Wave atoms and sparsity of oscillatory patterns

Abstract

We introduce "wave atoms" as a variant of 2D wavelet packets obeying the parabolic scaling wavelength ∼ (diameter) 2 . We prove that warped oscillatory functions, a toy model for texture, have a significantly sparser expansion in wave atoms than in other fixed standard representations like wavelets, Gabor atoms, or curvelets. We propose a novel algorithm for a tight frame of wave atoms with redundancy two, directly in the frequency plane, by the "wrapping" technique. We also propose variants of the basic transform for applications in image processing, including an orthonormal basis, and a shift-invariant tight frame with redundancy four. Sparsity and denoising experiments on both seismic and fingerprint images demonstrate the potential of the tool introduced.

A multiresolution framework for local similarity based image denoising

“…Three levels of wavelet transform (for MRM-Wav) and Laplacian pyramid (MRM-Lap) were computed, as one risks blurring the lowpass band and introducing ringing artifacts if a higher number of levels of the transforms is used. For comparison of objective quality, denoising results in terms of the PSNR are presented for three methods presented in this paper, namely Multilateral E or MF-E (Section 2), MRM-Wav, and MRM-Lap (Section 3) in comparison to six other published methods: (1) the original bilateral filtering (BF) [29], (2) one of its recent variants known as saliency bilateral filtering (SBF) [43], two of the recently proposed transform-domain shrinkage methods in (3) Contourlet MD or C-MD [13] in the multi-scale contourlet domain with sharp localization in frequency and (4) wave atoms (WA) by Demanet & Ying [19] which aim to achieve good localization in space and frequency using Villemoes' wavelet packets [44] in the frequency domain, (5) a multiresolution version of bilateral filtering (MRB) [35], and (6) the standard non-local means filtering (NL-Means or NLM) [21] algorithm.…”

“…The wavelet variant of multiresolution multilateral (MRM) filtering was shown to produce good denoising results on images containing oscillatory patterns of anisotropic features, while the latter variant in the Laplacian domain reconstructed isotropic contents well and performed better on images containing smooth regions. Both the variants compare favorably with more sophisticated and computationally expensive shrinkage methods such as [13,19] which are designed to capture oscillatory patterns in images. A possible future direction of this work is the development of a data-driven approach to automatically select the σ f parameter.…”

“…Two major limitations of the shrinkage methods are that they exhibit (a) pseudo-Gibbs and (b) fake feature types of artifacts in images corrupted with medium to high levels of noise. The pseudo-Gibbs or ringing artifacts are generated due to there being insufficient number of coefficients for reconstruction of image features, while the fake feature artifacts such as those visible in contourlets [13] and wave atoms [19] result from the association of noisy image pixels with perceived image features.…”

“…Shrinkage methods in the transform domain approximate the image by modeling and efficiently representing important image features such as discontinuities [5], edges [6,7,8], curves [9,10,11], contours [12,13], ridges [14,15], and textured regions [16,17,18,19] present in the image or locally linear planes [20] in video sequences. Generally speaking, shrinkage methods first transform the image into some other domain, highlighting important image features, and thresholding the transform coefficients.…”

Morphological Diversity and Sparsity for Multichannel Data Restoration

“…For instance, in image processing, a number of dictionaries have been designed that can capture very different features in an image: discrete cosine basis for globally oscillating patterns, wave atoms for local oscillatory textures [7], wavelets for pointwise singularities [8], curvelets for edges and contours [9,10].…”