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.