➗Approximation Theory Unit 6 – Wavelets and Multiresolution Analysis

Introduction to Wavelets

• Wavelets are mathematical functions used to analyze and represent signals or data at different scales and resolutions
• Provide a powerful tool for signal processing, image compression, and data analysis across various fields (mathematics, engineering, physics)
• Wavelets are localized in both time and frequency domains, allowing for effective analysis of non-stationary signals
• Offer a multi-scale approach to signal analysis, enabling the capture of both high-frequency details and low-frequency trends
• Wavelet transforms decompose a signal into a set of basis functions called wavelets, which are derived from a single prototype function (mother wavelet) through scaling and translation
• Key properties of wavelets include compact support, vanishing moments, and orthogonality or biorthogonality
• Wavelets have found applications in diverse areas such as image and video compression (JPEG2000), denoising, feature extraction, and numerical analysis

Fourier Analysis vs. Wavelet Analysis

• Fourier analysis decomposes a signal into a sum of sinusoidal functions with different frequencies, providing a frequency-domain representation
• Fourier transform is global, meaning that the basis functions (sinusoids) are not localized in time, making it less suitable for analyzing non-stationary signals
• Wavelet analysis overcomes the limitations of Fourier analysis by using localized basis functions (wavelets) that are both time and frequency-localized
• Wavelets allow for a multi-scale analysis of signals, capturing both high-frequency details and low-frequency trends simultaneously
• Fourier analysis is well-suited for studying stationary signals with periodic or infinite duration, while wavelet analysis is more effective for non-stationary and transient signals
• Wavelet transforms provide a time-frequency representation of signals, enabling the identification of localized features and anomalies
• Wavelet analysis offers better time resolution at high frequencies and better frequency resolution at low frequencies compared to Fourier analysis

Multiresolution Analysis Fundamentals

• Multiresolution Analysis (MRA) is a mathematical framework that provides a systematic way to construct wavelet bases and analyze signals at different scales
• MRA decomposes a signal into a hierarchy of approximation and detail spaces, each associated with a specific resolution level
• The approximation spaces contain the low-frequency information of the signal, while the detail spaces capture the high-frequency information
• MRA is based on the concept of nested subspaces, where each approximation space is contained within the next higher resolution approximation space
  • This nesting property allows for efficient computation and storage of wavelet coefficients
• The scaling function $\phi(t)$ and the wavelet function $\psi(t)$ are the building blocks of MRA
  • The scaling function generates the approximation spaces, while the wavelet function generates the detail spaces
• MRA satisfies certain properties, such as the existence of a scaling function, the two-scale relation, and the orthogonality or biorthogonality of the wavelet basis
• The two-scale relation expresses the scaling function and wavelet function in terms of scaled and translated versions of themselves, enabling efficient computation of wavelet coefficients

Wavelet Bases and Properties

• Wavelet bases are sets of functions that span the space of square-integrable functions $L^2(\mathbb{R})$ and provide a complete representation of signals
• Orthonormal wavelet bases are constructed using the scaling function $\phi(t)$ and the wavelet function $\psi(t)$ through scaling and translation
  • Orthonormality ensures that the basis functions are perpendicular to each other and have unit norm, enabling efficient signal representation and reconstruction
• Biorthogonal wavelet bases relax the orthonormality condition, allowing for more flexibility in wavelet design and better symmetry properties
• Compact support is a desirable property of wavelets, meaning that the wavelet function has a finite duration and is non-zero only within a limited interval
  • Compact support enables efficient computation and localization of signal features
• Vanishing moments of a wavelet refer to the number of polynomial functions up to a certain degree that the wavelet can represent exactly
  • Higher vanishing moments lead to better approximation of smooth signals and faster decay of wavelet coefficients
• Regularity of wavelets measures the smoothness of the wavelet function and affects the decay rate of wavelet coefficients
  • Smoother wavelets with higher regularity are better suited for representing smooth signals
• Symmetry or antisymmetry of wavelets is important for certain applications (image processing) and can be achieved using biorthogonal wavelet bases

Discrete Wavelet Transform

• The Discrete Wavelet Transform (DWT) is a digital signal processing technique that decomposes a signal into a set of wavelet coefficients using a discrete set of scales and translations
• DWT is based on the concept of multiresolution analysis and uses a pair of filters (low-pass and high-pass) to decompose the signal into approximation and detail coefficients
• The low-pass filter captures the low-frequency information (approximation), while the high-pass filter captures the high-frequency information (details) of the signal
• DWT employs a dyadic sampling scheme, where the scale and translation parameters are powers of two, enabling efficient computation and storage
• The forward DWT recursively applies the filtering and downsampling operations to the approximation coefficients, generating a hierarchical decomposition of the signal
• The inverse DWT reconstructs the original signal from the wavelet coefficients by upsampling and filtering the approximation and detail coefficients at each level
• DWT has several advantages over the continuous wavelet transform, including reduced computational complexity and the ability to handle discrete-time signals
• Fast wavelet transform algorithms, such as the Mallat algorithm, have been developed to efficiently compute the DWT and its inverse

Wavelet Families and Their Applications

• Wavelet families are groups of wavelets with similar properties and characteristics, designed for specific applications
• Haar wavelet is the simplest wavelet, consisting of a single scale and translation, and is useful for piecewise constant signal approximation
• Daubechies wavelets are a family of orthogonal wavelets with compact support and a specified number of vanishing moments
  • Daubechies wavelets are widely used in signal and image processing applications (denoising, compression)
• Symlets are a modified version of Daubechies wavelets with increased symmetry, making them suitable for image processing tasks
• Coiflets are another family of orthogonal wavelets with additional vanishing moments for both the scaling and wavelet functions
  • Coiflets are used in numerical analysis and solving partial differential equations
• Biorthogonal wavelets, such as the Cohen-Daubechies-Feauveau (CDF) wavelets, offer more flexibility in terms of symmetry and vanishing moments
  • Biorthogonal wavelets are commonly used in image compression standards (JPEG2000)
• Discrete Meyer wavelet is an orthogonal wavelet with compact support in the frequency domain, providing good frequency localization
• Wavelet families are chosen based on the specific requirements of the application, considering factors such as smoothness, symmetry, and computational efficiency

Computational Aspects of Wavelets

• Efficient algorithms and data structures are crucial for the practical implementation of wavelet transforms and their applications
• The Fast Wavelet Transform (FWT) is an efficient algorithm for computing the DWT and its inverse, leveraging the multiresolution structure of wavelets
  • FWT has a computational complexity of $O(n)$, where $n$ is the length of the input signal
• Lifting scheme is an alternative approach to construct wavelet bases and perform the DWT, offering increased flexibility and reduced computational complexity
  • Lifting consists of three steps: split, predict, and update, allowing for in-place computation of wavelet coefficients
• Wavelet packet transforms extend the standard DWT by recursively applying the filtering operations to both the approximation and detail coefficients
  • Wavelet packets provide a more flexible and adaptive signal representation, enabling better frequency resolution and feature extraction
• Boundary handling techniques, such as periodic extension or symmetric extension, are employed to deal with finite-length signals and avoid artifacts at the signal boundaries
• Thresholding methods, such as hard thresholding and soft thresholding, are used to denoise signals by setting small wavelet coefficients to zero or shrinking their values
• Quantization and encoding schemes are applied to wavelet coefficients for efficient storage and transmission, particularly in compression applications
• Parallel and distributed algorithms have been developed to accelerate wavelet-based computations on multi-core processors and distributed systems

Advanced Topics and Current Research

• Wavelet-based methods have been extended and applied to various domains beyond signal and image processing, leading to active research areas
• Nonlinear wavelet transforms, such as the Nonlinear Wavelet Transform (NWT) and the Empirical Mode Decomposition (EMD), have been developed to handle non-stationary and nonlinear signals
  • These transforms adapt to the local characteristics of the signal and provide more accurate time-frequency representations
• Multidimensional wavelet transforms extend the concept of wavelets to higher dimensions, enabling the analysis and processing of multidimensional signals (images, videos, volumetric data)
  • Separable and non-separable multidimensional wavelet bases have been constructed to capture spatial and temporal dependencies
• Wavelet-based methods have been applied to solve partial differential equations (PDEs) and integral equations, leveraging the multiresolution properties of wavelets
  • Wavelet-Galerkin methods and wavelet collocation methods have been developed for efficient numerical solutions of PDEs
• Compressed sensing is a paradigm that exploits the sparsity of signals in a wavelet domain to recover them from undersampled measurements
  • Wavelets provide a suitable basis for sparse signal representation, enabling efficient compression and reconstruction
• Wavelet-based machine learning and deep learning approaches have been explored, using wavelets as feature extractors or incorporating wavelet transforms into neural network architectures
  • Wavelet-based methods have shown promise in tasks such as classification, regression, and anomaly detection
• Wavelet scattering networks are a class of deep learning models that use wavelet transforms as a feature extractor, providing invariance to translations and deformations
• Ongoing research focuses on the development of new wavelet bases, the integration of wavelets with other mathematical tools (shearlets, curvelets), and the application of wavelets to emerging domains (quantum computing, computational biology)