11.2 Wavelet and Scaling Coefficients

Wavelet and scaling coefficients are key components of the Discrete Wavelet Transform. They capture high-frequency details and low-frequency trends in signals, respectively. Understanding their meaning and significance is crucial for effective signal analysis and processing.

Computing these coefficients involves applying high-pass and low-pass filters to the signal, followed by downsampling. The choice of wavelet function and number of decomposition levels affects the resulting coefficients' characteristics, influencing their ability to represent different signal features.

Wavelet Coefficients: Meaning and Significance

Physical Interpretation of Wavelet Coefficients

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• Wavelet coefficients represent the correlation between the analyzed signal and the wavelet function at different scales and translations
  • Capture the high-frequency, detail information of the signal
  • The magnitude of wavelet coefficients indicates the strength of the signal's similarity to the wavelet function at a particular scale and translation
    • Large coefficients suggest a strong presence of the corresponding wavelet pattern in the signal (sharp edges, transient features)
  • The sign of wavelet coefficients indicates the phase or orientation of the wavelet function relative to the signal
    • Positive coefficients suggest a positive correlation (signal matches the wavelet shape)
    • Negative coefficients suggest an inverted correlation (signal is inverted relative to the wavelet shape)

Significance of Wavelet Coefficient Location and Scale

• The location of significant wavelet coefficients in the time-scale plane provides information about the temporal location and scale of the signal's features or singularities
  • Time location of coefficients corresponds to the position of signal features (spikes, discontinuities)
  • Scale of coefficients relates to the frequency content or resolution of signal features
    • Finer scales capture high-frequency details (sharp transitions)
    • Coarser scales capture low-frequency trends (smooth variations)
• Scaling coefficients at each scale represent a coarse approximation of the signal, capturing its overall trend or average behavior
  • Provide a low-resolution representation of the signal
  • Useful for analyzing the signal's global characteristics or long-term behavior (baseline, slow variations)

Computing Wavelet Coefficients

Discrete Wavelet Transform (DWT) Process

• The discrete wavelet transform (DWT) decomposes a signal into wavelet and scaling coefficients using a hierarchical, multi-resolution approach
  • Applies a series of high-pass and low-pass filters to the signal, followed by downsampling at each level of decomposition
  • High-pass filter is derived from the wavelet function and captures the high-frequency information, resulting in the wavelet coefficients
  • Low-pass filter is derived from the scaling function and captures the low-frequency information, resulting in the scaling coefficients
  • Filtering and downsampling process is repeated iteratively on the scaling coefficients to obtain wavelet and scaling coefficients at multiple scales

Wavelet Function and Filter Choice

• The choice of wavelet function and its associated filters affects the characteristics and properties of the resulting coefficients
  • Different wavelet families have distinct properties (symmetry, smoothness, vanishing moments)
  • Commonly used wavelet functions include Haar, Daubechies, Symlets, and Coiflets
  • The number of vanishing moments of a wavelet determines its ability to represent polynomial trends in the signal
  • Wavelets with more vanishing moments can capture higher-order polynomial behavior more efficiently
• The number of levels of decomposition depends on the desired resolution and the length of the signal
  • Each level of decomposition reduces the resolution by a factor of 2
  • The maximum number of levels is limited by the signal length (signal length must be divisible by $2^{level}$)

Wavelet Coefficients: Scale and Translation

Relationship Across Scales

• Wavelet and scaling coefficients at different scales represent the signal's information at varying levels of resolution or frequency bands
  • Finer scales capture high-frequency details and short-term variations
  • Coarser scales capture low-frequency trends and long-term behavior
• The scaling coefficients at a given scale serve as the input for computing the wavelet and scaling coefficients at the next finer scale
  • Scaling coefficients at scale $j$ are used to compute wavelet and scaling coefficients at scale $j-1$
  • This hierarchical relationship allows for efficient computation and reconstruction of the signal
• The number of wavelet and scaling coefficients decreases by a factor of 2 at each coarser scale due to the downsampling operation in the DWT
  • Downsampling reduces the number of coefficients while maintaining the essential information
  • Helps in reducing computational complexity and storage requirements

Translation and Localization

• Translations of the wavelet and scaling functions at each scale allow for localized analysis of the signal's features in time or space
  • Wavelets are shifted along the signal to capture local variations
  • Different translations capture information at different time locations
• The wavelet coefficients at a particular scale capture the high-frequency details that are lost when transitioning from a finer scale to a coarser scale
  • Provide a detailed representation of the signal's local features
  • Enable detection and characterization of transient events or singularities
• The scaling coefficients at a coarser scale can be reconstructed from the scaling and wavelet coefficients at the next finer scale using the inverse discrete wavelet transform (IDWT)
  • IDWT performs upsampling and filtering operations to reconstruct the signal from its wavelet and scaling coefficients
  • Allows for perfect reconstruction of the original signal from its wavelet representation

Wavelet Coefficient Sparsity vs Energy Compaction

Sparsity of Wavelet Coefficients

• Sparsity refers to the property of wavelet coefficients where a large number of coefficients have small or zero values, while only a few coefficients carry significant information
  • Most of the signal's energy is concentrated in a small number of large coefficients
  • Enables efficient compression and denoising by focusing on the significant coefficients
• Signals with sparse wavelet representations have most of their energy concentrated in a small number of large coefficients
  • Examples include smooth signals with few discontinuities or singularities (sinusoids, polynomial functions)
  • Wavelet functions can efficiently capture the local behavior of such signals
• Signals with abrupt changes, edges, or transient features may have less sparse wavelet representations
  • More coefficients are required to capture these localized phenomena
  • Examples include signals with sharp transitions (square waves) or impulses (delta functions)

Energy Compaction Property

• Energy compaction refers to the ability of the wavelet transform to concentrate most of the signal's energy into a small number of coefficients
  • Wavelets with good energy compaction properties can represent the signal's information with fewer coefficients
  • Leads to efficient compression and storage of the signal
• The degree of sparsity and energy compaction depends on the choice of wavelet function and its compatibility with the characteristics of the analyzed signal
  • Wavelets with more vanishing moments generally provide better sparsity and energy compaction for smooth signals
  • Daubechies wavelets are known for their good energy compaction properties
• Techniques such as thresholding or coefficient selection can be applied to exploit the sparsity and energy compaction properties
  • Thresholding sets small coefficients to zero, reducing noise and achieving compression
  • Coefficient selection retains only the most significant coefficients for signal representation
  • These techniques are used in applications like signal denoising, compression, and feature extraction (image compression, audio denoising)