6.1 Wavelet transforms

Wavelet transforms are powerful tools for analyzing signals in both time and frequency domains. They offer a multi-resolution approach, allowing us to extract local features at different scales. This versatility makes wavelets useful for various applications in signal processing and data analysis.

Wavelets come in different families, each with unique properties suited for specific tasks. From the simple Haar wavelet to more complex Daubechies wavelets, these mathematical functions provide flexible ways to represent and manipulate signals efficiently.

Wavelet transform basics

• Wavelet transforms are mathematical tools used to analyze and represent signals or functions in both time and frequency domains simultaneously
• They provide a multi-resolution analysis of the signal, allowing for the extraction of local features at different scales

Continuous wavelet transform

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• Decomposes a signal into a set of scaled and translated versions of a mother wavelet function
• Involves computing the inner product of the signal with the scaled and translated wavelets
• Provides a redundant representation of the signal, as the wavelets are not orthogonal
• Allows for a fine-grained analysis of the signal's time-frequency characteristics (Morlet wavelet, Mexican hat wavelet)

Discrete wavelet transform

• Decomposes a signal into a set of orthogonal wavelet basis functions
• Uses a dyadic grid of scale and translation parameters, resulting in a non-redundant representation
• Efficiently implemented using a hierarchical filter bank structure (Mallat algorithm)
• Commonly used in practical applications due to its computational efficiency and compact representation (image compression, signal denoising)

Wavelet families

• Various wavelet families have been developed to suit different signal characteristics and application requirements
• Each wavelet family has unique properties, such as support size, smoothness, and number of vanishing moments

Haar wavelet

• The simplest and oldest wavelet family, consisting of a single scale and wavelet function
• Haar wavelet is a step function with compact support and one vanishing moment
• Suitable for piecewise constant signals and fast computations
• Lacks smoothness and may introduce artifacts in some applications (edge detection, binary image processing)

Daubechies wavelets

• A family of orthogonal wavelets with compact support and a specified number of vanishing moments
• Daubechies wavelets are constructed to have the highest number of vanishing moments for a given support size
• Increasing the number of vanishing moments leads to smoother wavelets but larger support sizes (db4, db8)
• Widely used in various signal and image processing tasks (compression, denoising, feature extraction)

Biorthogonal wavelets

• A family of wavelets where the analysis and synthesis wavelets are different but form a biorthogonal system
• Allows for the construction of symmetric wavelets, which is not possible with orthogonal wavelets (except for Haar)
• Symmetric wavelets are desirable for some applications, as they avoid phase distortions (image compression, signal reconstruction)
• Examples include the Cohen-Daubechies-Feauveau (CDF) wavelets and the spline wavelets (CDF 9/7, spline 5/3)

Multiresolution analysis

• A mathematical framework that formalizes the concept of analyzing a signal at different scales or resolutions
• Multiresolution analysis consists of a sequence of nested subspaces, each representing the signal at a particular scale

Scaling functions

• Generate a basis for the approximation subspaces in the multiresolution analysis
• Scaling functions are designed to capture the low-frequency or coarse-scale information of the signal
• They satisfy a two-scale relation, which relates the scaling functions at different scales (dilation equation)
• Examples include the Haar scaling function and the Daubechies scaling functions

Wavelet functions

• Generate a basis for the detail subspaces in the multiresolution analysis, which capture the high-frequency or fine-scale information
• Wavelet functions are derived from the scaling functions and satisfy a similar two-scale relation
• They are designed to have zero mean and to be orthogonal or biorthogonal to the scaling functions
• Examples include the Haar wavelet and the Daubechies wavelets

Decomposition and reconstruction

• The process of decomposing a signal into its approximation and detail coefficients using the scaling and wavelet functions
• Decomposition is performed by applying a series of low-pass and high-pass filters followed by downsampling (Mallat algorithm)
• Reconstruction is the inverse process, where the approximation and detail coefficients are upsampled and filtered to obtain the original signal
• Perfect reconstruction is achieved when the analysis and synthesis filters satisfy certain conditions (quadrature mirror filters, biorthogonal filters)

Wavelet properties

• Different wavelet families and individual wavelets within a family possess various mathematical properties that influence their performance and suitability for specific applications
• These properties include vanishing moments, compact support, and regularity

Vanishing moments

• A wavelet has $m$ vanishing moments if its inner product with any polynomial of degree less than $m$ is zero
• Vanishing moments determine the wavelet's ability to represent smooth signals compactly and to suppress polynomial trends
• Wavelets with a higher number of vanishing moments can capture more complex signal features and provide sparser representations
• However, increasing the number of vanishing moments typically increases the support size of the wavelet

Compact support

• A wavelet has compact support if it is non-zero only over a finite interval
• Compact support is desirable for efficient computation and for capturing local signal features
• Wavelets with smaller support sizes lead to sparser representations and faster algorithms
• Examples of wavelets with compact support include the Haar wavelet and the Daubechies wavelets

Regularity

• Regularity refers to the smoothness or differentiability of a wavelet
• Smoother wavelets have higher regularity and can provide better approximations of smooth signals
• Regularity is related to the number of vanishing moments and the decay rate of the wavelet coefficients
• Wavelets with higher regularity are less sensitive to noise and artifacts but may have larger support sizes
• Examples of wavelets with different regularity properties include the Daubechies wavelets and the Symlets

Wavelet applications

• Wavelet transforms have found numerous applications in various fields, including signal processing, image processing, and data analysis
• The multi-resolution and time-frequency localization properties of wavelets make them particularly suitable for certain tasks

Signal denoising

• Wavelet-based denoising techniques exploit the sparsity of wavelet representations to separate signal from noise
• Noisy signals are decomposed using a wavelet transform, and the resulting coefficients are thresholded to remove noise-related components
• The denoised signal is then reconstructed from the thresholded coefficients
• Wavelet denoising is effective for removing Gaussian noise and preserving signal features (wavelet shrinkage, SureShrink)

Image compression

• Wavelet transforms are used in several image compression standards, such as JPEG 2000
• Images are decomposed using a 2D wavelet transform, and the resulting coefficients are quantized and encoded
• The multi-resolution nature of wavelets allows for progressive transmission and scalable compression
• Wavelet-based compression achieves high compression ratios while maintaining good image quality (embedded zerotree wavelet coding, set partitioning in hierarchical trees)

Feature extraction

• Wavelet transforms can be used to extract meaningful features from signals or images for pattern recognition and classification tasks
• The wavelet coefficients at different scales and locations capture local signal characteristics and can serve as discriminative features
• Wavelet-based features are often more robust and informative than traditional Fourier-based features
• Applications include texture analysis, object detection, and biomedical signal classification (wavelet energy features, wavelet packet features)

Wavelet vs Fourier transform

• The wavelet transform and the Fourier transform are both tools for analyzing signals, but they have different properties and are suited for different types of signals
• The Fourier transform decomposes a signal into a sum of sinusoids, while the wavelet transform decomposes a signal into a sum of scaled and translated wavelets

Time-frequency localization

• The Fourier transform provides frequency information but lacks temporal localization, as the basis functions (sinusoids) are infinite in extent
• Wavelets offer good time-frequency localization, as they are localized in both time and frequency domains
• The wavelet transform can capture both the frequency content and the temporal location of signal features
• This property makes wavelets suitable for analyzing non-stationary signals and detecting transient events (EEG analysis, fault detection)

Computational efficiency

• The Fast Fourier Transform (FFT) algorithm enables efficient computation of the Fourier transform, with a complexity of $O(n \log n)$ for a signal of length $n$
• The discrete wavelet transform can be computed using the Mallat algorithm, which has a complexity of $O(n)$
• Wavelet transforms are computationally efficient, especially for sparse signals, due to the hierarchical filter bank structure
• However, the Fourier transform may be more efficient for certain tasks, such as convolution and frequency-domain filtering

Suitability for non-stationary signals

• The Fourier transform assumes that the signal is stationary, meaning that its frequency content does not change over time
• Non-stationary signals, such as speech, music, and biomedical signals, have time-varying frequency components
• Wavelets are well-suited for analyzing non-stationary signals, as they can capture both the time-varying frequency content and the local signal features
• The short-time Fourier transform (STFT) can also be used for non-stationary signals, but it has a fixed time-frequency resolution, while wavelets offer a multi-resolution analysis

Wavelet thresholding

• Wavelet thresholding is a technique used in signal and image denoising to remove noise while preserving important signal features
• It involves applying a threshold to the wavelet coefficients and setting the coefficients below the threshold to zero or shrinking them towards zero

Hard thresholding

• In hard thresholding, wavelet coefficients with absolute values below a given threshold are set to zero, while coefficients above the threshold remain unchanged
• Hard thresholding function: $\delta_H(x) = x \cdot \mathbb{1}(|x| > \lambda)$, where $\lambda$ is the threshold and $\mathbb{1}$ is the indicator function
• Hard thresholding results in a sparse representation of the signal but may introduce discontinuities and artifacts

Soft thresholding

• In soft thresholding, wavelet coefficients with absolute values below a given threshold are set to zero, while coefficients above the threshold are shrunk towards zero by the threshold value
• Soft thresholding function: $\delta_S(x) = \text{sign}(x) \cdot \max(0, |x| - \lambda)$
• Soft thresholding results in a smoother signal representation and avoids discontinuities but may lead to some signal attenuation

Threshold selection methods

• The choice of the threshold value is crucial for the performance of wavelet thresholding
• Various threshold selection methods have been proposed, based on different criteria and assumptions about the noise and signal characteristics
• Universal threshold: $\lambda = \sigma \sqrt{2 \log n}$, where $\sigma$ is the noise standard deviation and $n$ is the signal length (VisuShrink)
• Minimax threshold: minimizes the maximum risk over a class of signals, assuming a certain noise distribution (SureShrink)
• Cross-validation threshold: selects the threshold that minimizes the cross-validation error, estimating the noise level from the data (SURE-based methods)

Wavelet packets

• Wavelet packets are a generalization of the standard wavelet transform that allows for a more flexible decomposition of signals
• In the wavelet packet decomposition, both the approximation and detail coefficients are further decomposed, creating a full binary tree of subspaces

Wavelet packet decomposition

• The wavelet packet decomposition recursively splits both the approximation and detail subspaces using the same set of filters as in the standard wavelet transform
• This results in a tree-structured decomposition, where each node represents a subspace with a specific frequency band and time localization
• The depth of the tree and the number of subspaces can be adapted to the signal characteristics and the desired time-frequency resolution
• Wavelet packets offer a richer signal representation compared to the standard wavelet transform

Best basis selection

• The wavelet packet decomposition generates a large number of possible bases, each corresponding to a different partitioning of the time-frequency plane
• Best basis selection aims to find the optimal basis that provides the most compact or meaningful representation of the signal
• Common criteria for best basis selection include entropy minimization, cost function minimization, and discrimination power maximization
• Examples of best basis selection algorithms include the Coifman-Wickerhauser entropy-based algorithm and the local discriminant basis algorithm

Wavelet packet applications

• Wavelet packets have been applied in various fields, exploiting their flexible time-frequency representation and best basis selection capabilities
• Signal compression: wavelet packets can achieve higher compression ratios than standard wavelets by adaptively selecting the best basis for each signal segment
• Feature extraction and classification: wavelet packet features, such as subband energies or coefficients, can capture discriminative information for pattern recognition tasks
• Denoising and signal enhancement: wavelet packet thresholding can be applied in the best basis domain to effectively remove noise while preserving signal details

Lifting scheme

• The lifting scheme is an alternative approach to constructing and implementing wavelet transforms, based on a spatial interpretation of the wavelet coefficients
• It provides a flexible framework for designing custom wavelets and allows for in-place computation of the wavelet transform

Lifting steps

• The lifting scheme consists of three main steps: split, predict, and update
• Split: the input signal is divided into two disjoint subsets, usually the even and odd indexed samples
• Predict: the samples in one subset are predicted from the samples in the other subset using a prediction operator
• Update: the samples in the other subset are updated using the prediction errors and an update operator to maintain certain properties (e.g., vanishing moments)
• The predict and update steps are repeated for multiple levels of decomposition

Advantages of lifting

• The lifting scheme has several advantages over the traditional filter bank implementation of wavelet transforms:
  • It allows for in-place computation, reducing memory requirements
  • It is computationally efficient, requiring fewer arithmetic operations
  • It provides a simple and flexible way to design custom wavelets adapted to specific signal characteristics or application needs
  • It facilitates the construction of non-linear wavelet transforms and the incorporation of additional properties (e.g., integer-to-integer mapping)

Integer wavelet transforms

• The lifting scheme enables the construction of integer wavelet transforms, where the wavelet coefficients are integers instead of real numbers
• Integer wavelet transforms are useful for lossless compression and for applications that require exact reconstruction and reversibility
• The lifting steps are modified to use integer arithmetic and rounding operators, ensuring that the transform maps integers to integers
• Examples of integer wavelet transforms include the S+P transform, the TS transform, and the integer Haar transform

Boundary handling

• Boundary handling refers to the treatment of the signal boundaries when applying the wavelet transform
• The wavelet transform is defined for infinite-length signals, but in practice, signals have finite length, and the transform must be adapted to handle the boundaries

Periodic extension

• Periodic extension assumes that the signal is periodic and extends it by wrapping around the boundaries
• The signal is treated as if it repeats itself indefinitely in both directions
• Periodic extension is simple to implement and maintains perfect reconstruction, but it may introduce discontinuities at the boundaries
• It is suitable for signals that are naturally periodic or when the boundary effects are not critical

Symmetric extension

• Symmetric extension mirrors the signal at the boundaries, creating a symmetric signal that is twice the length of the original signal
• There are two types of symmetric extension: half-sample symmetric (HSS) and whole-sample symmetric (WSS)
• HSS extension: $x(-i) = x(i-1)$ and $x(n+i) = x(n-i)$, where $n$ is the signal length
• WSS extension: $x(-i) = x(i)$ and $x(n+i) = x(n-i-1)$
• Symmetric extension preserves the continuity of the signal and its derivatives at the boundaries, reducing boundary artifacts

Zero-padding

• Zero-padding extends the signal by adding zeros at the boundaries
• The signal is assumed to be zero outside its original domain
• Zero-padding is simple to implement and does not introduce discontinuities, but it may lead to large wavelet coefficients near the boundaries
• It is suitable when the signal is naturally zero or close to zero at the boundaries or when the boundary effects are not critical
• The number of zeros added depends on the support size of the wavelet and the desired level of decomposition