〰️Signal Processing Unit 12 – Wavelet Bases and Frames

What's the Deal with Wavelets?

• Wavelets are mathematical functions used for analyzing and representing signals or data
• Provide a way to decompose complex signals into simpler, more manageable components
• Enable multi-resolution analysis, allowing examination of signal details at different scales
• Offer localized analysis in both time and frequency domains, unlike traditional Fourier analysis
  • Captures both frequency and temporal information simultaneously
• Wavelets have compact support, meaning they are non-zero only within a finite interval
  • Allows for efficient representation of signals with discontinuities or sharp changes
• Widely used in various fields such as signal processing, image compression, and data analysis
• Wavelet transforms can be discrete (DWT) or continuous (CWT), depending on the application

The Basics: Wavelets vs. Fourier

• Fourier analysis decomposes signals into sine and cosine waves of different frequencies
  • Provides frequency information but lacks temporal localization
  • Assumes signal is stationary (statistical properties do not change over time)
• Wavelets offer a more flexible and adaptive approach to signal analysis
• Wavelet basis functions are localized in both time and frequency domains
  • Allows for capturing transient or non-stationary features in signals
• Wavelets come in various shapes and sizes (e.g., Haar, Daubechies, Morlet)
  • Can be chosen based on the characteristics of the signal being analyzed
• Wavelet transforms provide a multi-resolution representation of signals
  • Enables analysis at different scales, from coarse to fine details
• Fourier transforms are global, while wavelet transforms are local
  • Wavelets can identify the location of specific frequency components in time

Wavelet Transforms Explained

• Wavelet transforms decompose signals into a set of wavelet coefficients
• Continuous Wavelet Transform (CWT) uses a continuous set of scale and translation parameters
  • Provides a highly redundant representation of the signal
  • Useful for signal analysis and feature extraction
• Discrete Wavelet Transform (DWT) uses a discrete set of scale and translation parameters
  • Provides a non-redundant representation of the signal
  • Commonly used for signal compression and denoising
• DWT is implemented using a filter bank approach
  • Signal is passed through a series of high-pass and low-pass filters
  • Outputs are downsampled at each level, resulting in a multi-resolution decomposition
• Inverse Wavelet Transform (IWT) reconstructs the original signal from wavelet coefficients
• Wavelet transforms can be extended to higher dimensions (e.g., 2D for images)

Types of Wavelets: A Quick Tour

• Haar wavelet: the simplest wavelet, resembles a step function
  • Discontinuous and non-differentiable
  • Used in early wavelet applications due to its simplicity
• Daubechies wavelets: a family of orthogonal wavelets with compact support
  • Designated by the number of vanishing moments (e.g., db4, db6)
  • Widely used in signal and image processing applications
• Morlet wavelet: a complex-valued wavelet with a Gaussian envelope
  • Provides good time-frequency localization
  • Often used in continuous wavelet transform analysis
• Mexican Hat wavelet: a real-valued wavelet derived from the second derivative of a Gaussian function
  • Symmetric and has good time-frequency localization properties
• Coiflets: a family of orthogonal wavelets with additional vanishing moments
  • Designed to have more symmetry than Daubechies wavelets
• Symlets: a modified version of Daubechies wavelets with increased symmetry
• Biorthogonal wavelets: a class of wavelets with separate decomposition and reconstruction filters
  • Allows for more design flexibility and better symmetry properties

Frames: More Than Just a Pretty Picture

• Frames are a generalization of bases in a Hilbert space
• A frame is a set of vectors that span the space, but may be linearly dependent
  • Provides a redundant representation of signals
• Frame bounds $A$ and $B$ satisfy: $A ||f||^2 \leq \sum_{i} |\langle f, \phi_i \rangle|^2 \leq B ||f||^2$
  • $A$ and $B$ are positive constants, $f$ is a signal, and $\phi_i$ are frame elements
• Tight frames have equal frame bounds ($A = B$), providing a more stable representation
• Redundancy in frames allows for more robust signal representation and reconstruction
  • Resilient to noise and data loss
• Frames are used in various applications, such as signal denoising and compressed sensing
• Wavelet frames are constructed by oversampling the wavelet basis functions
  • Provide a redundant multi-resolution representation of signals

Wavelet Bases: Building Blocks of Analysis

• Wavelet bases are sets of functions that form a basis for a function space
• Constructed by dilating and translating a mother wavelet $\psi(t)$ and a scaling function $\phi(t)$
  • Dilation controls the scale (frequency) of the wavelet
  • Translation controls the position (time) of the wavelet
• Orthonormal wavelet bases ensure perfect reconstruction and energy preservation
  • Inner product of basis functions is zero (orthogonal) and norm is one (normalized)
• Biorthogonal wavelet bases use different functions for decomposition and reconstruction
  • Allows for more design flexibility and better symmetry properties
• Wavelet bases can be constructed using the Multiresolution Analysis (MRA) framework
  • Nested sequence of approximation spaces, each spanned by scaled and translated versions of $\phi(t)$
  • Detail spaces capture the differences between successive approximation spaces
• Wavelet bases provide a sparse representation of signals with localized features
  • Useful for signal compression, denoising, and feature extraction

Real-World Applications

• Image compression: wavelets are used in JPEG2000 and other image coding standards
  • Provide better compression performance and quality than traditional DCT-based methods
• Signal denoising: wavelet thresholding techniques remove noise while preserving important features
  • Commonly used in audio, speech, and biomedical signal processing
• Seismic data analysis: wavelets help identify and characterize subsurface features
  • Used in oil and gas exploration and geophysical studies
• Medical imaging: wavelets are used in MRI, CT, and ultrasound image processing
  • Enhance image quality, reduce artifacts, and extract relevant features
• Pattern recognition: wavelet-based features capture multi-scale information for classification tasks
  • Applied in face recognition, handwriting analysis, and texture classification
• Numerical analysis: wavelets are used in solving partial differential equations and integral equations
  • Provide efficient and adaptive discretization schemes
• Data compression: wavelet-based techniques are used in compressing large datasets
  • Applicable in remote sensing, scientific computing, and big data analytics

Coding with Wavelets: Hands-On Practice

• Wavelet toolboxes are available in various programming languages (e.g., MATLAB, Python)
  • Provide functions for wavelet decomposition, reconstruction, and analysis
• Discrete Wavelet Transform (DWT) can be implemented using the

  dwt()

  function
  • Specify the input signal, wavelet type, and number of decomposition levels
• Inverse Discrete Wavelet Transform (IDWT) is performed using the

  idwt()

  function
  • Reconstructs the original signal from wavelet coefficients
• Wavelet denoising involves thresholding the wavelet coefficients
  • Soft thresholding:

    wthresh(coeffs, 's', lambda)

  • Hard thresholding:

    wthresh(coeffs, 'h', lambda)

• Wavelet compression is achieved by retaining only the significant coefficients
  • Set small coefficients to zero and quantize the remaining coefficients
• Wavelet-based feature extraction can be done by computing statistical measures on coefficients
  • Energy, entropy, mean, standard deviation, etc.
• Visualizing wavelet decompositions helps understand the multi-scale structure of signals
  • Use

    wavedec()

    and

    wrcoef()

    functions to compute and reconstruct coefficients at different levels
• Experimenting with different wavelet types and parameters is crucial for optimal performance
  • Choose wavelets based on the characteristics of the signal and the desired analysis