Glossary

5.5 Wavelet filter banks

8 min read•august 20, 2024

Wavelet filter banks are a powerful tool in signal processing, enabling multiresolution analysis and efficient signal representation. They decompose signals into frequency subbands using high-pass and low-pass filters, allowing for localized time-frequency analysis.

These filter banks offer several advantages over traditional Fourier methods, including better handling of non-stationary signals and transient components. They're widely used in applications like compression, denoising, and feature extraction, making them essential in modern signal processing.

Wavelet filter banks overview

  • Wavelet filter banks are a powerful tool in advanced signal processing that enable multiresolution analysis and efficient representation of signals
  • They decompose a signal into a set of frequency subbands using a series of high-pass and low-pass filters, allowing for localized time-frequency analysis
  • Wavelet filter banks have several key properties and advantages compared to traditional Fourier-based methods, making them suitable for various applications such as compression, denoising, and feature extraction

Key properties of wavelet filter banks

Top images from around the web for Key properties of wavelet filter banks

  • GI - Multiresolution wavelet analysis applied to GRACE range-rate residuals View original

    Is this image relevant?

  • Angular Parameterization of Orthonormal Wavelet Filter Banks in Power Quality: A Method for ... View original

    Is this image relevant?

  • GI - Multiresolution wavelet analysis applied to GRACE range-rate residuals View original

    Is this image relevant?

1 of 3

Top images from around the web for Key properties of wavelet filter banks

  • GI - Multiresolution wavelet analysis applied to GRACE range-rate residuals View original

    Is this image relevant?

  • Angular Parameterization of Orthonormal Wavelet Filter Banks in Power Quality: A Method for ... View original

    Is this image relevant?

  • GI - Multiresolution wavelet analysis applied to GRACE range-rate residuals View original

    Is this image relevant?

1 of 3
  • Provide a time-frequency representation of signals, enabling analysis of both temporal and spectral information simultaneously
  • Offer multiresolution analysis, allowing for examination of signal details at different scales
  • Utilize a set of high-pass and low-pass filters to decompose the signal into frequency subbands
  • Exhibit perfect reconstruction property, ensuring that the original signal can be reconstructed from its wavelet coefficients without loss of information

Comparison to Fourier transform

  • Fourier transform provides frequency information but lacks temporal localization, while wavelet transform offers both frequency and time localization
  • Wavelet transform is better suited for analyzing non-stationary signals and signals with transient components
  • Fourier transform uses a fixed window size, whereas wavelet transform employs a variable-sized window, adapting to different frequency components

Wavelet decomposition

  • Wavelet decomposition is the process of breaking down a signal into its constituent frequency subbands using a wavelet filter bank
  • It involves applying a series of high-pass and low-pass filters to the signal, followed by downsampling, to obtain wavelet coefficients at different scales and orientations
  • The decomposition process is based on the concepts of multiresolution analysis, , and wavelet function

Multiresolution analysis

  • Multiresolution analysis is a mathematical framework that forms the basis for wavelet decomposition
  • It involves representing a signal at multiple scales or resolutions, with each scale providing a different level of detail
  • The signal is decomposed into a coarse approximation and a set of at each scale
  • The capture the low-frequency information, while the detail coefficients capture the high-frequency information

Scaling function

  • The scaling function, denoted as ϕ(t)\phi(t), is a fundamental component of wavelet analysis
  • It is a low-pass filter that generates the approximation coefficients in the wavelet decomposition
  • The scaling function satisfies a two-scale equation, relating its values at different scales
  • Examples of scaling functions include the Haar scaling function and the Daubechies scaling functions

Wavelet function

  • The wavelet function, denoted as ψ(t)\psi(t), is another essential component of wavelet analysis
  • It is a high-pass filter that generates the detail coefficients in the wavelet decomposition
  • The wavelet function is derived from the scaling function and satisfies certain mathematical properties
  • Examples of wavelet functions include the and the Daubechies wavelets

Wavelet reconstruction

  • Wavelet reconstruction is the process of reconstructing the original signal from its wavelet coefficients obtained during decomposition
  • It involves upsampling the wavelet coefficients, applying reconstruction filters, and combining the results to obtain the reconstructed signal
  • The reconstruction process ensures that the original signal can be perfectly reconstructed, provided certain conditions are met

Upsampling in reconstruction

  • Upsampling is a crucial step in wavelet reconstruction, where the wavelet coefficients are interpolated to increase their sampling rate
  • It involves inserting zeros between the wavelet coefficients to expand their length
  • Upsampling is necessary to match the dimensions of the wavelet coefficients with the reconstruction filters

Filtering in reconstruction

  • Filtering in reconstruction involves applying a set of reconstruction filters to the upsampled wavelet coefficients
  • The reconstruction filters are designed to be the inverse of the decomposition filters used in the wavelet decomposition process
  • The low-pass reconstruction filter is applied to the approximation coefficients, while the high-pass reconstruction filter is applied to the detail coefficients

Perfect reconstruction condition

  • Perfect reconstruction is a desirable property of wavelet filter banks, ensuring that the original signal can be reconstructed without loss of information
  • It requires the decomposition and reconstruction filters to satisfy certain mathematical conditions
  • The perfect reconstruction condition ensures that the reconstructed signal is identical to the original signal, up to a possible delay and scaling factor

Subband coding with wavelets

  • Subband coding is a technique used in signal compression and processing, where the signal is divided into frequency subbands
  • Wavelet filter banks are commonly used for subband coding, as they provide an efficient means of decomposing the signal into subbands
  • Subband coding with wavelets involves decomposing the signal using a wavelet filter bank, processing the subbands independently, and then reconstructing the signal

Subband decomposition

  • Subband decomposition using wavelets involves applying a series of high-pass and low-pass filters to the signal, followed by downsampling
  • The signal is decomposed into a set of frequency subbands, each representing a specific range of frequencies
  • The decomposition process can be repeated on the low-frequency subband to obtain a hierarchical representation of the signal

Subband reconstruction

  • Subband reconstruction is the process of reconstructing the original signal from its subband components
  • It involves upsampling the subband coefficients, applying reconstruction filters, and combining the results
  • The reconstruction filters are designed to cancel out the aliasing introduced during the decomposition process

Aliasing cancellation in subband coding

  • Aliasing is a distortion that occurs when the signal is not properly bandlimited before sampling or when the sampling rate is insufficient
  • In subband coding with wavelets, aliasing can be introduced during the decomposition process due to the downsampling operation
  • To achieve perfect reconstruction, the aliasing components introduced in the subbands must be canceled out during the reconstruction process
  • This is achieved through careful design of the decomposition and reconstruction filters, ensuring that the aliasing terms cancel each other out

Wavelet families

  • Wavelet families are sets of wavelets that share common properties and are defined by specific mathematical equations
  • Different wavelet families have distinct characteristics and are suitable for different applications
  • Some popular wavelet families include Haar wavelets, Daubechies wavelets, and biorthogonal wavelets

Haar wavelet

  • The Haar wavelet is the simplest and oldest wavelet family, named after Alfréd Haar
  • It consists of a piecewise constant scaling function and a wavelet function
  • The Haar wavelet has and is orthogonal, making it computationally efficient
  • However, it lacks smoothness and has limited ability to represent smooth functions

Daubechies wavelets

  • Daubechies wavelets, introduced by , are a family of orthogonal wavelets with compact support
  • They are characterized by a maximum number of vanishing moments for a given support width
  • Daubechies wavelets are widely used in signal processing and numerical analysis due to their good approximation properties
  • Examples of Daubechies wavelets include DB2, DB4, and DB6, where the number indicates the number of vanishing moments

Biorthogonal wavelets

  • Biorthogonal wavelets are a class of wavelets where the decomposition and reconstruction filters are not identical, but are biorthogonal to each other
  • They offer more flexibility in design compared to orthogonal wavelets, as the decomposition and reconstruction filters can have different properties
  • Biorthogonal wavelets allow for perfect reconstruction while having different vanishing moments and regularity for the decomposition and reconstruction filters
  • Examples of biorthogonal wavelets include the CDF (Cohen-Daubechies-Feauveau) wavelets and the spline wavelets

Design of wavelet filter banks

  • The design of wavelet filter banks involves selecting appropriate wavelet filters that satisfy certain desired properties
  • Key considerations in wavelet filter bank design include the number of vanishing moments, regularity, and compact support of the wavelets
  • The choice of wavelet filters depends on the specific application and the characteristics of the signal being analyzed

Vanishing moments

  • Vanishing moments are a property of wavelets that determines their ability to represent polynomial signals
  • A wavelet with NN vanishing moments can perfectly represent polynomials up to degree N−1N-1
  • Higher number of vanishing moments leads to better approximation of smooth signals and faster decay of wavelet coefficients
  • However, increasing the number of vanishing moments also increases the support width of the wavelet filters

Regularity of wavelets

  • Regularity refers to the smoothness of the wavelet functions
  • Wavelets with higher regularity are smoother and have better approximation properties for smooth signals
  • Regularity is related to the number of vanishing moments and the decay rate of the wavelet coefficients
  • Smooth wavelets are desirable for applications such as compression and denoising, as they can represent smooth signals more efficiently

Compact support of wavelets

  • Compact support refers to the property of wavelets having a finite duration in the time domain
  • Wavelets with compact support are zero outside a finite interval, making them computationally efficient
  • Compact support is important for local analysis and for implementing wavelet transforms with finite impulse response (FIR) filters
  • Examples of wavelets with compact support include the Haar wavelet and the Daubechies wavelets

Applications of wavelet filter banks

  • Wavelet filter banks have found numerous applications in various fields, including signal processing, image processing, and data analysis
  • They provide a powerful tool for representing and analyzing signals at different scales and resolutions
  • Some notable applications of wavelet filter banks include , denoising, and feature extraction

Image compression with wavelets

  • Wavelet-based image compression techniques exploit the multiresolution representation and compact support of wavelets
  • The image is decomposed using a wavelet filter bank, and the resulting wavelet coefficients are quantized and encoded
  • Wavelets can efficiently represent the local features and discontinuities in images, leading to high compression ratios with good visual quality
  • Examples of wavelet-based image compression standards include JPEG 2000 and DjVu

Denoising with wavelets

  • Wavelet-based denoising techniques leverage the ability of wavelets to separate signal and noise components in different frequency subbands
  • The noisy signal is decomposed using a wavelet filter bank, and the wavelet coefficients are thresholded or modified to suppress noise
  • Wavelets can effectively capture the local features of the signal while minimizing the impact of noise
  • Denoising with wavelets has applications in various domains, such as image denoising, audio denoising, and biomedical signal processing

Feature extraction with wavelets

  • Wavelet filter banks can be used for extracting meaningful features from signals or images
  • The wavelet coefficients obtained from the decomposition process capture the local time-frequency information of the signal
  • Relevant features can be extracted by analyzing the wavelet coefficients at different scales and orientations
  • Wavelet-based feature extraction has applications in pattern recognition, signal classification, and machine learning tasks
  • Examples include texture analysis, edge detection, and audio feature extraction for speech recognition or music classification

Key Terms to Review (18)

Approximation coefficients: Approximation coefficients are the output of the analysis stage in wavelet transformations that capture the low-frequency information of a signal. These coefficients are obtained by applying a low-pass filter to the input signal, effectively representing the smooth, general trend while discarding high-frequency details. This makes them essential for signal compression and reconstruction processes, allowing for a hierarchical decomposition of the signal into various frequency components.
Biorthogonal filter bank: A biorthogonal filter bank is a type of wavelet filter bank that uses two sets of filters—one for analysis and another for synthesis—allowing for perfect reconstruction of the original signal. This system provides greater flexibility in designing wavelets, enabling asymmetrical wavelet shapes which can result in improved signal representation and reduced artifacts in the transformed data.
Compact support: Compact support refers to a function that is non-zero only within a bounded interval or region and is zero outside that region. This property is significant in signal processing because it allows for localized analysis, enabling the efficient representation and manipulation of signals in various transforms, including wavelet transforms, discrete wavelet transforms, and filter banks.
Continuous Wavelet Transform: The continuous wavelet transform (CWT) is a mathematical tool used to analyze signals by breaking them down into wavelets, which are localized oscillatory functions. It allows for the representation of a signal in both time and frequency domains, making it particularly useful for examining non-stationary signals that change over time. The CWT provides a way to visualize how the frequency content of a signal evolves, which is essential in various applications including signal processing, image analysis, and even geophysics.
Daubechies wavelet: The Daubechies wavelet is a family of wavelets that are designed to provide efficient data representation and analysis by offering compact support and orthogonality. They are particularly significant in the context of signal processing, enabling multi-resolution analysis and allowing for precise signal reconstruction through the use of scaling and wavelet functions. These wavelets are essential for discrete wavelet transforms and play a key role in designing wavelet filter banks.
Detail Coefficients: Detail coefficients are the components of a wavelet transform that capture the high-frequency information in a signal. They represent how much detail or fluctuation exists at various scales and are essential for reconstructing the original signal from its transformed representation. In wavelet analysis, these coefficients provide insights into transient features, making them crucial in applications like signal processing and data compression.
Discrete Wavelet Transform: The Discrete Wavelet Transform (DWT) is a mathematical technique used to decompose a signal into its constituent wavelets at different scales and positions, providing both time and frequency localization. This transformation is particularly powerful for analyzing non-stationary signals, allowing for the extraction of temporal features while preserving frequency information. The DWT uses filter banks to process the data, making it essential for various applications in signal processing, image compression, and data analysis.
Dyadic filter bank: A dyadic filter bank is a specific type of multiresolution analysis framework used in signal processing, particularly in wavelet transforms. It operates by recursively splitting a signal into different frequency bands, allowing for the analysis of the signal at multiple scales. This structure enables efficient representation of signals while preserving important details and is particularly useful in applications like image compression and feature extraction.
Haar wavelet: The haar wavelet is the simplest type of wavelet, used for various signal processing applications, particularly for its ability to analyze and represent data with discontinuities. It forms the basis for many wavelet transforms and is characterized by its step-like function, which helps in capturing changes in signals effectively. Haar wavelets are especially important in discrete wavelet transforms and filter banks, as they facilitate efficient data compression and noise reduction techniques.
Image compression: Image compression is the process of reducing the amount of data required to represent a digital image while maintaining its visual quality. This is achieved by removing redundancies and unnecessary information in the image data. Techniques such as wavelet transforms and filter banks are often employed to analyze the image and minimize storage requirements, making image processing and transmission more efficient.
Ingrid Daubechies: Ingrid Daubechies is a renowned mathematician known for her pioneering work in wavelet theory, specifically for developing compactly supported wavelets and their applications in signal processing. Her contributions have revolutionized how signals can be analyzed and processed, leading to advancements in various fields, including image compression and data analysis. Daubechies' wavelets provide efficient ways to represent data at different resolutions, making her work essential in understanding modern wavelet transforms, discrete wavelet transforms, and filter banks.
Mother wavelet: A mother wavelet is a fundamental waveform used in wavelet transform to analyze signals at various scales. It serves as the basis for generating a family of wavelets through scaling and translating operations, making it essential for breaking down complex signals into simpler components while preserving important features. The choice of mother wavelet significantly impacts the analysis results, influencing how effectively different frequencies and time-localized information are captured.
Orthogonality: Orthogonality refers to the property of two functions or vectors being perpendicular to each other in an inner product space, which leads to the idea that their inner product is zero. This concept is crucial in signal processing because it allows for the separation and reconstruction of signals without interference, making it fundamental in analyzing and synthesizing signals using techniques such as wavelet transforms and filter banks.
Scaling Function: A scaling function is a mathematical tool used in wavelet theory to represent signals at different scales, enabling the analysis of data across various resolutions. It plays a crucial role in constructing wavelet bases and decomposing signals into different frequency components, allowing for multi-resolution analysis. This function essentially captures the low-frequency information of a signal, providing a foundation for understanding its structure and behavior across time and frequency.
Signal denoising: Signal denoising is the process of removing noise from a signal to recover the original, cleaner version of the signal. It involves various techniques that enhance the quality of the signal, making it easier to analyze or interpret, while retaining the essential characteristics of the original data. Effective denoising can significantly improve performance in tasks such as feature extraction, classification, and further processing.
Wavelet packet decomposition: Wavelet packet decomposition is a sophisticated method that extends traditional wavelet transforms by allowing for a more detailed analysis of signals through the division of both the approximation and detail coefficients. This technique provides a multi-resolution representation of a signal, enabling efficient signal processing by capturing both high-frequency and low-frequency information. By using a complete binary tree structure, it generates various levels of detail and offers enhanced flexibility in analyzing signals compared to standard wavelet transforms.
Wavelet thresholding: Wavelet thresholding is a signal processing technique used for noise reduction and data compression, leveraging the unique properties of wavelet transforms. It works by manipulating the wavelet coefficients of a signal, applying a threshold to filter out noise while preserving important features of the signal. This technique is particularly effective in removing noise from signals represented in a multi-resolution framework, allowing for enhanced analysis and interpretation.
Yves Meyer: Yves Meyer is a prominent French mathematician recognized for his groundbreaking contributions to the field of wavelet theory, particularly the development of the mathematical framework that supports wavelet transforms. His work significantly advanced the understanding and applications of both continuous and discrete wavelet transforms, influencing various areas such as signal processing, image compression, and data analysis. Meyer's insights into multiresolution analysis have become foundational in constructing wavelet filter banks and in generating scalograms for time-scale representations.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide