The Pairing of a Wavelet Basis with a Mildly Redundant Analysis via Subband Regression
M. Unser, D. Van De Ville
IEEE Transactions on Image Processing, in press.
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A distinction is usually made between wavelet bases and wavelet frames. The former are associated with a one-to-one representation of signals, which is somewhat constrained but most efficient computationally. The latter are over-complete but they offer advantages in terms of flexibility (shape of the basis functions) and shift-invariance. In this work, we propose a framework for improved wavelet analysis based on an appropriate pairing of a wavelet basis with a mildly redundant version of itself (frame). The processing is accomplished in four steps: (1) redundant wavelet analysis, (2) wavelet-domain processing, (3) projection of the results onto the wavelet basis, and (4) reconstruction of the signal from its non-redundant wavelet expansion. The wavelet analysis is pyramid-like and is obtained by simple modification of Mallat's filterbank algorithm (e.g., suppression of the down-sampling in the wavelet channels only). The key component of the method is the subband regression filter (Step 3) which computes a wavelet expansion that is maximally consistent in the least squares sense with the redundant wavelet analysis. We demonstrate that this approach significantly improves the performance of soft-threshold wavelet denoising with a moderate increase in computational cost. We also show that the analysis filters in the proposed framework can be adjusted for improved feature detection; in particular, we present new quincunx Mexican-hat-like wavelet transforms that are fully reversible and essentially behave like the (γ⁄2)th Laplacian of a Gaussian.