Fractal and Wavelet Image Compression Techniques by Stephen Welstead

By Stephen Welstead

Curiosity in photograph compression for net and different multimedia functions has spurred examine into compression suggestions that might elevate garage functions and transmission pace. This instructional presents a realistic advisor to fractal and wavelet approaches--two options with intriguing capability. it really is meant for scientists, engineers, researchers, and scholars. It offers either introductory info and implementation information. 3 Windows-compatible software program platforms are incorporated in order that readers can discover the recent applied sciences extensive. entire C/C++ resource code is equipped, permitting readers to head past the accompanying software program. The mathematical presentation is out there to complex undergraduate or starting graduate scholars in technical fields.

Contents

- Preface
- Introduction
- Iterated functionality Systems
- Fractal Encoding of Grayscale Images
- rushing Up Fractal Encoding
- basic Wavelets
- Daubechies Wavelets
- Wavelet picture Compression Techniques
- comparability of Fractal and Wavelet photo Compression
- References
- Appendix A: utilizing the Accompanying Software
- Appendix B: application home windows Library (UWL)
- Appendix C: association of the Accompanying software program resource Code

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Extra info for Fractal and Wavelet Image Compression Techniques

Example text

N k Note also that d ( x, f ° ( x )) ≤ d ( x, f ( x )) + d ( f ( x ), f ° ( x )) + ... + d ( f ° k 2 ≤ (1 + s + s + ... + s 1 ≤ d ( x, f ( x )), 1− s 2 k −1 ( k −1) ( x ), f ° ( x )) k )d ( x, f ( x )) where the final inequality follows from the series expansion of (1-s)-1 which is valid here because 0 ≤ s < 1. So, for example, if n < m, we have d ( f ° n ( x ), f ° m ( x )) ≤ sn d ( x, f ( x )). 1− s Since s < 1, the expression on the right approaches 0 as n,m → ∞. In other words, the sequence {f°n(x)} is a Cauchy sequence in (X,d).

3) holds. 3 Contraction mapping theorem for grayscale images Partition the unit square I2 into a collection of range cells {Ri} that tile I2: 46 Fractal Encoding of Grayscale Images I2 = UR , i Ri I R j = ∅ . ~ } be a PIFS such that Let {w i ~ :D → R w i i i for some collection of domains Di ⊂ I2 (the Di’s may overlap, and need not cover I2). Fig. 3 shows this configuration. ~ w i Di Ri ~ maps domain D to range R . The Fig. 3 Transformation w i i i domains may overlap, while the ranges tile the unit square.

Note that f(x), f°2(x), f°3(x),… forms a sequence in X. Suppose f is a contraction mapping with contractivity factor s. Note that d ( f ° ( x ), f ° n (n+k ) ( x )) ≤ s d ( f ° ( n −1) ( x ), f ° ( n + k− 1) ( x )) ≤ s d ( x, f ° ( x )). n k Note also that d ( x, f ° ( x )) ≤ d ( x, f ( x )) + d ( f ( x ), f ° ( x )) + ... + d ( f ° k 2 ≤ (1 + s + s + ... + s 1 ≤ d ( x, f ( x )), 1− s 2 k −1 ( k −1) ( x ), f ° ( x )) k )d ( x, f ( x )) where the final inequality follows from the series expansion of (1-s)-1 which is valid here because 0 ≤ s < 1.

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