Fractal Image Compression. Kyle Patel EENG510 Image Processing Final project

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1 Fractal Image Compression Kyle Patel EENG510 Image Processing Final project

2 Introduction Extension of Iterated Function Systems (IFS) for image compression Typically used for creating fractals Images tend have some repetitive segments Why not exploit this self similarity Extends well to binary, grayscale, and color images A form of lossy image compression Often compared to JPEG However, can be decoded to a resolution higher than the original image Allows for absurdly high compression ratios However, rather computationally expensive game/sierp det.gif

3 Introduction FIC takes advantage of portions of the image that, through affine transformations, can be made to look like other regions of the image

4 Introduction To perform FIC: Select an arbitrary number of domain cells, that could overlap, that have block sizes greater than each range cell Partition the image completely into non overlapping range cells Perform a search, find affine transformations wi that would map each Di approximately to each and every Ri Pick the Di and wi pair that best approximates a given Ri and save it in a file Welstead, Stephen. Fractal and Wavelet Image Compression Techniques. Bellingham: SPIE, 1999.

5 Introduction pg To undo FIC: Simply read in your FIC codebook You do have to know the block sizes you used in the FIC step Iteratively apply the transformations to your starting image More iterations the smoother the image, but more time Guaranteed a convergence due to the domain cells being larger than the range cells Contractiveness Theorem and Collage Theorem for IFS

6 Previous Work Many implementations of FIC exist Typically vary by the block encoding step Wavelet Quadtree decomposition, or other adaptive methods Non rectangular decomposition Particle Swarm Optimization Genetic Algorithms Discrete Cosine Transform Number of and type of affine parameters Methods of approximating the affine parameters Tolerances when comparing approximation error Parallel methods Fewer variations in decoding schemes Run length encoding Variable length encoding pg

7 Detailed Description How does one approximate the affine parameters for each Di? One method is to use a least squares approach In this project, the two knobs for grayscale images were contrast (scalar multiplier) and intensity (scalar translation). Could also account for the other operations, such as rotation and shearing min Eqns from: Welstead, Stephen. Fractal and Wavelet Image Compression Techniques. Bellingham: SPIE, 1999.

8 Detailed Description FIC done on 512x512 Lena grayscale image (~262 KB) Non overlapping constant grid size imposed for both range and domain decomposition Various sizes and ratios utilized (R,D) Block sizes: (4,4);(4,8);(4,16);(8;32);(16:32);(32,64); Watching run times and the size of the FIC codebook Stores 3 integers and one floating point number E.g. [Di Dj s o] Observe if the decompressed image behaves as it should compared to other Lena examples in literature Compared two implementations of FIC encode/decode One method used the same block size for range and domain cell, but used a downsampled version of the range image for the domain image The other is the more standard method discussed in this presentation, where the domain and range image are inherently the same size

9 Results For (4,4) case: Runtime of 5 hours Initial encoding file was ~2x the size of source file (~400KB) with no repetition of Di mapping For (16,32) Runtime of 15 minutes Poor image quality Initial encoding file was ~1x the size of the source file(~200kb) with some repetition of Di mapping For (4,8) Runtime of 10 hours Very good image quality Initial encoding file was ~1.5x the size of the source file (~300KB) with some repetition of Di mapping All decodes took ~15 seconds per iteration Literature Results (400x400) (DCT (8,16)) Dhawan, S., A Review of Image Compression and Comparison of its Algorithms, IJECT, vol. 2, Mar

10 Discussion Achieved Two basic implementations of a FIC encoder/decoder Some comparisons between the run time performance Issues Some sort of integer/floating point conversion issue No RLE/VLE on encoded file Decoder has some issues with converging properly

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