the number of i/o opera¬ tions necessary to perform common processing operations on the digitized image 120. For exam¬ ple, the resolution of a typical CRT monitor is 75 pixels per inch whereas the typical resolution of a digitized 8 1/2 in. by 11 in. image is 300 pixels per inch. Thus, typically only 1/16 of a stored image 120 can be displayed at one time on an 8 1/2 in. by 11 in. CRT monitor. In order to display the image 120, it is typically abstracted by a reduction factor r, that is, the displayed image includes every r th pixel of every r th row. For the abovementioned example, the reduction factor r is set to 4. According to a preferred embodiment of the invention, the permutation P is typically defined such that the full-size image is stored as r 2 smaller images, known as abstracted images, that each have Mr the number of both rows and columns. The full-size image is stored in the band structure described hereinabove where every band is organized into r 2 abstracted bands and where each abstracted band is written to the storage medium 118 as a continuous whole. An abstracted image can thus be read by reading only those blocks belonging to the abstracted bands comprising the abstracted image. It will be appreciated that the abovementioned embodiment presents a savings of generally r 2 , neglecting disk granularity, over reading the entirety of the image and subsequently abstracting it. Each abstracted image comprises every r th pixel of every r th row of the full-size image, but the initial pixel of each abstracted image differs among abstracted images. If the leftmost and topmost pixel is defined two-dimensionally as pixel (0,0) where the location numbers increase to the right and down, the initial pixels are chosen from the pixels belonging to the square whose comers are the pixels (0,0), (0,r-l), (r-1,0) and (r-l,r-l)- In other words, the abstractions differ by a two- dimensional phase. As mentioned hereinabove, each band 122 of the digitized image 120 is typically divided into r 2 abstracted bands where an abstracted band comprises only those elements of the band 122 belo n ging to one of the r 2 abstractions. Moreover, the number of rows and columns of the band 122 are typically multiples nr and mr, respectively, of r. The size of the abstracted band is mn and the size of band 122 is r 2 times the size of the abstracted band, or mnr 2 . If enough memory 114 exists to store the entire image, n can be chosen so the entire image will occupy one band. For each band, the incoming pixels belonging to a single abstracted band are located, in order, in a single continuous area of memory 114. Equations (10a-e) present the permutation equation P x of the first band 122 of image 120. θ c = ι' mod r (10a) . i θ r = mod r (10b) rm i mod rm λ c = r (10c) • V- -^- (10d) mr* Pi = mn c +rQ r ) + λ c + λ-, (10e) Where L ' " J is the floor function. In -equations (11a-e) θ e is the column geometric phase of pixel ., θ r is the row geometric phase of pixel ., λ £ is the column within the abstracted band of pixel i, . is the row within the abstracted band of pixel i. Equations (11a-e) comprise the inversion formula for converting permuted indices into sequential indices. Pi θ r = (11b) rmn λ c =pι mod m (11c) Pi mod mn = m did) i = θ c + mrθ r + rλ e + r 2 mλ r (He) Software for the address generating routine, the permutation formula, and the inversion for¬ mula, written in the C++ language, are found in Annex A. The address generating routine is called next and the routine that produces the permutation function is called q. The address inversion generating routine is called inext and the inversion permutation function is presented in qinv. A test program, called gatest, to check the the equality of the address generating routines and the address permutation functions, and code to check that the address inversion permutation function correctly invert the address permutation function, are also included in Annex A. Note that, although the address generating routine is complicated, it requires no multiplica¬ tion operations, other than of constants which can be precomputed Thus, the address generating routine is typically fast enough to permute the address for a first pixel before scanner 110 acquires a second pixel. Fig. 2A illustrates a sequentially received image and Fig. 2B illustrates the abovedescribed permutation definition for some abstractions of one 12x16 band of an image 130 abstracted by 4. The number of rows nr in a single band is 12, where n is 3, and the number of columns mr is 16, where m is 4. Thus, the size mn of an abstracted band is 12 pixeis. The band 131 will be abstracted into 16 abstracted bands where each of the 16 abstracted bands is referred to by two indices x,y representing its initial pixel, or geometric phase. If each abstracted image is denoted by A, then A(0,0) is the abstracted image which includes the pixels (0,0), (0,4), (4,0), etc.. Fig. 2B shows some of the 16 abstracted bands. Five of the abstracted bands are labelled according to the abstracted images A(x,y) to which they belong and the pixels stored in them are indicated. In the interests of darity, the remaining abstracted bands are not filled in nor are they labelled. Fig. 2B illustrates the storage order of the abstracted bands on the storage medium 118. Fig 2C illustrates an alternative view of the abstracted bands shown in Fig. 2B wherein each abstracted band is shown as a rectangle corresponding to the original full scale image shown in Fig. 2A. As mentioned hereinabove, the pixels belonging to an abstracted band are stored in a con¬ tinuous area of memory 114. If the pixel 0, the first pixel of the abstracted band belonging to A(0,0), is stored in memory location 0 labelled 0, then the second pixel of the abstracted band of A(0,0), pixel 4, is stored in memory location 1 labelled 1. The memory locations for some indicative pixels are marked on Fig. 2B. if pixels are stored in more than 1 memory location then the memory loca¬ tions indicated is understood to be multiplied by the number of memory locations needed to store a pixel in actual memory address calculations. The first pixel of the abstracted band of A(1,0), pixel 1, is typically stored in the first location after the end of the abstracted band belonging to A(0,0). Since the size of each abstracted band is 12, the first location of the abstracted band belonging to A(1,0) is location 12. Accordingly, the abstracted band of A(2,0) begins in location 24, and the abstracted band belonging to A(0,1) begins in location 48. As mentioned hereinabove, abstracted images A(x,y) each comprise 1/r 2 of the pixels in image 130 but differ by a two-dimensional phase. It can be seen from Fig. 2B that the pixels of abstracted band A(0,0) come from the first, fifth, ninth and thirteenth columns of the first, fifth and ninth rows of band 130 whereas the pixels of abstracted band A(3,2) come from the fourth, eighth, twelfth and sixteenth columns of the third, seventh and twelfth rows of band 130. For the example in Fig. 2B, for the pixel 73, ι=73. Equations (10a-e) will give the results θ e =ι, θ r =0, λ*=2, =l and pτ 3 =18. Pixel 73 is in fact in the abstraction having the phase (1,0) and within its abstracted it is in column 2, row 1 (see Fig.3C). Pixel 73 is in the 18th location sequentially from the first stored pixel stored at location 0. Using the inverse equations (11a-e) with />,=18 we obtain θ c =l, Qr=0, λ e =2, λ^l and ι=73. It will be appreciated that if image 130 comprised more than one band, then the pixels belong¬ ing to the other bands would be stored in memory 114 in a manner similar to that described hereinabove. If double buffering is used, the pixels belonging to a second band are stored in the second buffer of memory 114 while the pixels already stored in the first buffer are being written to storage medium 118. For image 130 of more than one band, A(x,y) comprises more than the one abstracted band shown with respect to Fig.2B. An alternative embodiment of the invention defines the permutation P such that the digi¬ tized image is organized according to a multiplicity of reduction factors. For example, it may be desired to be able to abstract an image at a range of small reduction factors 2, 3, 4, 5, and 6, rather than the fixed factor 4 in the previous embodiment. To do so, a large reduction factor is found which is the least common multiple of the entirety of small reduction factors. For the example given, the large reduction factor might be 60 since each of 2, 3, 4, 5, and 6 are factors of 60. If it is desired to store the image according to the previous embodiment, the image would have to be stored as 3600 abstracted images. However the abstracted images of the different small reduction factors are not disjoint with respect to each other. Thus, an alternative embodiment of the present invention discloses a permutation P which facilitates access to the multiplicity of abstracted images at the small reduction factors by utilizing regions of overlap between them. The following discussion uses an example of a large reduction factor of 60 but should not be construed to limit the invention to that example reduction factor. To design the permutation P, the abstracted images at the smaller reduction factors are calculated from the 3600 possible abstracted images. An abstraction set S r is defined such that its elements are pδcels belonging to an abstracted image at reduction factor r which begins at the (0,0) pixel. The intersections of the five abstraction sets S 2 , S 3 , S 4 , S s and S 6 are shown in a Venn diagram in Fig. 3. A multiplicity of regions or subsets, denoted S β , ■ • ■ S k on Fig. 3, are shown. Subsets S a , S b , S e , S d , S t , S f , S t and S h are regions of overlap and belong to a multiplicity of sets; the remaining subsets, _?,-, S j and S k , are independent regions which belong to the sets S 3 , S 5 and _> 2 , respectively. Subsets S a , • • • ,S k are defined as follows: S b = (S 3 n S 4 )- S s S c = 'S s nS 4 )- S 3 S<_ = (£ 4 ~ Ss) - S3 S e = ((S 2 - S 4 ) n S 5 ) -S z S f = <(S 2 -S 4 ) n S 3 ) -S s (12) S g = ((S 3 nS 5 ) nS 2 )-S 4 5 A = ( l S 3 n 5 s )- 5 2 Si = ($3 — S 2 ) — Ss Sj = (S5 - S 2 ) - 5 3 S t = ((S 2 -S 4 )-S 3 )-S 5 Si = Si — S 2 — S 3 — Ss For example: the subset that is common to all sets is S a which is defined by equation 13, S β ={A(0,0)} (13) The subset S c is defined in equation 14, S c = {A (0,20), Λ(0,40), Λ(20,0), Λ (20,20), Λ(20,40), Λ(40,0), Λ(40,20), .4(40,40)} (14) As can be seen from equations 13 and 14, the size of a subset is typically the number of abstracted images A(x,y) at the large reduction factor which belong to it. It will be appreciated that the f sets S a , • • • ,S k are not of equal size, nor do they as a whole include the entirety of pixels of the in 3120. Table 1 presents the sizes of the subsets. The abstraction sets S 2 , S 3) S 4 , S 5 and S 6 can be written in terms of the subsets of Fig.3 as follows: 5 2 = S a u S b u S c u S d u S e u S f S g S k S 4 = S a vS b vS c υS d ιo = S 0 u5 c u5 r uS g 12 The subsets S β , • • • ,S k enable the Venn diagram to be be ordered in a linear fashion to allow fast access to specific abstraction sets. A variety of permutations utilizing the subsets S a , - - - ,S k can be performed; the optimal one will depend on the expected frequency of access to the various abstraction sets. One such ordering ensures that abstraction sets at smaller reduction factors which have more elements to them be less fragmented than the ones at larger reduction factors. According to a preferred embodiment of the present invention, the multiplicity of abstraction sets, _ 2 . * ' " .S 6 in the example, are stored in the band structure described hereinabove. The band size is typically at least mnr 2 where r is the least common multiple used to create the abstracted images A(x,y). Fig. 4A illustrates an example permutation of a band 122 which typically enables abstracted bands of the example abstraction sets S 2 , S 3 , S 4 and S 6 to be read sequentially. In Figs 4B- 4F the bold lines specify the subsets to be read to create abstracted sets S , ■ ■ • ,S 6 respectively. Abstraction set S s typicaly cannot be read sequentially. An alternative embodiment of the inven¬ tion, shown in Fig. 5, enaβles abstracted bands of abstraction set Ss to be read in one sequen¬ tial read. A subset S q is created by replacing the pixels in the set S q = S c uS α uS/uS, with pixels from the set 5 which are located close to the pixels of set S q . For example, the approximation to the abstracted image A(0,0) is the abstracted image A(1,0) which is in the subset Sj. It will be appreciated that Figs. 4 and 5 are not to scale. The entirety of the image 120 is stored in bands 122. Thus, the abstracted band belonging to subset Si for each band 122 is typically stored in an area of both a buffer of memory 114 and of the storage medium 118 after the abstracted bands of subsets S a , • • ■ ,S k . Alternatively, the abstracted band of subset S, can be used to block align the abstracted bands of subsets S a , • ■ ■ ,S k . In the alternative embodiment, elements in one abstracted band of subset S t are located before and after the elements of the abstracted band subsets S a , • • • ,S k , located according to either of Figs. 4 or 5, such that at least one frequently accessed abstracted band begins at the beginning of a storage medium block. The image typically contains a multiple of r rows and r columns. An image that does not con¬ form to these constraints typically is expanded to conform. For an image which is 8.5 inches wide, scanned at 300 pixels per inch where each pixel is 32 bits, the minimal number of rows for the example given herein needed to make a band is 60 rows. The resultant band typically occupies 597K bytes. The maximal expansion of the image due to the abovementioned constraint is 1/5 inch (60 pixels) in height and width and the average expansion is half this amount. In accordance with the alternative embodiment of the invention, an abstracted image with reduction factor n, where n is a product of at least two of the factors r, can be read by reading the subsets which are common to the factors which make up n. Thus, the abstracted image at reduction factor 10 comprises the subsets S a , S c , S t , and S g . If n is a multiple of one of the factors r, its abstracted image can be read by reading the abstraction set S r of which n is an integral multiple and further abstracting S r . For example, the abstracted image with reduction factor 9 is read by reading the abstracted image S 3 and further abstracting it by a factor of 3. An abstracted image with a reduction factor of n, where n is relatively prime with reference to the factors r or is not an integral scale factor, can be obtained by reading an abstracted image with a smaller reduction factor and the resampling the lower order abstracted image to obtain the desired abstracted image. For example, the abstracted image with a reduction factor of 7 can be obtained by reading the abstracted image S 4 interpolating by 7/4. It will be appreciated that the abovedescribed subset ordering and its inverse can be precomputed and stored in the scanner acquisition unit 112 in a lookup table. The lookup table will enable a fast calculation of the locations of the permuted pixels. Annex B contains a program to generate the permutation and its inverse for the multiple abstracted image permutation described hereinabove. This program makes use of the base class GABS defined in Annex A. Another example of a multiplicity of reduction sizes that is possible with the abovedescribed embodiment is fixed power reductions. Typically a set of reductions that are successive powers of 2 are needed, that is multiple reductions of order 2,4,8,16,32,... For example, if r is 64 the 2,4,8,16,32,64 order reductions can be read from the storage medium in one access. Reduction factors of powers of 2 are useful in the pyramid transformation. The pyramid transformation and its use is described in detail in the book, A. Rosenfeld, ed. Multiresolution Image Processing and Analysis. Springer-Verlag, Berlin, 1984. It should be noted that, in general, each pixel in the reduced order image of the pyramid transformation may be a complex mathematical transformation of the full reso¬ lution image data. The abovedescribed embodiment derives the reduced image pixels from a simple subsampling of the original image. The permutation equations for the abovedescribed transformation are slightly different than the equations (lOe) and (11e). Since the r 2 abstracted bands are not stored in a simple order dictated by the geometric phase, a mapping M(θ r ,θ e ) of each abstracted band is needed in order to compute the transformation. This mapping returns the sequential index of the location of the abstracted band _4(θ r ,θ c ) within the subset ordering as illustrated in Fig. 4A. Equation (10e) is then replaced by equation (16) for . .is transformation. Pi = mnM- i (θ c ,Q r ) + λ c + mλ r (16) Equation (11e) for the inverse is replaced by three new equations (17a-c). c = M c ,Q r ) mod r (17a) M(Θ C ,Θ Γ ) Yr = (17b) i = γ c + mry r + rλ c + r^mX, (17c) If the digitized image 120 is stored in a format defined by either of the permutations dis¬ closed hereinabove, or in any other non-sequential format, it typically can be transformed to another, more common, format for additional processing. A typical processing format is the sequential format. For instance, the image 120 may be stored as abstracted images of reduction factor 4 and it may be desired to perform a convolution on the entirety of the image 120. The sequential format is the natural format for this calculation. Thus, the stored abstracted image is typically transformed into the sequential format. This can be accomplished by simply transforming the data read from the storage medium 118 via the inverse of the transformation by which it was stored, as shown in equation 18. P ~l =P x l - P5 1 • ■ • PJ 1 ' (18) P" 1 is typically performed on a first band of data in parallel with the reading of a subsequent band of stored data from the storage medium 118. Since the time to transform the data is typically less than the time to read the stored data from the storage medium 118, the conversion from one format to the other generally does not take longer than reading the stored data without any format conversion. The size of memory 114 needed to transform the data for a double buffering scheme is bounded by twice the size of the largest band of the transformed image. According to another embodiment of the invention, an image can be transformed from one band structure format F x to another band structure format F 2 . The method comprises the fol¬ lowing steps: a. Read as many bands of stored image data stored in the Pi format as necessary to produce enough data to fHf a band in format F 2 . b. Convert the data in format F i to sequential format. c. Convert the data in sequential format to format F 2 . Convert only as much data as necessary to produce an integral number of bands of format F 2 . d. Store the integral number of bands of F 2 on storage medium 118. e. Free the memory 114 of the integral number of bands in format F 2 . f . Return to step a. and continue until no more data is available. Fig. 6 illustrates an example of the method of transforming between two formats as described hereinabove. An image 150 is stored in format F x whose band structure comprises equal bands of 100K pixels each. An image 152 is stored in format F 2 whose band structure comprises equal bands of 150K pixels each. To transform image 150 to image 152, the following steps are typically followed. 1. Read the first two bands B a and B b of image 150 and transform them into sequential format. 2. Transform the entirety of band B a and hatf of band B b into band B x from sequential format to format F 2 . 3. Write band B x to image 152. 4. Free the space in memory 114 currently occupied by band B a and by the first half of band B b . 5. Read in band 5,. of image 150 and convert to sequential format. 6. Transform the second half of band B b and the entirety of band B e into band B 2 from sequential format to format F 2 . 7. Write band B 2 to image 152. 8. Free up the space in memory 114 currently occupied by the second half of band B b and by the entirety of band B e . 9. Repeat until no more data remains in image 150. The maximum space in memory 114 required for the method of converting between two for¬ mats described hereinabove, when using a double buffering scheme, is typically twice the sum of the band size required for each of the formats. It is a feature of the invention that the pixels of image 150 are read only once. If the band structure of the two formats Pi and F 2 are equivalent, the data can be easily transformed without first transforming the data into the sequential format. The permutation is -PF 1 where is the transformation from format P 2 to sequential format and P Fχ is the transfor¬ mation from sequential format to format i. Alternatively, if the transformation between the two for¬ mats Pi and F 2 is frequent, the permutation can be precomputed, thus reducing the complexity of the transformation and the time needed to execute the transformation. The maximum space in memory 114 required for this alternative method when using a double buffering scheme is typ¬ ically twice the size of the band. According to an embodiment of the present invention, the image data can be rotated at an arbitrary angle using a tile transformation which is well known in the art. The tile transformation is described in the Pixar™ manual, Scope, Release 1.1, Genera 7.1 by The Graphics Division of Symbol¬ ics Inc., p. TU 2, which is incorporated herein by reference. The tile transformation consecu¬ tively stores pixels which are elements of a square section of the image in a single block on the storage medium 118. A tile of size « 2 comprises u rows of pixels each having « columns. Typically, a tile will occupy a block. For a block size of 4096 bytes and pixels of 4 bytes each, a 32x32 tile is typically chosen. Fig. 7 illustrates the transformation for sequential format image 130 of Fig. 2A. Shown are 5 of the 124x4 tiles in which the first 192 pixels are stored. The first tile 160, comprising the first four pixels of the first four rows, is stored consecutively in an area of memory 114. The second tile 162, comprising the second four pixels of the first four rows, is stored after the end of the first tile 160. Thus, if first tile 160 begins at memory location 0, then second tile 162 begins at memory loca¬ tion 16. If pixels are stored in more than 1 memory location then the memory locations indicated is understood to be multiplied by the number of memory locations needed to store a pixel in actual memory address calculations. The permutation P for the abovementioned example is given in equation 19. Pi = i mod u + u 2 mod m + « urn (19) Where m is the number of tiles in a row of the image. To rotate, by 90°, the image which is stored in tile format a block size tile is read into a buffer of memory 114, moved to the block it will occupy when it is rotated, and then ' rotated. The operation is shown in Fig. 8. The operation can alternatively be performed in one read/write pass of the image by reading a tile is read into a buffer of memory 114 and directly writing it to the corresponding destination rotated by 90°. This alternative method typically requires enough space in memory 114 to hold two tiles, the tile read in and the rotated tile. If double buffering is desired, to increase the speed of the computation, four buffers are typically allocated. According to a preferred embodiment of the invention, rotation at an arbitrary angle using a tile transformation can be performed. Rotation by 45°, being the rotation which typically uses the most buffer space in memory 114, is illustrated in Fig. 9 for an image organized into a 5x5 tile for¬ mat. The fact that the resultant rotated image is non-rectangular is irrelevant to the current discus¬ sion. Well know techniques can be used to convert the rotated image to a rectangular form. For example, the rotated image can either be clipped to rectangular form, or expanded to contain the rotated image. For the example in Fig 9, the rotation operation takes a 5x5 image 172, shown rotated in Fig. 9, rotates it and writes it to storage 118 as a 5x5 image 174. The tiles of .source image 172 are denoted with the double capital letters AA-YY, those of destination image 174 are denoted with the double small letters aa-yy. Image 172 is organized into bands 176, 178, 180, 182 and 184 where each band comprises five tiles. Typically a source image 172 tile affects number of destination image 174 tiles. This depen¬ dency is denoted by δ=D (σ), where δ is a set of destination image 174 tiles and σ is a set of source image 172 tiles. For example, D({AA})={cc) ami D({BB} ={cc,dd,ϋ). The function D is dependent on the details of the calculation of each destination pixel since each destination pixel typically is dependent on a multiplicity of source image pixels. According to a preferred embodiment of the rotation operation tiles of source image 172 are buffered as follows. 1. Read into memory 114 the first two rows of tiles from the source image 172. 2. Read into memory 114 the next row of tiles from source image 172. 3. Calculate and write to destination image 174 all destination tiles that only dependent on the current three rows of source image 172 tile. 4. Free the memory occupied by the last recently read row of source image 172 tiles. 5. Repeat steps 2-4 until the entirety of the image has been written. An example of the abovementioned embodiment reads from the storage medium 118 bands of tiles labelled 176, 178, 180 from image 172 and writes out all rotated tiles that are fully defined by the tiles of image 172 currently in memory 114, being tiles ee, dd, jj, cc, ii, oo, bb, hh, nn, and tt. The buffers for the first band of tiles 176 can then be freed and the band 182 read from the storage medium 118 into the location previously occupied by band 176. All previously unwritten rotated tiles that are fully defined by the unrotated tiles currently in memory 114, being tiles αα, gg, mm, ss and yy, are written to storage medium 118. The last recently unrotated band 178 is freed and the next unro¬ tated band 184 is read in from storage medium 118. All previously unwritten rotated tiles that are fully defined by the unrotated tiles currently in memory 114, being tiles ff, 11, rr, xx, kk, qq, ww, pp, w, and mi, are written to storage medium 118. This procedure is summarized in Table 2. It should be appreciated that the only 3 bands of unrotated tiles need be stored in memory at any one time for any size image that is to be rotated and for any angle of rotation. This is because any destination trie can only be dependent on source tiles which span three sequential rows of tiles. The total memory 114 requirements for this method is 3 bands of tiles, to buffer the unrotated tiles, plus one tile to hold the rotated tile. If double buffering is to be used the total memory 114 require¬ ments is 4 bands of tiles to buffer the unrotated tiles, plus 2 tiles for the space to store rotated tiles before they are written to storage medium 118. An alternative embodiment of the rotation operation utilizes virtual memory to store the entirety of the rotated image 174. A Least Recently Used (LRU) page replacement policy causes the last three bands to remain resident in memory 114. If there are at least enough pages available to store 4 bands concurrently, on the order of 0(f) page faults typically occur, where /is the number of pages in the rotated image 172. Upon termination of the rotation of rotated image 172, the entirety of the rotated image 172 is written to storage medium 118. Alternatively, if the operating system of CPU 116 supports direct memory mapped disk files, the rotated image can be directly mapped to the virtual memory of CPU 116 and the rotated image written to storage medium 118 via standard virtual memory mechanisms. This alternative embodiment saves system swap area and the final writing of the image from virtual memory to the storage medium 118. According to an embodiment of the present invention, images stored in any band format can be rotated. The steps 1-6 are as follows. 1. Read in and convert to tile format a sufficient number of bands 122 of the image so that after conversion to tile format at least two bands of tiles are in memory. 2. Read in and convert to tile format a sufficient number of bands 122 of the image so that after conversion an additional band of tiles is present in memory 3. Calculate the entirety of rotated tiles that are only dependent on the converted three bands of unrotated tiles. Write these tiles to a disk file in tile format. 4. Free the storage for the least recently read band of unrotated tiles. 5. Repeat steps 2, 3 and 4 until the entirety of image 120 has been read in and converted to tile format. 6. Use the method mentioned hereinabove to convert the rotated image currently stored on disk in tile format to a rotated image in the original band format. It should be noted that in the abovementioned rotation methods the image 120 is rotated by reading and writing the entirety of two image-sized files. Reference is now made back to Fig. 1. The scanner 110 is connected to the scanner acquisi¬ tion unit 112 via a parallel interface, such as a Digital Equipment Corporation (DEC) DR11 interface defined by DEC and documented in the reference Microcomputer Interface Handbook, Digital Equip¬ ment Corporation, 1980. The CPU 116, memory 114 and scanner acquisition unit 112 are con¬ nected together on a CPU data and address buses 202. A similar CPU bus structure and DMA scheme is described in Intel technical manual 82380 High Performance 32-bit DMA Controller with Integrated System Support Peripherals, Intel Corp 1988. As mentioned hereinabove, the system illus¬ trated in Fig. 1 is a variant of standard DMA architectures. The system operates as follows: A NEW_DATA_READY line of the scanner 110 strobes, indicating to the scanner acquisition unit 112 that a valid pixel value /,- (or a color separated pixel value representing one color of a color separated pixel) is available on the DATA lines of the scanner 110. Since there is typically no handshake between the scanner 110 and the scanner acquisition unit 112, the acquisition unit 112 must be able to receive the pixel value at a rate greater than or equal to the rate that the scanner 110 acquires the pixel value. Upon pulsation of the NEW_DATAJREADY line, the HOLD control line typically is asserted, thus requesting control of the data and address buses 202 from the CPU 116, or other permanent bus master. When CPU 116 relinquishes control of the data and the address buses 202, it asserts the HLDA control line and -enters a hold-state until the scanner acquisition unit 112 drives the HOLD signal false. With the HLDA signal asserted, the scanner acquisition unit 112 places an address on the address bus and the valid pixel value on the data bus 202. The address is the permuted memory location z. for the pixel value and is calculated according to any of the methods of the present invention. The scanner acquisition unit 112 generates a memory write signal and the pixel value /,- is written to the proper address in memory 114. According to a preferred embodiment of the present invention, the permutations described hereinabove are implemented as address generator routines typically stored in the scanner acquisition unit 112 as microcode programs. The programs in Annex A and B are typically compiled into microcode to implement an address generator routine G(P). In an alternative embodiment of the present invention the permutations are generated by precomputed lookup tables stored in ROM (read only memory) or lookup tables computed just prior to the image acquisition. The lookup tables are computed by the address generator routine G(P). It will be appreciated that the address generator routine G(P) of the present invention is significantly different from those of standard DMA controllers which merely generate a new address by incrementing the previous address by a predetermined value. Once the pixel value is written to its location in memory 114, the scanner acquisition unit 112 deasserts the HOLD signal. Subsequently, the CPU 116 deasserts the HLDA signal and continues processing. The above described process is repeated for each new pixel value / / . After a band 122 has been completely written into a first buffer of memory 114, the scanner acquisition unit 112 signals this fact to the CPU 116 via the EOP signal which typically causes a CPU 116 interrupt. The CPU 116 then instructs the scanner acquisition unit 112, by writing to i o or memory mapped registers on the scanner acquisition unit 112, to switch the base address to a second buffer of memory 114, contain¬ ing the location of the next buffer. The band of data stored in the first buffer is transferred to storage medium 118 via the storage medium DMA controller 119, which may typically contain an 82380 DMA controller from Intel corporation. The storage medium DMA controller 119 and the storage medium 118 are typically chosen to transfer the band of data at a faster rate than that by which the scanner 110 acquires image data. Thus, the first buffer will be emptied before the second buffer is completely filled. It should be noted that this technique must be slightly modified for systems with more more than one DMA unit on the system bus. A bus arbitration protocol (for example, daisy chaining or priorit y resolution) must be instituted for the HOLD signal. These well known techniques are described in Microprocessor System Design Concepts, Nikitas A, Alexandridis, Computer Science Press, 1984, specifically sections 9.7.3 and 9.8.3. it will be appreciated by persons skilled in the art that the present invention is not limited by what has been particularly shown and described hereinabove. The scope of the present invention is defined only by the claims which follow: ANNEX A The following file named gabs .cc defines a base class GABS that is used in both Annex A and Annex B. 1 // This file i called gabs , cc 2 // It is writen in the C++ language as describled in the book: 3 // "The C++ programming Language" by Bjarne Stroustrup, 4 // 1986, Addϊson-Wesley. 5 class GABS { / / The abstract class for reduction permutations 6 protected: ^ 7 int last_ptr; / / The pointer to the begining of the last 8 / / abstracted block. 9 int column; // The column in the image. 10 int row__in_abs_ image; / / A flag to indicate when to go back to 11 / / the first abstraction. 12 int ad; / / Addressing within an abstracted band. 13 int grow; / / Flag to signify when to start a new 14 / / row in the abstracted band, e. 15 / / after r image rows. 16 int inv_ad; / / Addressing for inverse transformation 17 int inv_count; / / Counter to keep track of abstracted band 18 int inv_rowcoun ; / / Counter to keep track of which row we are in. 19 int inv_ptr; / / The index of the current abstracted band. 20 int n; 11 n, number of rows in each abstracted band. 21 int m; / / m, number of columns in each abstracted band. 22 int r; / / r, order of the abstraction. 23 int r2; // r 2 24 int b_size; // nmr 2 , size of band in pixels 25 int phase; // The phase column. 26 pixel** abs_bases; // Lookup table for abstract block addresses. 27 pixel** iabs_i>ases; / / Lookup table for inverse abstract block addresses. 28 public: 29 GABS (int nn, int mm, int rr ) // A constructor for the GABS class. 30 { 31 n= nn; 32 ιtι= mm; 33 r= rr; 34 r2= r*r; 35 b_size-*= n*m*r2; 36 abs_bases= new pixel*[r2] ; // The table size is r 2 . 37 iabs_bases= new pixel*[r2]; // The table size is r 2 . 38 init() ; 39 } 40 // This is the part of the initialization that is repeated 41 // For each band. 42 void init ( ) 43 { inv_ad= -r; inv_count= 0; inv__rowcount= 0; inv_ptr= 0; last_ptr= -1; coluιtιn= -1; phase= -1; row_in_abs_image= 0; ad= 0; grow=0; inline int band size O // Return band size. { return (b size) ; } / / The next function is called to generate the next / / address when the next pixel value is presented // by the scanner. pixel* next() { column++; / / Increment the current column. phase++; / / Increment the current phase. last_ptr++; / / Increment to the next abstract block.. if(phase >•= r) / / The variance of phase within a row is r. { ad++; phase**-* 5 0; // Reset the phase. last_ptr-= r; / / Jump bade r abstracted blocks. } if(column >= (r*m) ) // Are we at the end of a scanner row. { column= 0; // Reset column. last_ptr+= r; / / Jump forward r abstracted blocks. row_in_abs_image++; / / We are now in the next row of abstracted blocks. / / We have only nr rows of abstracted blocks. // After each r scanner row jump the next row. if (row_in_abs_image >= r) { row_in_abs_image= 0 ; last_ptr= 0; // Start again at abstract block zero. } grow++; // Keep track of rows. // If we have read r rows go. / / to next row in the abstract image. if (grow >= r) { grow=0 ; / / Go to the next row in the next abstracted band. 92 } 93 else 94 { 95 ad-= m; // Continue on the same row in the next abstracted band. 96 } 97 } 98 // Return the sum of the abstract block pointer 9 / / and the addressing within the block. 100 return(abs_bases[last_ptr] + ad); 101 } 102 pixel* inext() { 103 inv_ad+= r; 104 if(inv_rowcount++ >= m) 105 { 106 inv_ad+= m*r*r - m*r; 107 inv_rowcount= 1; 108 } 109 if (inv_count++ >= m*n) 110 { 111 inv_ptr++; 112 inv_ad= 0; 113 inv_count**= 1; 114 } 115 return (iabs_bases [inv_ptr] + inv_ad) ; 116 } 117 // qa is the routine to compute the 118 // addressing within an abstracted band 119 int qa (irtt abs_col, int abs_row) 120 { 121 return (abs_col + abs_row*m) ; // Map pixels to sequential positions. 122 } 123 // qinvabs is the routine to compute the 124 // inverse addressing within an abstracted band 125 int qainv(int abs_col, int abs_row) 126 { 127 return (r*abs_col + r2*m*abs_row) ; 128 } 129 130 // This routine computes the location of the abstracted band 131 // having the phase (phase_row,phase_col) 132 // This routine is overridden in the actual class. 133 // It is an error to call the qabs routine from 134 // the class GABS . 135 virtual int qabs (int phase_col, int phase_row) 137 {{ 138 cerr « "Shouldn' t get here"; 139 } 140 141 / / This routine computes the base location of first pixel of the abstracted 142 // band having the phase (phase_row,phase_col) . 143 // This routine is overridden in the actual class. 14 / / It is an error to call the qinvabs routine from 145 // the class GABS . 146 virtual int qinvabs (int phase_col, int phase_row) 147 { 148 cerr « "Shouldn't get here"; 149 } 150 151 / / Compute the mapping to abstracted format given a scanner row index. 152 int q(int scanner_row_index) 153 { 154 // phase_col is the column phase of the pixel. 155 // phase_row is the row phase of the pixel. 156 // abs_col is the column number within the abstracted band. 157 // abs_row is the row number within the abstracted band. 158 int phase_col, phase_row, abs_col, abs_row; 159 phase_col= scanner_row_index%r; 160 phase_row= (scanner_row_index/ (r*m) ) %r; 161 abs_col= scanner_row_index% (r*m) /r; 162 abs_row= scanner_row_index/ (r2*m) ; 163 // The mapping is the sum of the location of the abstracted band (qabs) 164 / / and the location with that abstracted band ( qa ) . 165 return (m*n*qabs (phase_col, phase_row) + qa (abs_col, abs_row) ) ; 166 } 167 168 / / Compute the mapping to scanner row index given a abstracted format index 169 int qinv(int abs_format_index) 170 { 171 int phase_col, phase_row, abs_col, abs_row; 172 phase_col= abs_format_index%(r*m*n) / (m*n) ; 173 phase_row= abs_format_index/(r*m*n) ; 174 abs_col= abs_format_index%m; 175 abs_row= (abs_format_index%(m*n) ) /m; 176 return (qinvabs (phase_col, phase_row) + qainv(abs_col, abs_row) ) 177 } 178 }; The following file named g . cc defines a derived class of GABS called GA that implements the fixed size reduction embodiment. 1 // Thisfileis called ga .ee 2 #include <stream.h> 3 #include "gabs . ee" // the file that defines class GABS 4 // Class GA is derived from class GABS it defines an ordering 5 // of the abstracted bands 6 class GA: public GABS { 7 public: 8 // Initialize an instance of class GA 9 / / The array abs_bases is a table that gives the 10 // starting location of each abstracted band 11 / / The array iabs_bases is a table that gives the 12 / / upper left hand corner of the abstracted band in the original band 13 GA(pixel* offset, int nn, int mm, int rr) : GABS(nn,mm,rr) 14 { 15 for(int i=0;i<r2;i++) 16 { 17 abs_bases[i]= offset + i*m*n; 18 iabs_bases[i]= offset + i%r + r*m*(i/r) ; 19 } 20 // No te: the base address of the band in memory should be added to abs bases . 21 } 22 // This defines the location of each abstracted band 23 int qabs (int phase_col, int phase__row) 24 { 25 return (phase_col + r*phase_row) ; 26 } 27 28 // This defines the location of the first pixel of the indicated abstracted band 29 int qinvabs(int phase_col, int phase_row) 30 { 31 return (phase_col + m*r*phase_row) ; 32 } 33 } ; The following file named gatest . cc tests the class GA. 1 typedef char pixel; // Use an example 32 bits per pixel 2 #include "ga. cc" 3 pixel* offset; // Should point to the begining of the band buffer 4 main ( ) 5 { 6 GA ga (offset, 3, 4, 4) ; //Create an instance of a mapping having n=β,m=3,r=4 7 int j, i; 8 9 // Check that the address generator and the mapping function return the 10 / / same address. The multiplication by the size of a pixel is because 11 / / the generator returns addresses while the mapping functions returns 12 // the permutation 13 for (i= 0; i < ga.band_size () ; i++) 14 { 15 if (ga.next () != offset + ga.q(i) ) 16 { 17 cout « "Error in assertion : " « i; 18 } 19 } 20 / / Reinitialize the instance of the mapping-generator 21 ga. init () ; 22 / / Check that the inverse address generator and the inverse mapping function 23 // return the same address. 24 for(i= 0; i < ga.band_size() ; i++) 25 { 26 // cout « int (ga.inext() ) « " " « ga.qinv(i) « "0; 27 if(ga.inext () != offset + ga.qinv(i)) 28 { 29 cout « "Error in inverse assertion: " « i ; 30 } 31 } 32 / / Reinitialize the instance of the mapping-generator 33 ga . init ( ) ; 3 / / Check that the mapping function and the inverse mapping function 35 // are in fact inverses of each other. 36 for (i=0; i < ga.band_size ( ) ; i++) 37 { 38 int j ; 39 if (i != ga. qinv (ga. q(i) ) ) // Check that i=q ~l (q(i)) 40 cout « "Inversion error : " « i; 41 } 42 } ANNEX B The foltowing file named gm.cc defines a derived class of GABS called GM that implements the multiple size reduction embodiment. 1 // This fileiscalled gm. ee 2 #include <stream.h> 3 #include "gabs . ee" // the file that defines class GABS 4 #include "cluster . hh" 5 extern int mapping [ ] ; 6 static int* imapping; 7 // Class GM is derived from class GABS it defines an ordering 8 // of the abstracted bands 9 // This is the multiple order abstraction format 10 class GM: public GABS { 11 public: 12 // Initialize an instance of class GM 13 // The array abs_bases is a table that gives the 14 // starting location of each abstracted band 15 // The array iabs_bases is a table that gives the 16 / / upper left hand corner of the abstracted band in the original band 17 // We here take the specific case of r=60 and we execute the function 18 // maO which defines the array mapping . 19 GM (pixel* offset, int nn, int mm) : GABS (nn, m, 60 ) 20 { 21 if (ca. starting_location == 0) 22 ma ( ) ; 23 imapping= new int [60*60] ; // The inverse mapping 24 for (int i=0; i< 2; i++) 25 { 26 abs_bases [mapping [i] ] = offset + i*m*n; , 27 iabs_bases [i] = offset + mapping[i] %r + r*m* (mapping [i] /r) ; 28 imapping [mapping [ i] ] = i; // Create inverse mapping array 29 } 30 } 31 / / The function qabs and qinvabs are used in the generating functions. 32 33 // qabs defines the location of each abstracted band 34 int qabs(int phase_col, int phase_row) 35 { 36 return imapping[ (phase_col + r*phase_row) ] ; 37 } 38 39 // Tills defines the location of the first pixel of the indicated abstracted band 40 int qinvabs (int phase_col, int phase_row) 41 { int mapped_abs_band= π..oping[phase_col + r*phase__row] ; int true__phase_col= maρped_abs_band % r; int true_ρhase__row= mapped_abs_band / r; return (true_phase_col + m*r*true_phase_row) ; } The following file named set ^ hh defines the basic operations on sets. 1 // This FOe contains support routines for the class Abs Set . 2 / / It uses the B it S et class as described in the manual 3 // "User's Guide to theONU C++ Library" by Doug Lea. 4 // Copyright 1988 Free Software Foundation.. 5 // This library and the C++ compiler can be obtained from the 6 // Free Software Foundation, Inc., 675 Mass AVE, Cambridge, MA 02139. 7 tinclude <BitSet.h> 8 const int n-= 60; 9 // the function rc2can converts row, column coordinates to 10 // canonical form. 11 inline int rc2can(int r, int c) { return r + n * c; } 12 class AbsSet 13 { 14 BitSet* bs; // This provides the basic Set functionality. 15 int* ref_count; 16 public: 17 AbsSet(); 18 AbsSet(int,int =0, int =0); 19 AbsSet: :AbsSet(BitSetδ) ; 20 "AbsSet (); 21 void operator = (AbsSet&) ; 22 AbsSet operator & (AbsSetδ x) ; 23 AbsSet operator I (AbsSet& x) ; 24 AbsSet operator - (AbsSet& x) ; 25 void operator |= (AbsSet& x) ; 26 int operator •== (AbsSetδ x) ; 27 int operator != (AbsSet& x) ; 28 int operator <= (AbsSetδ x) ; 29 void set(int r, int c) { bs->set(rc2can(r, c) 1 );} 30 int count() { return bs->count(1) ; } 31 cluster append(int*&) ; 32 }; 33 AbsSet: :AbsSet(int s, int r, int c) 34 { 35 bs= new BitSet; 36 ref_count= new int; 37 *ref_count= 1; 38 for(int i=0;i<60;i+=s) 39 for(int j=0;j<60;j+=s) 40 set(i+r, j+c) ; 41 } 42 AbsSet: -.AbsSet() { bs= new BitSet; ref_count= new int; *ref_count= 1; } AbsSet: :AbsSet(BitSetδ b) { bs= new BitSet(b) ;; ref_count= new int; *ref count= 1; } AbsSet: :"AbsSet() { if(*ref_count == 1) delete bs; else (*ref_count)—; } void AbsSet::operator = (AbsSetS x) { if(*ref_count == 1) delete bs; else *ref_count—; (*x.ref_count)++; ref_count= x.ref_count; bs= x.bs; } AbsSet AbsSet::operator & (AbsSet& x) { return(AbsSet( (*bs) & *(x.bs))); } AbsSet AbsSet::operator | (AbsSet& x) { return(AbsSet((*bs) | *(x.bs))); } AbsSet AbsSet::operator - (AbsSetδ x) { 85 return(AbsSet((*bs) - *(x.bs) ) ) ; 86 } 87 int AbsSet: :operator — (AbsSetfi x) 88 { 89 return( (*bs) — *(x.bs)); 90 } 91 int AbsSet: :operator != (AbsSetS x) 92 { 93 return((*bs) != *(x.bs)); 94 } 95 int AbsSet: :operator <= (AbsSet& x) 96 { 97 retur ( (*bs) <= *(x.bs)); 98 } 99 void AbsSet::operator |= (AbsSetδ x) 100 { 101 *bs |= *x.bs; 102 } 103 cluster AbsSet::append(int*& ip) 104 { 105 cluster el; 106 el.size_of_set= 0; 107 el.starting_location= cl.total_size_of_sets; 108 for(int i= bs->first(); i >= 0; i=bs->next(i) ) 109 { 110 *ip++= i; 111 el.size_of_set++; 112 } 113 cl.total_size_of_sets+= cl.size_of_set; 114 return el; 115 } The following file named cluster. hh defines the data structure that allows the program to access the linear ordering of the sets S a , • • ■ ,S k . 1 II This defines the clustering of each set within the linear ordering of sets. 2 struct cluster 3 { 4 static total_size_of_sets; // How big is the mapping so far. 5 int start ing_location; // Where does this cluster start. 6 int size_of_set; // How big is the set. 7 } ; 8 extern cluster ca, cb, cc, ed, ce, cf, eg, ch, ci, cj , ck, cspl; The following file named ma .cc calculates the subsets S a , • • • ,S k . 1 #include "cluster. hh" 2 #include "set .hh" 3 // This function creates the sets a -k. I also clusters the sets in the 4 // order detailed in the example. 5 int mapping[3600] ; 6 cluster ca, cb, cc, cd, ce, cf, eg, ch, ci, cj, ck, cspl; 7 ma ( ) 8 { 9 int *ip; 10 // Create the set S t . 11 AbsSet sl(l) // Si 12 AbsSet s2(2) II s 2 13 AbsSet s3(3) II s 3 14 AbsSet s4(4) // s 4 15 AbsSet s5 5) II S s 16 AbsSet s6(6 11 s 6 17 AbsSet spl(((sl - s2) - s3) - s5) II s 18 AbsSet a= s3 & s4 & s5; II These 5 ( 19 AbsSet b= (s3 & s4) - s5; 20 AbsSet c= (s5 & s4) - s3; 21 AbsSet d= (s4 - s5) - s3; 22 AbsSet e= ( (s2 - s4) & s5) - s3; 23 AbsSet f= ((s2 - s4) & s3) - s5; 24 AbsSet g= ( (s3 & s5) & s2) - s4; 25 AbsSet h= (s3 & s5) - s2; 26 AbsSet i= (s3 - s2) - s5; 27 AbsSet j- (s5 - s2) - s3; 28 AbsSet k= ( (s2 - s4) - s3) - s5; 29 // The order of the sets a-k are as outlined in Fig.4 30 ip=mapping; 31 cj= j .append(ip) ; 32 ce= e.append(ip) ; 33 ck= k. ppend(i ) ; 34 cd= d.append(i ) ; 35 cc= c.append(ip) ; 36 ca= a.append(ip) ; 37 cb= b.append(ip) ; 38 cf= f.append(i ) ; 39 cg= g.append(ip) ; 40 ch= h.append(ip) ; 41 ci= i.append(i ) ; 42 cspl= spl.append(ip) ; 43 } The following file named gmtes . cc tests the class GM. 1 typedef int pixel; // Use an example 32 bits per pixel 2 #include "gm. cc" 3 const int nn= 1; 4 const int mm= 9; 5 pixel* offset; // Should point to the begining of the band buffer 6 void is_scaled_image (pixel*, cluster**, int, int, int) ; 7 main ( ) 8 { 9 int j , i; 10 11 GM gm (offset, nn, mm) ; //Create an instance of a mapping having n=l,m=Θ,r=60 12 #ifdef NOCHECK 13 // Check that the address generator and the mapping function return the 14 / / same address. The multiplication by the size of a pixel is because 15 / / the generator returns addresses while the mapping functions returns 16 // the permutation 17 f or (i= 0; i < gm.band_size ( ) ; i++) 18 { 19 if (gm.next () != offset + gm.q(i) ) 20 { 21 cout « "Error in assertion: " « i; 22 } 23 } 2 / / Reinitialize the instance of the mapping-generator 25 gm. init ( ) ; 26 / / Check that the inverse address generator and the inverse mapping function 27 // return the same address. 28 for(i= 0; i < gm.band_size() ; i++) 29 { 30 if(gm.inext() != offset + gm.qinv(i)) 31 { 32 cout « "Error in inverse assertion: " « i ; 33 } 34 } 35 / / Reinitialize the instance of the mapping-generator 36 gm. init () ; 37 / / Check that the mapping function and the inverse mapping function 38 // are in fact inverses of each other. 39 for (i=0; i < gm.band_size ( ) ; i++) 40 { 41 int j ; 42 if (i != gm. qinv (gm. q(i) ) ) // Check that i=q ~λ {q(i)) cout « "Inversion error: " « i; } #endif NOCHECK // Allocate room for a band pixel* image= new pixel [60*60 *nn*mm] ; GM gt (image, nn, mm); // Create an instance of a mapping with the base address image for(i=0;i < gt.band_size () ; i++) *gt.next()= i; // Write into the mapped band the pixel values // in order to know what pixel it is take as the // pixel value its sequential location in the band // The following are the subset makeup of each abstraction. // It is used to check that we can pull out an specific abstraction // from the band. See equation (7). cluster *cl[]= { &ca, &cb, &cc, &cd, &ce, &cf, &cg, &ch, &ci, &cj, Sck, Scspl, 0}; cluster *c2[]*= { Sea, &cb. Sec, &cd. See, &cf, scg, &ck, 0}; cluster *c3[]= { &ca, &cb, &cf, &cg, &ch, &ci, 0}; cluster *c4[]= { Sea, Scb, Sec, Scd, 0}; cluster *c5[]= { ea, Sec, See, &cg, sch, scj, 0}; cluster *c6[]= { Sea, Scb, Scf, Scg, 0}; cluster *cl0[]= { sea. Sec, See, Scg, 0}; // Check if the subset construction is correct. is_scaled_image(image, el, 1, nn, mm); is_scaled_image(image, c2, 2, nn, mm); is_scaled_image(image, c3, 3, nn, mm); is_scaled_image(image, c4, 4, nn, mm); is_scaled_image(image, c5, 5, nn, mm); is_scaled_image(image, c6, 6, nn, mm); is_scaled__image(image, clO, 10, nn, mm); } / / Check that the proper abstracted image can be extracted properly void is_scaled_image(pixel* pimage, cluster** pclusters, int divisor, int height, int width) { int i; int image_size= 5 0; pixel* pp; while(*pclusters) { cluster* pc= *pclusters; image_size+= pc->size_of_set; pp= pimage + width * height * ρc->starting_location; for(i=0; i < pc->size_of_set * height * width; i++) 87 { 88 if((*pp++ % divisor) != 0) 89 cerr « "out of order " « divisor 90 } 91 pclusters++; 92 } 93 if(image_size !•= 60*60/ (divisor*divisor) ) 94 cerr « "Bad total size " « divisor; 95 }

Claims: CLAIMS 1. An image processing system comprising: image acquisition means for acquiring pixels of a digital image; permutation means for receiving said pixels subsequent to their acquisition by said image acquisition means and for ordering said pixels thereby to reorder said digital image; and storage means for receiving and for storing said multiplicity of abstracted images. 2. An image processing system comprising: image acquisition means for acquiring pixels of a digital image; permutation means for receiving said pixels subsequent to their acquisition by said image acquisition means and for ordering said pixels thereby to reorder said digital image into a multiplicity of abstracted images at a reduction factor; and storage means for receiving and for storing said multiplicity of abstracted images. 3. An image processing system comprising: image acquisition means for acquiring pixels of a digital image; permutation means for receiving said pixels subsequent to their acquisition by said image acquisition ηaeans, for ordering said pixels thereby to reorder said digital image into a plurality of abstracted images at a relatively large reduction factor and for combining said abstracted images into a multiplicity of sets of said abstracted images combinable into a plural¬ ity of abstracted images at a plurality of different relatively small reduction factors; and storage means for receiving and for storing said digital image in said multiplicity of sets. 4. An image processing system according to claim 1 and wherein said permutation means comprise means for dividing said digital image into a plurality of bands comprising a predetermined number of rows of said digital image and means for operating on said plurality of bands in a band by band manner. 5. An image processing system according to claim 2 and wherein said permutation means comprise means for dividing said digital image into a plurality of bands comprising a predetermined number of rows of said digital image and means for operating on said plurality of bands in a band by band manner. 6. An image processing system according to claim 3 and wherein said permutation means comprise means for dividing said digital image into a plurality of bands comprising a predetermined numfaser of rows of said digital image and means for operating on said plurality of bands in a band by band manner. 7. An image processing system according to claim 4 and wherein said means for operating operate separately on each band. 8. An image processing system according to claim 5 and wherein said means for operating operate separately on each band. 9. An image processing system according to claim 6 and wherein said means for operating operate separately on each band. 10. An image processing system according to claim 7 and wherein portions of said bands are written by said means for operating to said storage means when said portions of said band fill at least one file system block. 11. An image processing system according to claim 8 and wherein portions of said bands are written by said means for operating to said storage means when said portions of said band fill at least one file system block. 12. An image processing system according to claim 9 and wherein portions of said bands are written by said means for operating to said storage means when said portions of said band fill at least one file system block. 13. An image processing method including the steps of: acquiring pixeis of a digital image; receiving said pixels subsequent to their acquisition in said step of acquiring and ord¬ ering said pixels thereby to reorder said digital image into a multiplicity of abstracted images; and storing said multiplicity of abstracted images. 14. An image processing method including the steps of: acquiring pixels of a digital image; receiving said pixels subsequent to their acquisition in said step of acquiring and order¬ ing said pixels thereby to reorder said digital image into a plurality of abstracted images at a relatively large reduction factor and combining said abstracted images into a multiplicity of sets of said abstracted images combinable into a plurality of abstracted images at a plurality of different relatively small reduction factors; and storing said digital image in said multiplicity of sets. 15. An image processing method including the steps of: acquiring pixels of a digital image; ordering said pixels thereby to reorder said digital image into a multiplicity of abstracted images at a single reduction factor; and storing said multiplicity of abstracted images. 16. An image transformation and rotation method including the steps of: receiving an image of a given image size stored in a permuted format; transforming said image to a tile format image via an inverse transformation of said per¬ muted format to a sequential format and via a forward transformation from said sequential format to said tile format; rotating said tile format image into a rotated tile format image by rotating the location of each tile of said image in said tile format, utilizing an amount of computer memory less than said image size; retransforming said rotated tile format image to a rotated permuted format image via an inverse transformation from said tile format to said sequential format and via a for¬ ward transformation from said sequential format to said permuted format; and storing said rotated permuted format image. 17. An image transformation and rotation method according to claim 16 and wherein said step of rotation includes the steps of: receiving and buffering a first number of tiles of a source image to be rotated; writing to a storage medium a second number of tiles of a destination image, said second number of tfles being different from said first number of tiles, which second number of tiles are fully defined by rotations of said first number of tiles of a source image; and repeating said steps of receiving and writing until said source image no longer contains tiles. 18. An image processing rotation method including the steps of: receiving and buffering a first number of tiles of a source image to be rotated; writing to a storage medium a second number of tiles of a destination image, said second number of tiles being different from said first number of tiles, which second number of tiles are fully defined by rotations of said first number of tiles of a source image; and repeating said steps of receiving and writing until said source image no longer contains tiles. 19. An image processing system comprising: image acquisition means for acquiring pixels of a digital image; means for dividing said digital image into a plurality of bands each comprising a predetermined number of rows; and permutation means for operating on said plurality of bands in a band by band manner and for reordering the location of said pixels; 20. A system according to claim 19 and also comprising storage means for receiving and for stor¬ ing said bands in a band by band manner.

Inventors: ISAACSON JOEL, ISAACSON, JOEL
Applicants: ISAACSON JOEL, ISAACSON, JOEL
Application year: 1990
Main classification: G06F
IPC codes: G06T1, G06T9