3-D Data Denoising and Inpainting with the Low-Redundancy Fast Curvelet Transform

Abstract

In this paper, we first present a new implementation of the 3-D fast curvelet transform, which is nearly 2.5 less redundant than the Curvelab (wrapping-based) implementation as originally proposed in [1, 2], which makes it more suitable to applications including massive data sets. We report an extensive comparison in denoising with the Curvelab implementation as well as other 3-D multi-scale transforms with and without directional selectivity. The proposed implementation proves to be a very good compromise between redundancy, rapidity and performance. Secondly, we exemplify its usefulness on a variety of applications including denoising, inpainting, video de-interlacing and sparse component separation. The obtained results are good with very simple algorithms and virtually no parameter to tune.

Learning sparse representations on the sphere

“…The subsampled version of the α-shearlet transform computes these coefficients. Internally, this is achieved by using the "frequency wrapping" approach outlined in Candès et al (2006, Sections 3.3 and 6), Woiselle (2010, Chapter 4), and Woiselle et al (2011) for the case of the curvelet transform. Since each convolution is sampled along a different lattice, the subsampled transform of a given image f is a list of rectangular matrices of varying dimension.…”

From restricted isometry property to observability under sparse measurement

“…Generally, the choice of bases in compressive sensing is very important task (Fourier analysis, wavelets [17]) in signal processing. Similarly, some information (for example the regrouping nonzero information) on the signal are very useful for the design of a decompression algorithm.…”

“…In fact, compressive sampling (CS) is based on the hypothesis that the signal is located in an extended space that is reconstructed in a suitable base [17] and the associated matrix verifies a restricted isometry property (RIP) [2]. Under these assumptions, the signal can be reconstructed using techniques based on regularised linear regression.…”

Seislet-based morphological component analysis using scale-dependent exponential shrinkage

“…Then, the signal is considered as a superposition of several morphological components. One has to choose a dictionary whose atoms match the shape of the geometrical structures to sparsify, while leading to a non-sparse (or at least not as sparse) representation of the other signal content, that is the essence of so-called morphological component analysis (MCA) (Starck et al, 2004(Starck et al, , 2007Woiselle et al, 2011).…”