Abstract:
In this paper we define a new coherence index, named the global 2-coherence, of a given dictionary and study its relationship with the traditional mutual coherence and the restricted isometry constant. By exploring this relationship, we obtain more general results on sparse signal reconstruction using greedy algorithms in the compressive sensing (CS) framework. In particular, we obtain an improved bound over the best known results on the restricted isometry constant for successful recovery of sparse signals using orthogonal matching pursuit (OMP).

Abstract:
We propose a scheme to measure the cross-correlations and mutual coherence of optical and matter fields. It relies on the combination of a matter-wave detector operating by photoionization of the atoms and a traditional absorption photodetector. We show that the double-detection signal is sensitive to cross-correlation functions of light and matter waves.

Abstract:
The letter presents a method for the reduction in the mutual coherence of an overcomplete Gaussian or Bernoulli random matrix, which is fairly small due to the lower bound given here on the probability of the event that the aforesaid mutual coherence is less than any given number in (0, 1). The mutual coherence of the matrix that belongs to a set which contains the two types of matrices with high probability can be reduced by a similar method but a subset that has Lebesgue measure zero. The numerical results are provided to illustrate the reduction in the mutual coherence of an overcomplete Gaussian, Bernoulli or uniform random dictionary. The effect on the third type is better than a former result.

Abstract:
As to block sparse signals, both theoretical analysis and experimental verification showed that sufficient condition for precise reconstruction is both block-coherence and sub-coherence of dictionary must be very small. This paper introduced a new block orthogonal matching pursuit algorithm using mutual alternating projection method MAP-BOMP. By exploiting the mutual alternating projection, the algorithm proposed to design the new measurement dictionary and sensing dictionary constantly for reducing the block-coherence and sub-coherence. The algorithm also gave the specific convergence conditions to reduce the complexity. Simulation results demonstrate this algorithm can provide more significant improvement for the recovery performance and speed than other existing algorithm.

Abstract:
The performance guarantees of generalized orthogonal matching pursuit (gOMP) are considered in the framework of mutual coherence. The gOMP algorithm is an extension of the well-known OMP greed algorithm for compressed sensing. It identifies multiple N indices per iteration to reconstruct sparse signals. The gOMP with N≥2 can perfectly reconstruct any K-sparse signals from measurement y=Φx if K< 1/N((1/μ)-1)+1, where μ is coherence parameter of measurement matrix Φ. Furthermore, the performance of the gOMP in the case of y=Φx+e with bounded noise ‖e‖2≤ε is analyzed and the sufficient condition ensuring identification of correct indices of sparse signals via the gOMP is derived, i.e., K<1/N((1/μ)-1)+1-((2ε)/(Nμxmin)), where xmin denotes the minimum magnitude of the nonzero elements of x. Similarly, the sufficient condition in the case of Gaussian noise is also given. The performance guarantees of generalized orthogonal matching pursuit (gOMP) are considered in the framework of mutual coherence. The gOMP algorithm is an extension of the well-known OMP greed algorithm for compressed sensing. It identifies multiple N indices per iteration to reconstruct sparse signals. The gOMP with N≥2 can perfectly reconstruct any K-sparse signals from measurement y=Φx if K< 1/N((1/μ)-1)+1, where μ is coherence parameter of measurement matrix Φ. Furthermore, the performance of the gOMP in the case of y=Φx+e with bounded noise ‖e‖2≤ε is analyzed and the sufficient condition ensuring identification of correct indices of sparse signals via the gOMP is derived, i.e., K<1/N((1/μ)-1)+1-((2ε)/(Nμxmin)), where xmin denotes the minimum magnitude of the nonzero elements of x. Similarly, the sufficient condition in the case of Gaussian noise is also given.

Abstract:
Compressed Sensing (CS) is a novel technique for simultaneous signal sampling and compression based on the existence of a sparse representation of signal and a projected dictionary $\PP\D$, where $\PP\in\mathbb{R}^{m\times d}$ is the projection matrix and $\D\in\mathbb{R}^{d\times n}$ is the dictionary. To exactly recover the signal with a small number of measurements $m$, the projected dictionary $\PP\D$ is expected to be of low mutual coherence. Several previous methods attempt to find the projection $\PP$ such that the mutual coherence of $\PP\D$ can be as low as possible. However, they do not minimize the mutual coherence directly and thus their methods are far from optimal. Also the solvers they used lack of the convergence guarantee and thus there has no guarantee on the quality of their obtained solutions. This work aims to address these issues. We propose to find an optimal projection by minimizing the mutual coherence of $\PP\D$ directly. This leads to a nonconvex nonsmooth minimization problem. We then approximate it by smoothing and solve it by alternate minimization. We further prove the convergence of our algorithm. To the best of our knowledge, this is the first work which directly minimizes the mutual coherence of the projected dictionary with a convergence guarantee. Numerical experiments demonstrate that the proposed method can recover sparse signals better than existing methods.

Abstract:
This paper provides a simple proof of the mutual incoherence condition $\mu < \frac{1}{2K-1}$ under which K-sparse signal can be accurately reconstructed from a small number of linear measurements using the orthogonal matching pursuit (OMP) algorithm. Our proof, based on mathematical induction, is built on an observation that the general step of the OMP process is in essence same as the initial step since the residual is considered as a new measurement preserving the sparsity level of an input vector.

Abstract:
Greed is good. However, the tighter you squeeze, the less you have. In this paper, a less greedy algorithm for sparse signal reconstruction in compressive sensing, named orthogonal matching pursuit with thresholding is studied. Using the global 2-coherence , which provides a "bridge" between the well known mutual coherence and the restricted isometry constant, the performance of orthogonal matching pursuit with thresholding is analyzed and more general results for sparse signal reconstruction are obtained. It is also shown that given the same assumption on the coherence index and the restricted isometry constant as required for orthogonal matching pursuit, the thresholding variation gives exactly the same reconstruction performance with significantly less complexity.

Abstract:
Image denoising is an important pre-processing step for many image analysis and computer vision system. It refers to the task of recovering a good estimate of the true image from a degraded observation without altering and changing useful structure in the image such as discontinuities and edges. In this paper, we propose a new approach for image denoising based on the combination of two non linear diffusion tensors. One allows diffusion along the orientation of greatest coherence while the other allows diffusion along orthogonal directions. The idea is to track perfectly the local geometry of the degraded image and applying anisotropic diffusion mainly along the preferred structure direction. To illustrate the effective performance of our method, we present some experimental results on a test and real photographic color images.

Abstract:
Cyclidic nets are introduced as discrete analogs of curvature line parametrized surfaces and orthogonal coordinate systems. A 2-dimensional cyclidic net is a piecewise smooth $C^1$-surface built from surface patches of Dupin cyclides, each patch being bounded by curvature lines of the supporting cyclide. An explicit description of cyclidic nets is given and their relation to the established discretizations of curvature line parametrized surfaces as circular, conical and principal contact element nets is explained. We introduce 3-dimensional cyclidic nets as discrete analogs of triply-orthogonal coordinate systems and investigate them in detail. Our considerations are based on the Lie geometric description of Dupin cyclides. Explicit formulas are derived and implemented in a computer program.