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10333 Fisher formulation Fisher [FIS 95a] described the collage operation of a source block onto a destination block using a unique formula: an bn en 0 x x 0 y + fn (1029) n y = cn d n (n) (n) z z 1 0 0 2 where (x, y) are the coordinates of a pixel pertaining to the source block d (n) and z is its gray-level The symbols an , bn , cn , dn , en and fn are the coef cients of the af ne spatial transformation which brings back the pixels of source block d (n) inside the (n) (n) destination block rn The quantities 1 and 2 are the transformation coef cients of the pixels gray-level Whereas the mass transformation coef cients used by Jacquin are chosen from a prede ned set of values, Fisher uses a uniform scalar quanti er Jacobs et al showed that for this type of quanti er, the quanti cation of the translation coef cients and of the scale coef cients, with seven and ve bits, respectively, is optimal in terms of visual quality of the rebuilt images [JACO 92] Optimal coef cients calculation of the mass transformation For a destination block r, the approximation is given by: r = 2 b2 + 1 b1 r The calculation of the mass transformation coef cients is an optimization 2 problem within a linear subspace X of the vector space RB The purpose is to nd, for a destination block r and for a decimated source block b2 , the optimal coef cients 1 and 2 minimizing the least square distance d between r and its collage r THEOREM 102 (P ROJECTION) The optimal approximation in the L2 norm of a 2 r vector r element of RB , in the linear subspace X is the vector element of X which yields the residual vector r orthogonal to all vectors spanning the subspace X r Let us consider, the mean square error (MSE) between two vectors x and y in the 2 space RB , de ned by the following expression:.

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Err (x, y) =

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Determining the two optimal coef cients 1 and 2 leading to the best approximation of vector r in the basis b1 , b2 , amounts to canceling the two mean r square errors: Err (r 1 b1 2 b2 , b1 ) = 0 Err (r 1 b1 2 b2 , b2 ) = 0 The optimal coef cients 1 and 2 thus calculated are not constrained and the contraction of the collage operator is not guaranteed Jacobs et al empirically show that thresholding the magnitude of the 2 coef cient to 15 ensures the nal convergence of the fractal transformation H rtgen proposes a detailed study of the contraction control of the fractal transformation by considering particular cases based on square partitioning [HURT 93a, HURT 94] and on the spectral radius associated with the linear term L introduced by Lundheim (see equation (1018)) 1034 Experimentation on triangular partitions It has been shown in section 103 that the fractal coding of a natural image consisted of approximating each area of the image from other areas of the same image, by means of local transformations The image is thus partitioned into N blocks rn , forming a partition R Each block is then put in correspondence with another block dn of the image, from which it is possible to approximate, by an elementary transformation n , the gray-level function of rn The operator W de ning a fractal transformation of the image is composed of the N transformations n It is nally contracting provided that a suf cient number of transformations n are contracting Fractal transformation coding requires the coef cients of the N transformations n to be stored It is thus all the more effective that the partition R contains a reduced number of blocks The rst works of Jacquin, mentioned in section 103, showed the advantage of using square regular partitions to build the operator W However, they did not reach large compression rates because the partition R entailed too large a number of blocks rn This issue could be circumvented by calculating the fractal transformation on square or rectangular partitionings [FIS 95a] adapted to the image contents (quadtree, HV) We shall now present an approach that consists of calculating the fractal transformation of the image on a triangular partitioning, which is exible and adapted to the contents of the image Various algorithms [CHAS 93, DAVO 97] allow us to build such a triangulation.

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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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