Region-based Fractal Image Compression using Deterministic Search

The paper introduces a new method based on deterministic search to fractal image compression. In order to find a good region-based partitioning, we propose a deterministic search method for finding the blocks to be merged. For each range, a list of best N domains is maintained. When two ranges are to be merged and their common edge disappears, for the new range the best N domains are selected only from the 2×N domain extension of the two ranges. At each step the edge with minimum collage error increase is deterministically selected and the two corresponding ranges are merged. The process starts with atomic blocks as ranges and ends when the desired number of ranges is achieved. In order to reduce the encoding time, a suboptimal initialization method is also considered. Experimental results prove that our method yields a better rate-distortion curve than the classic quad-tree partitioning scheme. 1. Deterministic search method A major problem that researchers in fractal image compression face, beside reducing the encoding time, is image partitioning. Because of the huge searching space, the optimal partitioning problem for a desired bit-rate, cannot be practically solved. Until now, deterministically hierarchical partitioning (quadtree scheme [1], HV partitioning [2], polygonal partitioning [3]) and split-andmerge methods ([4], Delaunay triangulations [5][6], quadrilateral [7], heuristic search [8], evolutionary [9]) have emerged as solutions for the problem. In order to comply with the spatial contraction of the fractal transform we consider the domains twice as large as the corresponding range. The spatial transform applied to a domain for matching the range size is the usual method of shrinking by pixel averaging. We also take in consideration all the 8 isometries (4 rotations and 4 flips) that can be applied to a block, which have the effect of enlarging the domain pool that has to be searched in order to find the best match. For a range R and a domain D we determinate the scale and offset coefficients s and o by minimizing the collage error as a function having s and o as parameters. The value obtained for s is then clamped to the [-1,1] interval in order to assure the contraction in luminance space as well. The collage error becomes, after applying an uniform quantization to the parameters s and o and yielding s and o , E R D R sD o ( , ) ( ) = − + 1 2 (1) where 1 is a uniform image block with each pixel having unit intensity. The bit stream transmitted to the decoder contains the codebook index of the best corresponding domain and the quantized values s and o . Even if the fractal encoding method previously described is not new, being almost a standard one, we propose a new method for block merging. Our method starts with an initialization phase consisting in: • splitting the image in small square image blocks called atomic blocks of the same size (e.g. 8 by 8 pixels). • building, for each atomic block, a list of best N corresponding domains, regarding the collage error, by an expensive full-search of the domain pool. The initialization phase is followed by a merging phase in which ranges, initially identical to the atomic blocks, are merged in order to find a good partitioning which has less ranges. We consider ranges as frontierconnected sets of atomic blocks, and define “edge” the entire common border of two adjacent ranges. In order to form an edge, the two adjacent ranges must share at least one atomic block border. For each edge we compute the collage error increase if this edge would disappear. When an edge disappears, the two corresponding ranges are removed and replaced by their union as a new range. To determine the edge’s collage error increase we need to find the best domain for the union range. To avoid fullsearch for getting the new best N domains, we follow [9] and restrict the search only to the 2×N domains obtained by correspondingly enlarging each of the N domains of both initial ranges, as shown in Figure 1 (small figure the two ranges which merge; large dark figures the corresponding domains from both lists, N domains for each range; large light figures the corresponding extensions of each domain). The best N=2 domains from these 2×N=4 are maintained in list for the merged range. At each step of the merging process, the collage error increase being already known for all the edges in the partitioning, we can deterministically select the edge with the minimum collage error increase. Now the merging phase can be described as follows: • compute, for each edge, the collage error increase in case of edge disappearance. • select the edge whose disappearance would give a minimal increase of the collage error. • merge the two corresponding ranges. • compute the collage error increase in case of edge disappearance for all the edges of the new range. • repeat the previous 3 steps until the desired number of ranges or the maximum value for the collage error is achieved. During the merging phase, at each step, when the number of ranges is decremented, the bit-rate also decreases while the collage error increases, the ratedistortion curve being, approximately, continuously parsed. An interesting feature of our deterministic merging algorithm is its simplicity, both in description and in parameters: beside the stop-condition it has only one parameter: the number N of the best domains maintained for each range.