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The preceding chapters have examined techniques for compressing images using fractal and wavelet approaches. This final chapter will compare these two approaches and discuss the relative advantages of each. The results shown here were generated with the accompanying software. As mentioned previously, this software was developed to illustrate the ideas of the book and was not developed with performance as a primary goal. Also, the systems compared here are not complete compression systems. In particular, there is no entropy coding on the output of the fractal or wavelet algorithms. The presence of entropy coding might alter the results presented here. For example, one or the other of these algorithms might produce output that is more compressible under entropy coding. The results shown here should be used to compare the relative merits of the fractal and wavelet algorithms presented here and should not be compared, for example, to research or commercial quality compression software.
Rate distortion compares the tradeoffs between compression and distortion of the decoded image in lossy compression schemes. Rate is defined as the average number of bits needed to represent each pixel value (Sayood 1996). It is usually expressed as bits per pixel (bpp). Distortion is usually measured in terms of PSNR, although this is not always a good measure of perceived image quality. Rate-distortion curves normally plot bpp versus PSNR. However, the fractal encoding literature more commonly reports rate distortion in terms of compression ratio versus PSNR, rather than bpp versus PSNR. This may be due to the fact that fractal encodings are not tied to an image size in pixels, as are other encoding methods. The discussion that follows will also report rate distortion in terms of compression ratio versus PSNR. For fractal methods, the encoded image size is determined by assuming 4 bytes for each range cell. The compression ratio is determined by dividing the size of the original bitmap image, in bytes, by the number of bytes in the encoded image. Distortion, as measured by PSNR, is determined by decoding the image at the same size as the original bitmap, and comparing the decoded image to the original. For wavelet methods, the encoded image size is the size of the actual binary zerotree file, as discussed in Chapter 7.
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