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If, for a parent block, three or four child collages are necessary, only the four child collages are coded If one or two child collages are necessary, the parent block is coded by parent collage complemented with child collages 10323 Coding of the collage operation on a destination block The storing of the collage of a source block (parent or child) d (n) on a destination block (parent or child) rn includes: the index of the source block d (n) retained among the Q blocks of the library, provided that those are arranged in a block list and that their organization on the image support is known Otherwise, it is necessary to store the coordinates (xk , yl ) of a reference pixel in block d (n) (for example, the upper left corner in the case of a square block); the isometry used during collage (one among eight); the coef cients 1 and 2 of the mass transformation This information is associated with each N destination block of the partition R It is coded on a variable number of bits since it is not always necessary to store the set of the three components It depends on the mass transformation that is used 10324 Contraction control of the fractal transformation The contraction control of the fractal transformation is a dif cult problem Jacquin shows that the contraction factor of a mass transformation depends on the scale factor (n) 2 The contraction factor of the other mass transformations (shift, absorption) is equal to 1 The author thus ensures the contraction of the fractal transformation by (n) imposing the condition ( 2 )2 < 1 whatever the value of n in [1, , N ] This too stringent constraint limits the quality of the collage A more detailed study of the contraction control of the fractal transformation will be presented in section 10332 and we shall see in section 10333 that it is possible to slightly slacken the constraint while preserving the nal contraction of the fractal transformation 1033 Algebraic formulation of the fractal transformation Immediately following the work by Jacquin, Lundheim proposed an algebraic formulation aimed at facilitating the comprehension of various theoretical and practical problems raised by the extension of IFS theory to the coding of natural images [LUN 92] Its formulation, applied to the 1D signal, was then extended to the case of 2D signals by ien [ IE 93] and Leps y [LEP 93] The formulation that is provided here is used in the simple case of a fractal transformation operating on the source and destination blocks of xed size and simple geometry (square, rectangular or triangular) The source blocks do not overlap A.

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block is seen as a vector by supposing that the pixels which make it up are connected inside the original image Let us consider an image as a column vector x made up of M 2 pixels The fractal transformation T of the image x, composed of a linear term L and of a translation vector t, takes on the following form: T x = Lx + t (1018)

By specifying this transformation at the level of each destination block of the partition R, equation (1018) can be written as follows:

The source sends out a RREQ packet Each RREQ packet has the following elds:

Close-out, 81, 82, 83

(1019)

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The elementary transformations associated with each matrix Ln and each vector tn are detailed below The transformation associated with the elementary matrix Ln : RM RM operates on the source vector d (n) to attach it to the destination vector rn The source 2 2 and destination vectors belong respectively to RD and RB , where D > B The matrix Ln reads: Ln = 2 Pn DF (n)

(n) 2 2

(1020)

ARREQ this is a label that speci es that this is a route request message; ARREQ_id this is a globally unique identi er for the ARREQ; Dest_id this is an identi er of the destination; Destseq this is the last known sequence number for the destination or an unknown ag if this information is not known; PS active pseudonym

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where: 2 2 1) F (n) : RM RD selects a source block d (n) of size D2 pixels of the image The isometries can be applied inside the block; 2) D : RD RB brings back the source block selected to the size of a destination block by sub-sampling or by averaging pixels The block thus obtained (n) is called block b2 It should be thought as the decimated source block described by Jacquin; 3) Pn : RB RM positions the decimated source block b2 destination block rn and cancels the other pixels of the image

2 2 2 2

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