A Parametric Family of Polynomial Wavelets for Signal and Image Processing

TL;DR

The paper develops a parametric family of polynomial wavelets based on de la Vallée Poussin interpolation on Chebyshev nodes, defined on a bounded interval to naturally handle boundaries and enable fast, DCT-based transforms. It introduces a triadic multiresolution-like structure on $[-1,1]$, along with explicit univariate and tensor-product constructions for signals and images. Through detailed algorithms and normalization strategies, the VP wavelets are applied to denoising and image compression, showing competitive performance against classical wavelets, particularly at higher compression and in low-SNR scenarios. The work highlights practical benefits such as boundary handling and reduced decomposition depth, while noting non-orthogonality and proposing directions for improved energy normalization and broader applications.

Abstract

This paper investigates the potential applications of a parametric family of polynomial wavelets that has been recently introduced starting from de la Vallée Poussin (VP) interpolation at Chebyshev nodes. Unlike classical wavelets, which are constructed on the real line, these VP wavelets are defined on a bounded interval, offering the advantage of handling boundaries naturally while maintaining computational efficiency. In addition, the structure of these wavelets enables the use of fast algorithms for decomposition and reconstruction. Furthermore, the flexibility offered by a free parameter allows a better control of localized singularities, such as edges in images. On the basis of previous theoretical foundations, we show the effectiveness of the VP wavelets for basic signal denoising and image compression, emphasizing their potential for more advanced signal and image processing tasks.

Figures (13)

• Figure 1: The mean and standard deviation of the sum of the squared norms for $100$ samples for a signal of length $3^7$, $\theta = 0.5$ and $7$ steps done for $f_{sca} = \sqrt{3}$, $f_{wav}=\sqrt{\frac{3}{2}}$ (left) and for $f_{sca} = f_{wav} = 1$ (right).
• Figure 2: Six test signals: Blocks, Bumps, Heavy Sine, Doppler, Quadchirp, Mishmash.
• Figure 3: The results for different values of $\theta = 0.1, 0.2, \ldots, 0.9$ for the six test signals: Blocks, Bumps, Heavy Sine, Doppler, Quadchirp, Mishmash.
• Figure 4: Comparison of our denoising method with the ones of "wdenoise" of Matlab for the six test signals: Blocks, Bumps, Heavy Sine, Doppler, Quadchirp, Mishmash.
• Figure 5: The standard deviation divided by the average in function of input SNR (dB) for each of the test function for $\theta = 0.1$