〰️Signal Processing Unit 9 – Wavelets and Multi-resolution Analysis Intro

What's the Big Deal?

• Wavelets revolutionized signal processing by providing a powerful tool for analyzing non-stationary signals
• Enable multi-resolution analysis, allowing signals to be examined at different scales and resolutions simultaneously
• Offer superior time-frequency localization compared to traditional Fourier analysis, capturing both frequency and temporal information
• Provide sparse representations of signals, leading to efficient compression and denoising techniques
• Wavelets have found widespread applications across various domains, including image processing, audio analysis, and biomedical signal processing
• Serve as a foundation for modern signal processing techniques and have inspired the development of related concepts like wavelet packets and lifting schemes

Key Concepts

• Wavelets are mathematical functions used to analyze and represent signals at different scales and resolutions
• Wavelet transforms decompose a signal into a set of basis functions called wavelets, which are localized in both time and frequency domains
• Scaling function $\phi(t)$ and wavelet function $\psi(t)$ form the building blocks of wavelet analysis
  • Scaling function captures the low-frequency or average behavior of the signal
  • Wavelet function captures the high-frequency or detail information of the signal
• Dilation and translation operations are applied to the wavelet function to generate a family of wavelets at different scales and positions
• Discrete Wavelet Transform (DWT) is a computationally efficient implementation of wavelet analysis, using dyadic scales and translations
• Wavelet coefficients represent the contribution of each wavelet basis function to the original signal

Mathematical Foundations

• Wavelet analysis is built upon the concept of multi-resolution analysis (MRA), which decomposes a signal into a hierarchy of approximation and detail spaces
• MRA is characterized by a sequence of nested subspaces $V_j$ and $W_j$, where $V_j$ represents the approximation space at scale $j$ and $W_j$ represents the detail space at scale $j$
• The scaling function $\phi(t)$ and wavelet function $\psi(t)$ satisfy the two-scale equations:
  • $\phi(t) = \sum_{k} h[k] \phi(2t-k)$
  • $\psi(t) = \sum_{k} g[k] \phi(2t-k)$
• The coefficients $h[k]$ and $g[k]$ are called the scaling and wavelet coefficients, respectively, and form the low-pass and high-pass filters in the wavelet transform
• Orthogonality and biorthogonality conditions ensure perfect reconstruction and efficient computation of the wavelet transform
• Vanishing moments of the wavelet function determine its ability to represent polynomial signals and impact the sparsity of the wavelet coefficients

Types of Wavelets

• Haar wavelet is the simplest wavelet, with a rectangular shape and one vanishing moment
  • Suitable for piecewise constant signals but lacks smoothness
• Daubechies wavelets are a family of orthogonal wavelets with compact support and a specified number of vanishing moments
  • Daubechies 4 (db4) is commonly used in practice, balancing smoothness and support size
• Symlets are a modified version of Daubechies wavelets with increased symmetry
• Coiflets are designed to have vanishing moments for both the scaling and wavelet functions
• Biorthogonal wavelets relax the orthogonality condition, allowing for more degrees of freedom in design
  • Often used in image compression (JPEG2000) due to their symmetric nature and perfect reconstruction property
• Meyer wavelet is an infinitely differentiable wavelet with exponential decay, providing good regularity but lacking compact support

Multi-resolution Analysis Explained

• MRA provides a framework for analyzing signals at different scales and resolutions
• The approximation spaces $V_j$ capture the low-frequency content of the signal at scale $j$, while the detail spaces $W_j$ capture the high-frequency content
• The scaling function $\phi(t)$ and wavelet function $\psi(t)$ form an orthonormal basis for the approximation and detail spaces, respectively
• The wavelet decomposition of a signal $f(t)$ can be expressed as:
  • $f(t) = \sum_{k} c_{j_0}[k] \phi_{j_0,k}(t) + \sum_{j=j_0}^{\infty} \sum_{k} d_j[k] \psi_{j,k}(t)$
  • $c_{j_0}[k]$ are the approximation coefficients at the coarsest scale $j_0$
  • $d_j[k]$ are the detail coefficients at scale $j$
• The wavelet reconstruction formula allows the signal to be perfectly reconstructed from its wavelet coefficients
• The fast wavelet transform algorithm efficiently computes the wavelet coefficients using a filter bank approach, reducing computational complexity to $O(n)$

Applications in Signal Processing

• Denoising: Wavelet thresholding techniques effectively remove noise from signals while preserving important features
  • Soft and hard thresholding methods are commonly used, exploiting the sparsity of wavelet coefficients
• Compression: Wavelet-based compression schemes (JPEG2000, DjVu) achieve high compression ratios by discarding insignificant wavelet coefficients
  • The multi-resolution nature of wavelets allows for progressive transmission and scalable compression
• Feature extraction: Wavelet coefficients serve as discriminative features for pattern recognition and classification tasks
  • Wavelet-based features are used in applications like texture analysis, audio classification, and biomedical signal analysis
• Transient detection: Wavelets' time-frequency localization property makes them suitable for detecting and characterizing transient events in signals
  • Used in fault detection, seismic data analysis, and power quality monitoring
• Multiresolution analysis: Wavelets enable the analysis of signals at different scales, revealing hidden patterns and structures
  • Applied in image fusion, multiscale edge detection, and scale-invariant feature extraction

Pros and Cons

• Pros:
  • Provides excellent time-frequency localization, capturing both frequency and temporal information
  • Enables multi-resolution analysis, allowing signals to be examined at different scales and resolutions
  • Offers sparse representations of signals, leading to efficient compression and denoising
  • Suitable for analyzing non-stationary and transient signals
  • Has a wide range of applications across various domains
• Cons:
  • The choice of wavelet basis can significantly impact the analysis results, requiring careful selection
  • Wavelet coefficients are sensitive to shifts and translations of the input signal
  • The interpretation of wavelet coefficients may not be as intuitive as Fourier coefficients
  • Boundary effects can arise when applying wavelet transforms to finite-length signals
  • Computational complexity can be higher compared to traditional Fourier-based methods

Real-world Examples

• JPEG2000 image compression standard utilizes wavelet transforms to achieve high compression ratios while maintaining image quality
• Wavelet-based denoising is used in medical imaging (MRI, CT) to enhance image quality and improve diagnostic accuracy
• Seismic data analysis employs wavelet transforms to identify and characterize seismic events, aiding in oil and gas exploration
• Audio compression algorithms like WavPack and FLAC use wavelet-based techniques to efficiently compress audio signals
• Wavelet-based methods are used in power quality monitoring to detect and classify power disturbances, ensuring reliable electricity supply
• Biomedical signal processing, such as ECG and EEG analysis, relies on wavelets for denoising, feature extraction, and pattern recognition
• Astronomical image processing utilizes wavelets for noise reduction, image enhancement, and object detection in celestial images
• Wavelet-based techniques are employed in financial data analysis for trend detection, volatility estimation, and risk management