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.

1. FRACTAL IMAGE COMPRESSION Guided By Mrs. Sheena S Presented by NithinSinkaran Roll No:57

3. Why Fractal Image Compression

4. Mathematical Background

5. How does it work?

6. Examples

8. Self similarity between object part and whole object

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.

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.

17. Divide each parent block into 4 each blocks, or “child blocks.”

20. Compute affine transform.

21. Store location of parent block and child block, affine transform components, etc .into a file(Fractal code book).

22. Repeat for each child block.

23. Lots of comparisons andcalculations.For 256x256 original image and 16x16 sized parent blocks 241*241 = 58,081 block comparisons.

27. Each range and domain blocks are further divided in to 4 parts

28. Average intensities are calculated for each block

29. Based on the average intensities it falls into any one of the 3 major classes

30. Comparison is done with blocks belonging to similar class only.

31. Speed up factor:8

32. Nearest neighbour search scheme(D. Saupe and U. Freiburg)

33. fractal image compression is equivalent to the multidimensional nearest neighbour search.

34. optimal domain-range pairs is equivalent to solving nearest neighbour problems in a suitable Euclidean space

35. Multi-dimensional nearest neighbor searching operates in logarithmic time

37. Behaves almost like a fractal image

38. It can be zoomed at any magnitude without producing the jagged effect

40. fractal image compression EXAMPLES Image Details (JPEG) Original Size(KB) Compressed Size(KB) Lena 256X256(24bit) 84.3 17.5 Brick 256X256(24bit) 66.2 9.91 45.9 5.91 Leaf 256X256(24bit)

44. Used to create .FIF files from JPEG,PNG etc.. Fractal Imager showing 8:1 zooming of a Leaf. Original image(left), .FIF image (right)

46. Used as a plug-in to software like Adobe Photoshop, Adobe Light room

47. Images can be enlarged up to 1000 times its original sizeGenuine fractals in Adobe Photoshop CS5

48. fractal image compression QUESTIONS ?

49. fractal image compression THANK YOU!