ABSTRACT
Introduction ......................................................................................................... 118 Dyadic Multiresolution Analysis ...................................................................... 119 Axioms of Dyadic MRA ..................................................................................... 119
Ladder Axiom ................................................................................................. 119 Axiom of Perfect Reconstruction ................................................................. 121 The Intersection of All the Subspaces is the Trivial Subspace {0} ............ 121 If a Function x(t) ∈ V0 then, x(2it) ∈ Vi, i ∈ ℤ .............................................. 121 Similarly, if x(t) ∈ V0 then x(t − n) ∈ V0, n ∈ ℤ ........................................... 122 Axiom of Orthogonal Basis ........................................................................... 122
Theorem of MRA ................................................................................................. 122 Wavelet-Based Data Compression .................................................................... 124 Fractals and Self-Similar Functions .................................................................. 131
Fractals ............................................................................................................. 131 Self-Similar Functions ......................................................................................... 133 Wavelet Transform of Self-Similar Functions .................................................. 134 Application of Self-Similar Functions .............................................................. 135 Singularities and Noise Removal ...................................................................... 138
Singularities..................................................................................................... 138 Wavelets in Singularity Detection..................................................................... 139
Singularities in a Signal without Noise ....................................................... 139 Singularities in Noisy Signals ....................................................................... 140 Wavelets in Denoising ................................................................................... 141 Thresholding Techniques for Signal Denoising ......................................... 141
Two Case Studies of Singularity Detection ..................................................... 143 One-Dimensional Singularity in ECG Signals ........................................... 143 Two-Dimensional Singularities in Proteomic Images ............................... 148
Acknowledgments .............................................................................................. 153 References ............................................................................................................. 153
As we have seen in previous chapters, the wavelet transform has allowed scientists and engineers to analyze the time-varying and transient phenomena of a signal. The continuous wavelet transform (CWT) is used to measure the similarity between a signal and the analyzing function wavelet at various scales and locations. The CWT compares the signal to the shifted and compressed or stretched versions of the wavelet, also called translations and dilations of the wavelet, respectively. The discrete version of CWT is used to actually calculate the wavelet transform. This discrete version is called the discrete wavelet transform (DWT). Dyadic sampling of the time-frequency plane results in a very efcient algorithm for calculating the DWT. Dyadic multiresolution analysis (MRA) is one of the techniques that help us study how to analyze functions in space L2( ) at different scales. The different scales at which the functions are analyzed are related by powers of two.
