6.4 Wavelet transform

Wavelet transform is a powerful tool for analyzing signals in both time and frequency domains. It decomposes signals into wavelets, allowing for multi-resolution analysis and making it ideal for studying non-stationary signals in advanced signal processing.

This topic covers continuous and discrete wavelet transforms, wavelet families, and applications like signal compression and denoising. It explores how wavelets provide localized time-frequency analysis, offering advantages over traditional Fourier transforms for certain types of signals.

Wavelet transform basics

• Wavelet transform is a powerful tool for analyzing signals in both time and frequency domains simultaneously, making it well-suited for studying non-stationary signals in advanced signal processing applications
• Wavelet transform decomposes a signal into a set of basis functions called wavelets, which are localized in both time and frequency, allowing for multi-resolution analysis of the signal

Wavelet definition and properties

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• Wavelets are oscillatory, finite-duration functions with zero mean and varying frequency content
• Key properties of wavelets include:
  • Admissibility: Wavelets must satisfy certain mathematical conditions to ensure perfect reconstruction of the original signal
  • Regularity: Smooth wavelets lead to better frequency localization and sparse representation of signals
  • Vanishing moments: Higher vanishing moments result in better compression and denoising performance
• Examples of popular wavelets: Haar, Daubechies, Symlets, Coiflets

Continuous vs discrete wavelets

• Continuous wavelets are defined over a continuous range of scales and translations, providing a highly redundant representation of the signal
  • Useful for signal analysis and feature extraction
• Discrete wavelets are defined over a discrete set of scales and translations, forming an orthonormal basis for the signal space
  • Enables efficient computation and perfect reconstruction of the signal
  • Commonly used in compression and denoising applications

Wavelet families and types

• Wavelet families are groups of wavelets with similar properties and characteristics, such as support size, symmetry, and vanishing moments
• Common wavelet families include:
  • Daubechies wavelets: Orthogonal wavelets with compact support and maximal number of vanishing moments for a given support size
  • Biorthogonal wavelets: Pairs of scaling and wavelet functions that form biorthogonal bases, allowing for symmetric wavelets and perfect reconstruction
  • Gaussian wavelets: Derivatives of the Gaussian function, providing optimal time-frequency resolution but lacking compact support
• Wavelet types can be further categorized based on their properties, such as orthogonality, symmetry, and regularity

Continuous wavelet transform (CWT)

• CWT is a linear transformation that represents a signal as a weighted sum of scaled and translated versions of a mother wavelet, providing a continuous-time, multi-resolution analysis of the signal
• CWT is particularly useful for analyzing non-stationary signals and extracting time-frequency features in advanced signal processing applications

CWT definition and formula

• The CWT of a signal $x(t)$ with respect to a mother wavelet $\psi(t)$ is defined as: $W_x(a,b) = \frac{1}{\sqrt{|a|}} \int_{-\infty}^{\infty} x(t) \psi^* \left(\frac{t-b}{a}\right) dt$

where:

• $a$ is the scale parameter, controlling the width of the wavelet
• $b$ is the translation parameter, controlling the position of the wavelet
• $\psi^*(t)$ is the complex conjugate of the mother wavelet
• The scale parameter $a$ is inversely related to the frequency content of the wavelet, with smaller scales corresponding to higher frequencies and larger scales corresponding to lower frequencies

CWT scalogram and interpretation

• The scalogram is a visual representation of the CWT, displaying the absolute value of the wavelet coefficients as a function of scale and translation
• Scalogram interpretation:
  • Bright regions indicate high energy content at specific scales and translations
  • Vertical patterns suggest transient or localized events in the signal
  • Horizontal patterns indicate persistent frequency components
• The scalogram provides valuable insights into the time-frequency characteristics of the signal, making it a powerful tool for advanced signal processing tasks

CWT vs Fourier transform

• Fourier transform provides a global frequency-domain representation of the signal, assuming stationarity and losing temporal information
• CWT offers a local time-frequency analysis, capturing both frequency content and temporal variations in the signal
• CWT is better suited for analyzing non-stationary signals and extracting time-localized features, while Fourier transform is more appropriate for stationary signals and global frequency analysis

Discrete wavelet transform (DWT)

• DWT is a compact representation of the signal that captures both frequency and temporal information, making it a powerful tool for signal compression, denoising, and feature extraction in advanced signal processing
• DWT decomposes the signal into a set of orthogonal wavelet basis functions at discrete scales and translations, enabling efficient computation and perfect reconstruction

DWT definition and formula

• The DWT of a signal $x[n]$ is computed by passing it through a series of low-pass and high-pass filters followed by downsampling, resulting in a set of approximation and detail coefficients at each level of decomposition
• The decomposition formulas for the approximation ($a_j[k]$) and detail ($d_j[k]$) coefficients at level $j$ are: $a_j[k] = \sum_{n} x[n] h_j[2k-n]$ $d_j[k] = \sum_{n} x[n] g_j[2k-n]$

where:

• $h_j[n]$ is the low-pass filter at level $j$
• $g_j[n]$ is the high-pass filter at level $j$
• The reconstruction formula for the original signal $x[n]$ from the approximation and detail coefficients is: $x[n] = \sum_{k} a_J[k] \tilde{h}_J[n-2k] + \sum_{j=1}^J \sum_{k} d_j[k] \tilde{g}_j[n-2k]$

where:

• $\tilde{h}_j[n]$ is the reconstruction low-pass filter at level $j$
• $\tilde{g}_j[n]$ is the reconstruction high-pass filter at level $j$
• $J$ is the total number of decomposition levels

Multiresolution analysis with DWT

• Multiresolution analysis (MRA) is a framework for constructing and analyzing wavelet bases, representing the signal at multiple scales and resolutions
• Key components of MRA:
  • Scaling function $\phi(t)$: A low-pass function that captures the coarse-scale approximation of the signal
  • Wavelet function $\psi(t)$: A high-pass function that captures the fine-scale details of the signal
• MRA properties:
  • Nested subspaces: The approximation subspaces at different scales are nested, forming a multiresolution hierarchy
  • Orthogonality: The detail subspaces at different scales are orthogonal to each other and to the approximation subspaces
• DWT performs MRA by iteratively decomposing the approximation coefficients, resulting in a tree-like structure of wavelet coefficients

DWT decomposition and reconstruction

• DWT decomposition involves iteratively applying low-pass and high-pass filters followed by downsampling to compute the approximation and detail coefficients at each level
  • Approximation coefficients represent the low-frequency content and coarse-scale features of the signal
  • Detail coefficients capture the high-frequency content and fine-scale details of the signal
• DWT reconstruction is the inverse process, upsampling the coefficients and applying reconstruction filters to obtain the original signal
  • Perfect reconstruction is achieved when the decomposition and reconstruction filters satisfy certain conditions, such as orthogonality or biorthogonality

Wavelet filter banks and coefficients

• Wavelet filter banks are a set of low-pass and high-pass filters used in the DWT decomposition and reconstruction processes
• Properties of wavelet filter banks:
  • Orthogonality or biorthogonality: Ensures perfect reconstruction and energy preservation
  • Finite impulse response (FIR): Guarantees stability and linear phase
  • Vanishing moments: Controls the smoothness of the wavelet and the sparsity of the coefficients
• Wavelet coefficients are the output of the filter banks, representing the signal's content at different scales and translations
  • Approximation coefficients: Low-frequency content and coarse-scale features
  • Detail coefficients: High-frequency content and fine-scale details
• The number and distribution of wavelet coefficients depend on the chosen wavelet family, filter length, and decomposition level

Wavelet packet transform (WPT)

• WPT is an extension of the DWT that provides a more flexible and adaptive representation of the signal by allowing decomposition of both approximation and detail coefficients at each level
• WPT offers a richer analysis of the signal's time-frequency content, making it suitable for advanced signal processing tasks such as feature extraction, pattern recognition, and signal compression

WPT vs DWT

• DWT only decomposes the approximation coefficients at each level, resulting in a fixed time-frequency resolution
  • Suitable for signals with primarily low-frequency content or stationary characteristics
• WPT decomposes both approximation and detail coefficients, creating a complete binary tree of wavelet packet nodes
  • Provides a more balanced and adaptive time-frequency representation
  • Allows for better analysis of signals with significant high-frequency content or non-stationary behavior
• WPT offers greater flexibility in choosing the best basis for a specific signal or application, leading to improved performance in compression, denoising, and feature extraction

WPT decomposition tree

• WPT decomposition tree is a complete binary tree structure representing the wavelet packet coefficients at different levels and frequency bands
• Each node in the tree corresponds to a specific wavelet packet function, characterized by its scale, frequency band, and position
• The root node represents the original signal, while the leaf nodes contain the wavelet packet coefficients at the finest scale and narrowest frequency bands
• The depth of the tree determines the maximum level of decomposition and the time-frequency resolution of the WPT representation
• Pruning the WPT tree based on a chosen criterion (e.g., energy, entropy, or cost function) leads to an adaptive, signal-dependent decomposition

Best basis selection in WPT

• Best basis selection is the process of finding the optimal subset of wavelet packet nodes that best represents the signal according to a specific criterion or cost function
• Common criteria for best basis selection:
  • Shannon entropy: Minimizes the entropy of the wavelet packet coefficients, leading to a sparse and informative representation
  • Logarithmic energy: Maximizes the energy concentration of the wavelet packet coefficients, suitable for signal compression and denoising
  • Discriminant measures: Maximizes the separability between signal classes, useful for pattern recognition and classification tasks
• Best basis selection algorithms, such as the Coifman-Wickerhauser algorithm, efficiently search the WPT tree to find the optimal basis
• The selected best basis provides an adaptive, signal-dependent representation that captures the most relevant time-frequency features of the signal

Wavelet thresholding and denoising

• Wavelet thresholding is a powerful technique for removing noise from signals by applying a threshold to the wavelet coefficients and setting small coefficients to zero
• Wavelet denoising exploits the sparsity of the wavelet representation, assuming that signal information is concentrated in a few large coefficients while noise is spread across many small coefficients

Soft vs hard thresholding

• Hard thresholding sets all wavelet coefficients below a given threshold to zero and leaves the remaining coefficients unchanged
  • Produces a sharp cutoff and may introduce artifacts in the denoised signal
  • Defined as: $\hat{x}_H = x \cdot I(|x| > \lambda)$, where $\lambda$ is the threshold and $I(\cdot)$ is the indicator function
• Soft thresholding shrinks the wavelet coefficients towards zero by the threshold value, resulting in a smooth transition and reducing the risk of artifacts
  • Defined as: $\hat{x}_S = \text{sign}(x) \cdot \max(0, |x| - \lambda)$
  • Often preferred over hard thresholding due to its better statistical properties and smoother results
• The choice between soft and hard thresholding depends on the signal characteristics, noise level, and desired trade-off between noise reduction and signal preservation

VisuShrink and SureShrink methods

• VisuShrink is a universal threshold selection method that uses a fixed threshold based on the noise variance and the number of samples
  • Threshold is defined as: $\lambda = \sigma \sqrt{2 \log N}$, where $\sigma$ is the noise standard deviation and $N$ is the signal length
  • Ensures a high probability of removing all noise coefficients but may oversmooth the signal
• SureShrink is an adaptive threshold selection method based on Stein's Unbiased Risk Estimate (SURE)
  • Minimizes an estimate of the mean-squared error (MSE) between the true signal and the denoised signal
  • Threshold is chosen independently for each subband, adapting to the signal and noise characteristics
  • Provides a better balance between noise reduction and signal preservation compared to VisuShrink
• Both VisuShrink and SureShrink are widely used in wavelet-based denoising applications, with SureShrink often providing superior performance due to its adaptivity

Wavelet-based noise reduction applications

• Wavelet denoising has found numerous applications in various fields, including:
  • Image denoising: Removing Gaussian, Poisson, or speckle noise from digital images while preserving edges and details
  • Audio denoising: Enhancing speech signals by removing background noise, hum, or artifacts
  • Biomedical signal processing: Denoising ECG, EEG, or fMRI signals to improve diagnostic accuracy and feature extraction
  • Seismic data processing: Removing noise from seismic signals to enhance the interpretation of subsurface structures
  • Financial data analysis: Denoising economic or stock market time series to improve trend analysis and prediction
• The success of wavelet-based noise reduction in these applications stems from its ability to efficiently represent and separate signal and noise components in the time-frequency domain

Wavelet-based signal compression

• Wavelet-based signal compression exploits the sparsity and multi-resolution properties of the wavelet transform to efficiently represent and compress signals
• By concentrating signal energy into a few large coefficients and discarding or quantizing small coefficients, wavelet compression achieves high compression ratios while maintaining acceptable signal quality

Wavelet transform in JPEG2000

• JPEG2000 is an image compression standard that uses the discrete wavelet transform (DWT) as its core technology
• Key features of JPEG2000 wavelet compression:
  • Dyadic decomposition: The image is decomposed into a multi-resolution representation using the DWT, typically with 5-6 levels
  • Quantization: Wavelet coefficients are quantized using a uniform or adaptive quantizer to reduce the bit depth and achieve compression
  • Entropy coding: Quantized coefficients are encoded using context-adaptive binary arithmetic coding (CABAC) to further compress the data
  • Quality scalability: The compressed bitstream can be truncated at various points to obtain different quality levels or compression ratios
• JPEG2000 offers superior compression performance, scalability, and error resilience compared to the original JPEG standard, making it suitable for high-quality image applications

Embedded zerotree wavelet (EZW) coding

• EZW is a pioneering wavelet-based image compression algorithm that exploits the self-similarity and spatial correlation of wavelet coefficients across scales
• Key concepts in EZW coding:
  • Zerotree: A tree-like structure in the wavelet decomposition where all coefficients below a certain threshold are insignificant and can be represented by a single symbol
  • Significance map: A binary map indicating the positions of significant coefficients at each iteration
  • Successive approximation: The coefficients are encoded in multiple passes, progressively refining the quantization and improving the image quality
• EZW encoding steps:
  1. Perform wavelet decomposition of the image
  2. Initialize the threshold based on the maximum coefficient magnitude
  3. Scan the coefficients in a predefined order and encode their significance, sign, and refinement information using zerotrees and the significance map
  4. Update the threshold and repeat the process until the desired bitrate or quality is achieved
• EZW provides an efficient and embedded coding scheme that allows for progressive transmission and reconstruction of images

Set partitioning in hierarchical trees (SPIHT)

• SPIHT is an advanced wavelet-based image compression algorithm that improves upon the ideas of EZW by using a more efficient set partitioning and coding strategy
• Key features of SPIHT:
  • Spatial orientation trees: A hierarchical tree structure that groups wavelet coefficients across scales based on their spatial relationship
  • Set partitioning: The coefficients are partitioned into sets based on their significance and encoded using a recursive set splitting algorithm
  • Ordered bit plane coding: The coefficients are encoded bitplane by bitplane, from most significant to least significant, allowing for progressive refinement
  • Embedded bitstream: The compressed data is organized in an embedded manner, enabling rate or quality scalability
• SPIHT encoding steps:
  1. Perform wavelet decomposition of the image
  2. Initialize the lists of significant and insignificant coefficients
  3. Encode the coefficients using the set partitioning and ordered bit plane coding strategies
  4. Update the lists and repeat the process