Fractal compression is a lossy compression method for digital images, based on fractals. The method is best suited for textures and natural images, relying on the fact that parts of an image often resemble other parts of the same image.[citation needed] Fractal algorithms convert these parts into mathematical data called "fractal codes" which are used to recreate the encoded image.
10. Based on college theoremLet R²(Hausdorff space) be set of two real numbers, and L be an object Let w1,w2,w3… be some affine transforms which maps the entire image to its subsets W(L) =U wi(L) If distance h(L, U wi(L) )=ɛ (a small value) Then h(L,A)= ɛ/(1-c) where c is the contractility factor, A is the converging abstract set Now L can be approximated to the abstractor A
11. fractal image compression Significance in image compression Fern created using the fractal method(fig 1) The highlighted portion of the fern is similar to the entire Image. Application of different affine transformation on That portion produces the entire fern(fig 2). The fern is self similar The fern creation requires only 28 numbers and can Achieve a large amount of compression. The success of the compression depends on the amount of Self similarity found in that image.
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14. fractal image compression 3.performed the following affine transformation to each block (Di,j)=α Di,j + t0 where α - contrast scaling t0-luminance shift ([−255,255 ]). 4.Compare each domain block with each range block 5.Find Min Σ(Ri,j )m,n-T(Di,j))m,n 6.The transformed domain blockwhich is found to be the best approximation for the current range block is assigned to that range block 7. The coordinates of the domain block along with value of α, t0 describing the transformations. This is what is called the Fractal Code Book
15. fractal image compression Decoding Apply the transformations defined in fractal code book iteratively to some initial image Winit, until the encoded image is retrieved back. The transformation over the whole initial image can be described as follows W1 = h(Winit) W2 = h(W1) W3 = h(W2) ..... = ...... Wn = h(Wn-1) Wn will converge to a good approximation of original image after some iterations. Greater the number of iterations greater will be the decoded similarity.
Here we used similar sized blocks. This reduces the efficiency. For better compression the size of blocks should be non uniform . Hence we use quad tree partitioning
The compression can be further improved by multiple division of child blocks(Quad partition), but increases the number of iterations And comparisons