Zhihui Zhu

Approximating Sampled Sinusoids and Multiband Signals Using Multiband Modulated DPSS Dictionaries

Many signal processing problems—such as analysis, compression, denoising, and reconstruction—can be facilitated by expressing the signal as a linear combination of atoms from a well-chosen dictionary. In this paper, we study possible dictionaries for representing the discrete vector one obtains when collecting a finite set of uniform samples from a multiband analog signal. By analyzing the spectrum of combined time- and multiband-limiting operations in the discrete-time domain, we conclude that the information level of the sampled multiband vectors is essentially equal to the time–frequency area. For representing these vectors, we consider a dictionary formed by concatenating a collection of modulated discrete prolate spheroidal sequences (DPSS’s). We study the angle between the subspaces spanned by this dictionary and an optimal dictionary, and we conclude that the multiband modulated DPSS dictionary—which is simple to construct and more flexible than the optimal dictionary in practical applications—is nearly optimal for representing multiband sample vectors. We also show that the multiband modulated DPSS dictionary not only provides a very high degree of approximation accuracy in an MSE sense for multiband sample vectors (using a number of atoms comparable to the information level), but also that it can provide high-quality approximations of all sampled sinusoids within the bands of interest.

The non-convex geometry of low-rank matrix optimization

This work considers the minimization of a general convex function f (X) over the cone of positive semi-definite matrices whose optimal solution X∗ is of low-rank. Standard first-order convex solvers require performing an eigenvalue decomposition in each iteration, severely limiting their scalability. A natural nonconvex reformulation of the problem factors the variable X into the product of a rectangular matrix with fewer columns and its transpose. For a special class of matrix sensing and completion problems with quadratic objective functions, local search algorithms applied to the factored problem have been shown to be much more efficient and, in spite of being nonconvex, to converge to the global optimum. The purpose of this work is to extend this line of study to general convex objective functions f (X) and investigate the geometry of the resulting factored formulations. Specifically, we prove that when f (X) satisfies the restricted well-conditioned assumption, each critical point of the factored problem either corresponds to the optimal solution X∗ or a strict saddle where the Hessian matrix has a strictly negative eigenvalue. Such a geometric structure of the factored formulation ensures that many local search algorithms can converge to the global optimum with random initializations.

Global optimality in low-rank matrix optimization

This paper considers the minimization of a general objective function f (X) over the set of non-square n × m matrices where the optimal solution X∗ is low-rank. To reduce the computational burden, we factorize the variable X into a product of two smaller matrices and optimize over these two matrices instead of X. We analyze the global geometry for a general and yet well-conditioned objective function f (X) whose restricted strong convexity and restricted strong smoothness constants are comparable. In particular, we show that the reformulated objective function has no spurious local minima and obeys the strict saddle property. These geometric properties imply that a number of iterative optimization algorithms (such as gradient descent) can provably solve the factored problem with global convergence.

An efficient algorithm for designing projection matrix in compressive sensing based on alternating optimization

This paper considers the problem of optimally designing the projection matrix ? for a certain class of signals which can be sparsely represented by a specified dictionary Ψ . The optimal projection matrix is proposed to minimize the distance between the Gram matrix of the equivalent dictionary ? Ψ and a set of relaxed Equiangular Tight Frames (ETFs). An efficient method is derived for the optimal projection matrix design with a given Gram matrix. In addition, an extension of projection matrix design is derived for the scenarios where the signals cannot be represented exactly sparse in a specified dictionary. Simulations with synthetic data and real images demonstrate that the obtained projection matrix significantly improves the signal recovery accuracy of a system and outperforms those obtained by the existing algorithms. HighlightsAn efficient method is derived for the optimal projection matrix with a target Gram matrix.An innovated approach is developed to design the projection matrix when the signal is not exactly sparse.The simulations for natural images demonstrate the innovated approach can lead to a high PSNR.

Projection matrix optimization for block-sparse compressive sensing

Traditionally, the projection matrix in compressive sensing (CS) is chosen as a random matrix. In recent years, we have seen that the performance of CS systems can be improved by using a carefully designed projection matrix rather than a random one. In particular, we can reduce the coherence between the columns of the equivalent dictionary thanks to a well-designed projection matrix. Then, we can get a lower reconstruction error and a higher successful reconstruction rate. In some applications, the signals of interest have nonzero entries occurring in clusters - i.e., block-sparse signals. In this paper, we use the equiangular tight frame (ETF) to approach the Gram matrix of equivalent dictionary rather than the identity matrix used in [1]. Then, we minimize a weighted sum of the subblock coherence and the interblock coherence of the equivalent dictionary. The simulation results show that our novel method for projection matrix optimization significantly improves the ability of block-sparse approximation techniques to reconstruct and classify signals than the method proposed by Lihi Zelnik-Manor (LZM) [1].

Methods to enhance seismic faults and construct fault surfaces

Abstract Faults are often apparent as reflector discontinuities in a seismic volume. Numerous types of fault attributes have been proposed to highlight fault positions from a seismic volume by measuring reflection discontinuities. These attribute volumes, however, can be sensitive to noise and stratigraphic features that are also apparent as discontinuities in a seismic volume. We propose a matched filtering method to enhance a precomputed fault attribute volume, and simultaneously estimate fault strikes and dips. In this method, a set of efficient 2D exponential filters, oriented by all possible combinations of strike and dip angles, are applied to the input attribute volume to find the maximum filtering responses at all samples in the volume. These maximum filtering responses are recorded to obtain the enhanced fault attribute volume while the corresponding strike and dip angles, that yield the maximum filtering responses, are recoded to obtain volumes of fault strikes and dips. By doing this, we assume that a fault surface is locally planar, and a 2D smoothing filter will yield a maximum response if the smoothing plane coincides with a local fault plane. With the enhanced fault attribute volume and the estimated fault strike and dip volumes, we then compute oriented fault samples on the ridges of the enhanced fault attribute volume, and each sample is oriented by the estimated fault strike and dip. Fault surfaces can be constructed by directly linking the oriented fault samples with consistent fault strikes and dips. For complicated cases with missing fault samples and noisy samples, we further propose to use a perceptual grouping method to infer fault surfaces that reasonably fit the positions and orientations of the fault samples. We apply these methods to 3D synthetic and real examples and successfully extract multiple intersecting fault surfaces and complete fault surfaces without holes.

An efficient method for robust projection matrix design

Our objective is to efficiently design a robust projection matrix

On Joint Optimization of Sensing Matrix and Sparsifying Dictionary for Robust Compressed Sensing Systems

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A new framework for designing incoherent sparsifying dictionaries

for the Compressive Sensing (CS) systems when applied to the signals that are not exactly sparse. The optimal projection matrix is obtained by mainly minimizing the average coherence of the equivalent dictionary. In order to drop the requirement of the sparse representation error (SRE) for a set of training data as in [15] [16], we introduce a novel penalty function independent of a particular SRE matrix. Without requiring of training data, we can efficiently design the robust projection matrix and apply it for most of CS systems, like a CS system for image processing with a conventional wavelet dictionary in which the SRE matrix is generally not available. Simulation results demonstrate the efficiency and effectiveness of the proposed approach compared with the state-of-the-art methods. In addition, we experimentally demonstrate with natural images that under similar compression rate, a CS system with a learned dictionary in high dimensions outperforms the one in low dimensions in terms of reconstruction accuracy. This together with the fact that our proposed method can efficiently work in high dimension suggests that a CS system can be potentially implemented beyond the small patches in sparsity-based image processing.

Super-resolution in SAR imaging: Analysis with the atomic norm

Abstract This paper deals with joint design of sensing matrix and sparsifying dictionary for compressed sensing (CS) systems. Based on the maximum likelihood estimation (MLE) principle, a preconditioned signal recovery (PSR) scheme and a novel measure are proposed. Such a measure allows us to optimize the sensing matrix and dictionary jointly. An alternating minimization-based iterative algorithm is derived for solving the corresponding optimal design problem. Simulation and experiments, carried with synthetic data and real image signals, show that the PSR scheme and the CS system, obtained using the proposed approaches, outperform the prevailing ones in terms of reducing the effect of sparse representation errors.