Unser, MichaelVan De Ville, Dimitri2010-11-302010-11-302010-11-30200810.1109/TIP.2008.2004607https://infoscience.epfl.ch/handle/20.500.14299/60901WOS:000260465200005A 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 paper, 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 nonredundant 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, a new quincunx Mexican-hat-like wavelet transform that is fully reversible and essentially behaves the (gamma/2)th Laplacian of a Gaussian.Denoisingfeature detectionfractalsframesisotropyMexican-hat filterpyramidwaveletsReconstruction Filter BanksTransformShrinkageMammogramsAlgorithmsTrackingSplinesFramesCIBM-SPtext::journal::journal article::research article