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Figure 109 Fractal transformation calculation The left hand partition (D) encloses the source blocks and the right hand partition (R) contains the destination blocks r
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Let us assume that the triangulation R (adapted to the image contents) has N destination blocks ri and that the (regular) triangulation D contains Q source blocks (see Figure 109) The algorithm consists of associating each block ri with the block dj that minimizes the error d between the gray-level function of block ri and that of block dj transformed by The mass transformation to perform the collage of the source block dj onto the destination block ri is the same as the one proposed by Jacquin and Fisher It uses (i) (i) only two coef cients: the shift coef cient 1 and the scale coef cient 2 The decoding algorithm amounts to an iteration of operator W , after the partitions R and D were rebuilt, starting from an arbitrary image f0 After k iterations of operator W , the gray-level fk (xi , yi ) of a pixel in block r reads: fk (xi , yi ) = 2 fk 1 v 1 (xi , yi ) + 1 (xi , yi ) r (1030)
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In practice, the result converges towards the reconstructed image, an attractor of the fractal transformation, after ve to ten iterations (see Figure 1010) Primarily, the number of iterations depends on the surface ratio between the blocks of partition D and those of partition R As the number of iterations increases, the collage of a block d (n) (covering several blocks r) onto its corresponding block rn reduces the size of the details within blocks of partition R 1035 Coding and decoding acceleration 10351 Coding simpli cation suppressing the research for similarities Dudbridge proposed in 1995 [DUD 95b] a fast fractal compression method for images, based on a regular square partitioning The speed of this compression algorithm is due to the fact that no search for a similar interblock is made The
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Figure 1010 Decoding of Lena image 512 512 MSE At iteration 15: Tc = 112 : 1, PSNR = 10 log10 2552 = 3229 dB
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image is partitioned into a set of square xed size blocks, and each block is coded individually by a fractal transformation According to the author, the method gives less ef cient results than, for example, Jacquin s method The reasons for this will be explained at the end of the section
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Coding An image5 is coded using a set of contracting spatial transformations (IFS) { 1 , , N } de ned on R2 , associated with a contracting transformation G acting on the pixels luminance At resolution m, the square support of the IFS transformed image reads:
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N N N
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k (A) =
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km =1
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k1 km (A)
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(1031)
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It is noteworthy that the spatial transformation is applied to A and not to a subpart of A as it is the case in the traditional approach of coding de ned by Jacquin The quantity p = Ak1 km denotes an element of the image support at resolution m, which may contain several pixels of the original image At the maximum resolution, the size of element p is equal to that of an image pixel The set Pm = {Ak1 km ; k1 , , km = 1, , N } contains all the elements of the image at resolution m In the following section, we will consider that the IFS is composed of N = 4 af ne transformations de ned by equations (1010) In these conditions, equation (1031) is illustrated in Figure 1011
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Figure 1011 The square image A is divided into four square elements by four af ne contracting transformations k1 (k1 = 1 4) In the center, Ak1 = k1 (A) corresponds to one of the four elements of the image at resolution 1 On the right, Ak1 k2 = k1 k2 (A) corresponds to one of the 16 elements of the image at resolution 2
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5 Within this section, the term image stands for a square block resulting from the regular partitioning of the original image
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