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affine transformation approximation attractor average basis vectors Chapter classes classification cluster centers codebook blocks coefficients Collage Theorem complete metric space compression ratio compute Contractive Mapping convergence copy machine corresponding decoding method define denote discussed domain pool domain vectors domain-range domains and ranges edges Equation eventually contractive example fidelity fixed point Fixed-Point flag fractal code fractal dimension fractal image compression given grey-scale image compression inference algorithm initial image input image iterated function systems Lenna Lenna image lines matrix max_part method minimize node number of bits number of domains operator optimal original blocks original image orthogonal orthonormal basis output parameters piecewise PIFS code pixel values postprocessing printf PSNR quadrant quadtree partition quantization range block range vectors recursive resolution samples scheme Section self-similar shows Sierpinski triangle signal sm„x square sub-image subset subspace subsquare super-resolution wavelet xsize/2 y_exponent