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- 1. FRACTAL IMAGE COMPRESSION<br />Guided By<br />Mrs. Sheena S<br />Presented by<br />NithinSinkaran<br />Roll No:57<br />
- 2. fractal image compression<br />Overview<br /><ul><li>Basics of Fractal image compression
- 3. Why Fractal Image Compression
- 4. Mathematical Background
- 5. How does it work?
- 6. Examples
- 7. Possible Improvements</li></li></ul><li>fractal image compression<br />Some Important facts<br /><ul><li> Fractals have infinite details at every point
- 8. Self similarity between object part and whole object
- 9. Fractals are generated by iteratively applying transformation function to a region of space(initiator).</li></ul>Fractal image compression is NOT the compression of fractals<br /><ul><li>It uses the properties of fractals for compressing images.</li></li></ul><li>fractal image compression<br />Basics<br /><ul><li>Proposed by M.Barnsley
- 10. Based on college theorem</li></ul>Let R²(Hausdorff space) be set of two real numbers, and L be an object <br /> Let w1,w2,w3… be some affine transforms which maps<br /> the entire image to its subsets<br />W(L) =U wi(L)<br />If distance h(L, U wi(L) )=ɛ<br />(a small value)<br />Then h(L,A)= ɛ/(1-c) where c is the contractility factor, A is the converging abstract set<br />Now L can be approximated to the abstractor A<br />
- 11. fractal image compression<br />Significance in image compression<br />Fern created using the fractal method(fig 1)<br />The highlighted portion of the fern is similar to the entire <br />Image. Application of different affine transformation on <br />That portion produces the entire fern(fig 2).<br />The fern is self similar<br />The fern creation requires only 28 numbers and can <br />Achieve a large amount of compression.<br />The success of the compression depends on the amount of<br />Self similarity found in that image.<br />
- 12. fractal image compression<br />Limitation of basic theory<br /><ul><li>All images are not self similar
- 13. The direct application of affine transforms to the whole set L(image)</li></ul> will not always maps to its subsets due to lack of self similarities.<br /><ul><li>Now the affine transforms are applied not to L, but to the part of L</li></ul> Or subset of L and tries to map them to self similar parts of the <br /> Same image<br /><ul><li>This method can be used for compressing any common image</li></li></ul><li>fractal image compression<br />Jacquin’s Compression Algorithm<br />Suppose that we are dealing with a 128 x 128 image in(called Range image).<br />We then reduce by averaging (down sampling and low pass-filtering) the original image to 64 x 64(Domain Image).<br />Partitioned both images into blocks 4x4 pixels.<br />
- 14. fractal image compression<br />3.performed the following affine transformation to each block<br />(Di,j)=α Di,j + t0<br /> where α - contrast scaling<br />t0-luminance shift ([−255,255 ]).<br />4.Compare each domain block with each range block<br />5.Find Min Σ(Ri,j )m,n-T(Di,j))m,n<br />6.The transformed domain blockwhich is found to be the best approximation<br /> for the current range block is assigned to that range block<br />7. The coordinates of the domain block along with value of α, t0 describing <br />the transformations. This is what is called the Fractal Code Book<br />
- 15. fractal image compression<br />Decoding<br />Apply the transformations defined in fractal code book iteratively to some initial image Winit, until the encoded image is retrieved back.<br />The transformation over the whole initial image can be described as follows<br />W1 = h(Winit)<br />W2 = h(W1)<br />W3 = h(W2)<br />..... = ......<br />Wn = h(Wn-1)<br />Wn will converge to a good approximation of original image after some iterations.<br />Greater the number of iterations greater will be the decoded similarity.<br />
- 16. fractal image compression<br />Quad-tree partition method<br /><ul><li> Take a starting image and divide it into small, non-overlapping, square blocks, typically called “parent blocks”.
- 17. Divide each parent block into 4 each blocks, or “child blocks.”
- 18. Compare each child block against a subset of all possible parent blocks.</li></ul>(Need to reduce the size of the parent to allow the comparison to work.)<br /><ul><li>Determine which larger block has the lowest difference, according to some measure, between it and the child block.
- 19. Calculate a grayscale transform to match intensity levels between large block and child block precisely. Typically an affine transform is used (w*x = a*x + b) to match grayscale levels.</li></li></ul><li>fractal image compression<br />Encoding<br /><ul><li> Upper left corner child block, very similar to upper right parent block.
- 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.</li></ul>For 256x256 original image and 16x16 sized parent blocks<br />241*241 = 58,081 block comparisons.<br />
- 24. fractal image compression<br />Decoding<br /><ul><li> Use any blank starting image of same size as original image
- 25. For each child block apply stored transforms against specified</li></ul> transform block<br /><ul><li> Overwrite child block pixel values with transform block pixel </li></ul> values<br /><ul><li> Repeat until acceptable image quality is reached.</li></ul>Original image<br />Initial image<br />First Iteration<br />Second Iteration<br />Tenth Iteration<br />Fifth iteration<br />
- 26. fractal image compression<br />Compression of color images<br /><ul><li>Color images are often built of several stacked color channels, each of them </li></ul>representing value levels of the given channel.<br />For example, RGB images are composed of three independent channels for red, green and blue primary color components.<br />Here RGB image is separated <br />Into 3 color channels<br />Their gray scale equivalents<br />Are shown in the right side<br />The reverse transformation<br />Is also possible<br /><ul><li> These gray scale parts can be compressed separately</li></li></ul><li>fractal image compression<br />SPEED-UP TECHNIQUES<br /><ul><li>Boss, Fisher and Jacob's scheme
- 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
- 36. Speed up factor:1.3 up to 11.5</li></li></ul><li>fractal image compression<br />Properties<br /><ul><li>A fractal compressed image is actually stored as domain blocks and their</li></ul>Transformations. Its now resolution independent.<br /><ul><li>Can be de compressed into any resolution needed without loosing much details.
- 37. Behaves almost like a fractal image
- 38. It can be zoomed at any magnitude without producing the jagged effect
- 39. ‘ .FIF ’(Fractal Image File) is a commonly used format for fractal compressed images.</li></li></ul><li>fractal image compression<br />Zoom Using Fractal Decoding<br />Original image zooming<br />2x<br />2x<br />4x<br />4x<br />
- 40. fractal image compression<br />EXAMPLES<br /> Image Details (JPEG) Original Size(KB) Compressed Size(KB)<br />Lena<br />256X256(24bit)<br />84.3 17.5<br />Brick<br />256X256(24bit)<br />66.2 9.91<br />45.9 5.91<br />Leaf<br />256X256(24bit)<br />
- 41. fractal image compression<br />Application<br />Image enlargements<br />Automated Semiconductor Defect Detection<br /><ul><li>semiconductor devices always follows a definite patter. Fractal coding can be used to identify any variation from the self similar part of the image(defects).</li></ul>3. Image enhancement <br /><ul><li>Images have a large amount of affine redundancy
- 42. Any damaged part can be mapped into some other parts of the same image</li></ul>4.Texture compression<br /><ul><li>Textures are self similar images.
- 43. High compression can be achieved. </li></li></ul><li>fractal image compression<br />Some soft wares<br />Fractal Imager<br /><ul><li>developed by iterated systems Inc
- 44. Used to create .FIF files from JPEG,PNG etc.. </li></ul>Fractal Imager showing 8:1<br /> zooming of a Leaf.<br />Original image(left),<br /> .FIF image (right)<br />
- 45. fractal image compression<br />2. Genuine Fractals<br /><ul><li> Developed by onOne software Inc.
- 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 size</li></ul>Genuine fractals in Adobe Photoshop CS5<br />
- 48. fractal image compression<br />QUESTIONS ?<br />
- 49. fractal image compression<br />THANK YOU!<br />

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