13.4 Relationship Between Filter Banks and Wavelets

Filter banks and wavelets are like besties in signal processing. They work together to break down signals into different frequency parts, giving us a multi-level view of the data. It's like zooming in and out on a picture to see both the big picture and tiny details.

This relationship is super important for understanding how wavelets work in practice. By using filter banks, we can actually build and use wavelet transforms, making them a powerful tool for analyzing and processing all kinds of signals.

Filter banks and wavelet transforms

Relationship between filter banks and wavelet transforms

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• Filter banks decompose a signal into multiple frequency subbands using a set of bandpass filters, analogous to how wavelet transforms analyze signals at different scales and frequencies
• The output of a two-channel filter bank, consisting of a lowpass and highpass filter followed by downsampling, corresponds to the approximation and detail coefficients obtained from a wavelet transform
  • The lowpass filter captures the coarse-scale information (approximation coefficients)
  • The highpass filter captures the fine-scale details (detail coefficients)
• Perfect reconstruction filter banks, where the original signal can be exactly reconstructed from the subband components, exhibit the invertibility property of wavelet transforms
  • The analysis and synthesis filters in a perfect reconstruction filter bank are designed to cancel out aliasing and ensure perfect reconstruction
• Iterating two-channel filter banks on the lowpass subband results in a multiscale decomposition, similar to the dyadic scale arrangement in wavelet transforms (octave-band decomposition)

Properties of wavelet bases derived from filter banks

• The choice of filters in a filter bank determines the properties of the corresponding wavelet basis
  • Orthogonality: The wavelet basis functions are orthogonal to each other, enabling efficient signal representation and reconstruction (Daubechies wavelets)
  • Symmetry: Symmetric or antisymmetric wavelet basis functions are desirable for certain applications, such as image processing (biorthogonal wavelets)
  • Regularity: The smoothness of the wavelet basis functions affects the ability to represent smooth signals efficiently (higher-order Daubechies wavelets)
• The number of vanishing moments of the wavelet basis is related to the flatness of the frequency response of the highpass filter in the filter bank
  • Higher vanishing moments lead to better approximation of smooth signals and faster decay of wavelet coefficients

Wavelet bases from filter banks

Perfect reconstruction filter banks

• Perfect reconstruction filter banks satisfy specific conditions on the lowpass and highpass filters, ensuring perfect reconstruction of the original signal from the subband components
• The lowpass and highpass filters in a perfect reconstruction filter bank are related by the quadrature mirror filter (QMF) condition
  • The highpass filter is obtained by alternating signs and reversing the order of the lowpass filter coefficients: $h_{high}[n] = (-1)^n h_{low}[L-1-n]$, where $L$ is the filter length
• The QMF condition ensures that the aliasing introduced by downsampling in the analysis stage is canceled out during the synthesis stage
  • The synthesis filters are time-reversed versions of the analysis filters, with the lowpass and highpass filters swapped

Deriving wavelet bases from perfect reconstruction filter banks

• The scaling function and wavelet function of a wavelet basis can be derived from the lowpass and highpass filters of a perfect reconstruction filter bank, respectively
• The scaling function $\phi(t)$ is obtained by iterating the lowpass filter and computing the infinite product of the Fourier transforms of the upsampled lowpass filter
  • $\phi(t) = \sqrt{2} \sum_{n} h_{low}[n] \phi(2t-n)$
  • $\hat{\phi}(\omega) = \prod_{j=1}^{\infty} \frac{H_{low}(2^{-j}\omega)}{\sqrt{2}}$, where $H_{low}(\omega)$ is the Fourier transform of the lowpass filter
• The wavelet function $\psi(t)$ is obtained by applying the highpass filter to the scaling function and computing the infinite product of the Fourier transforms of the upsampled highpass filter
  • $\psi(t) = \sqrt{2} \sum_{n} h_{high}[n] \phi(2t-n)$
  • $\hat{\psi}(\omega) = \prod_{j=1}^{\infty} \frac{H_{high}(2^{-j}\omega)}{\sqrt{2}}$, where $H_{high}(\omega)$ is the Fourier transform of the highpass filter
• The derived scaling function and wavelet function form an orthonormal basis for the space of square-integrable functions $L^2(\mathbb{R})$, enabling the wavelet transform

Wavelet decomposition and reconstruction

Wavelet decomposition using filter banks

• Wavelet decomposition can be performed using a cascaded filter bank structure, where the lowpass subband is iteratively decomposed using two-channel filter banks
• The decomposition algorithm applies the lowpass and highpass filters to the input signal, followed by downsampling, to obtain the approximation and detail coefficients at each scale
  • Approximation coefficients: $a_j[n] = \sum_{k} h_{low}[k] a_{j-1}[2n-k]$
  • Detail coefficients: $d_j[n] = \sum_{k} h_{high}[k] a_{j-1}[2n-k]$
• The decomposition process is repeated on the approximation coefficients to obtain the coefficients at the next scale, resulting in a multiscale representation of the signal
• The number of decomposition levels depends on the desired frequency resolution and the length of the input signal

Wavelet reconstruction using filter banks

• The reconstruction algorithm upsamples the approximation and detail coefficients, applies the corresponding reconstruction filters (lowpass and highpass), and combines the results to obtain the reconstructed signal
  • Upsampled approximation coefficients: $\tilde{a}_j[n] = \begin{cases} a_j[n/2], & \text{if } n \text{ is even} \\ 0, & \text{if } n \text{ is odd} \end{cases}$
  • Upsampled detail coefficients: $\tilde{d}_j[n] = \begin{cases} d_j[n/2], & \text{if } n \text{ is even} \\ 0, & \text{if } n \text{ is odd} \end{cases}$
  • Reconstructed signal: $a_{j-1}[n] = \sum_{k} h_{low}[k] \tilde{a}_j[n-k] + \sum_{k} h_{high}[k] \tilde{d}_j[n-k]$
• The perfect reconstruction property of the filter bank ensures that the reconstructed signal is identical to the original signal in the absence of noise or quantization errors
• Efficient implementations of wavelet decomposition and reconstruction algorithms can be achieved using polyphase representations and lifting schemes
  • Polyphase representation separates the even and odd samples of the filters, reducing the computational complexity
  • Lifting schemes factorize the filter bank into a series of simple lifting steps, enabling in-place computation and reducing memory requirements

Multiscale signal representation with filter banks

Multiscale analysis of signals

• Wavelet filter banks provide a multiscale representation of signals, where each scale captures different frequency components and time localizations
• The approximation coefficients at each scale represent the low-frequency content of the signal, providing a coarse-scale approximation
  • The approximation coefficients capture the overall trend and smooth variations of the signal
• The detail coefficients at each scale represent the high-frequency content of the signal, capturing fine-scale details and transient information
  • The detail coefficients capture abrupt changes, edges, and local singularities in the signal
• The multiscale representation allows for the analysis of signal features across different scales, enabling the identification of scale-dependent patterns and structures
  • Scale-dependent features can be extracted by examining the magnitude and distribution of wavelet coefficients at each scale

Applications of multiscale signal representation

• The energy distribution across scales can provide insights into the signal's characteristics
  • The presence of singularities or edges in the signal results in large detail coefficients at the corresponding scales
  • Textured regions in the signal exhibit a more uniform energy distribution across scales compared to smooth regions
• Wavelet filter banks can be used for signal denoising by thresholding the detail coefficients at each scale
  • Assuming that noise is primarily present in the high-frequency components, thresholding the detail coefficients can effectively suppress noise while preserving important signal features
• The multiscale representation obtained from wavelet filter banks is useful for signal compression
  • The majority of the signal's energy is often concentrated in a few large coefficients, allowing for efficient encoding and storage
  • Compression algorithms, such as the Embedded Zerotree Wavelet (EZW) and Set Partitioning in Hierarchical Trees (SPIHT), exploit the multiscale structure of wavelet coefficients for efficient coding