Faster Fractal Image Compression Using Quadtree Recomposition Mahmoud, W.H. ; Jackson, D.J. ; A. Stapleton ; P. T. Gaughan Image and Vision Computing, 1997

Improved Quadtree Decomposition/Recomposition Algorithm for Fractal Image Compression Mahmoud, W.H. ; Jackson, D.J. Tennessee Technological University The University of Alabama Southeastcon '99. Proceedings. IEEE, 1999

Introduction Introduction to Fractals Introduction to Fractal Image Coding Quadtree Decomposition Partitioning (QD) Quadtree Recomposition Partitioning (QR) (1997) QD vs. QR Quadtree Decomposition / Recomposition Algorithm (QDR) (1999) Result

Introduction to Fractals Proposed by Mandelbrat ( 曼德保 ) in 1975 Infinite structure Self similar

Introduction to Fractals Mandelbrat set ( z 2 + c = 0 )

Introduction to Fractal Image Coding Attractor: Let W be a transformation and is its attractor, then for any p. (fixed point) Collage theorem (Barnsley ( 巴斯理 ), 1985) : i. Let W be a contractive transformation and is its attractor (if R, R’ are two similar images,, ie. ) ii., where R is a image => For any initial graph A,

Introduction to Fractal Image Coding Idea Divide the image R into a set of non-overlapping blocks {R i } (Range blocks) For each R i, we search a domain block D i, and a contractive transformation W i st. i. W i is a contractive transformation ii., let From Collage theorem, for any graph A. W i DiDi RiRi RiRi DiDi

larger, overlaping and R ∪ D i Introduction to Fractal Image Coding Encoding Algorithm 1. Get an image R 2. Cover the support of R with range blocks R i. 3. For each range block R i, find a domain block D i and a transformation W i st. The edge length of a domain block is usually twice the edge length of a range block 4. Write the transformations in an IFS form 5. Apply a lossless data compression for the IFSs small, non- overlaping and R = ∪ R i Range block Domain block

Introduction to Fractal Image Coding WiWi RiRi 1 4 DiDi δ x ^ δ y< δ y, δ x are search steps

Quadtree Decomposition Partitioning (QD) 1. Partition the image into a set of large range blocks 2. If a range is fail to find a match, the process is repeated after partitioning that particular range block into four quadrants c c c c cc c cc cc c cover Top-down c

Quadtree Decomposition Partitioning (QD) Grayshade blocks and edge blocks All range and domain blocks are compared with a constant grayshade block. Blocks that exhibit an MSE error less than the user-specified tolerance are classified as grayshade blocks. Remaining blocks are classified as edge blocks. Grayshade range blocks don’t require search for a matching domain block. Grayshade domain blocks are excluded from the domain pool used in the matching process

Quadtree Decomposition Partitioning (QD) QD algorithm may waste time on unnecessary computations during compression of high complex images or low MSE tolerance. Lena, medium complexityBoy, lowest complexity Goldhill, highest complexity Number of grayshade range blocks Number of matched range blocks

Quadtree Recomposition Partitioning (QR) (1997) 1. Partition the image into a set of small range blocks, and all blocks are initially uncovered 2. If a range finds its match, we assumed it uncovered. Remaining ranges are set covered. 3. At the next level, a block is uncovered if its four children are uncoverd. … …… Bottom-up

QD vs. QR Total image area α T The percentage areas covered by large, medium, and small edge range blocks, ρ l,ρ m,ρ s. The percentage of domain blocks which must be compared with large, medium, and small range blocks before an acceptable mach is found, σ l,σ m,σ s. The total domain area for each level, δ l,δ m,δ s. The percentage of large, medium, and small grayshade range blocks, γ R l, γ R m, γ R s. The percentage of large, medium, and small grayshade domain blocks, γ D l, γ D m, γ D s. QD search complexity for each level: Ο (S QD l ) = α T (1 -γ R l ) ∙δ l σ l (1 -γ D l ) Ο (S QD m ) = α T (1 -ρ l -γ R m ) ∙δ m σ m (1 -γ D m ) Ο (S QD s ) = α T (1 -ρ l -ρ m -γ R s ) ∙δ s σ s (1 -γ D s ) DR search complexity for each level: Ο (S QR l ) = α T (1 -γ R s ) ∙δ s σ s (1 -γ D s ) Ο (S QR m ) = α T (1 -ρ s -γ R m ) ∙δ m σ m (1 -γ D m ) Ο (S QR s ) = α T (1 -ρ s -ρ m -γ R l ) ∙δ l σ l (1 -γ D l ) Total search area for range blocks at first level Total search area for domain blocks at first level The range blocks which find matches at first level DiDi

QD vs. QR

Quadtree Decomposition / Recomposition Algorithm (QDR) (1999) QR: some unnecessary computations are used to recompose large or medium size range blocks that are then classified as grayshade blocks A range block that is classified as a grayshade block may have one or more of its quadrants are classified as edge blocks eg. Boy image, MSE = 10 The amount of wasted computations increases as the complexity of the image decreases. QDQR Numbers of grayshade large blocks at level g eg g ggge e

Quadtree Decomposition / Recomposition Algorithm (QDR) (1999) 1. Decompose the image in a top-down fashion in order to find and eliminate all grayshade range blocks at all quadtree levels 2. Use QR algorithm to process all remaining edge-classified range blocks … …

Result No computations are wasted on recomposing grayshade blocks

Result (QD, QR and QDR) Multiplication-accumulation is the total number of multiplications during searching domain block steps by applying MSE < threshold