Analysis of general weights in weighted ℓ1−2 minimization through applications

The weighted ℓ1−2 minimization has recently attracted some attention due to its capability to deal with highly coherent matrices. Notwithstanding the availability of its stable recovery guarantees, there appear to be some issues not addressed in the literature, which are (i). convergence of the solver for the weighted ℓ1−2 minimization analytically, and (ii). detailed analysis of relevance of general weights to applications. While establishing the convergence of the solver of the weighted ℓ1−2 minimization, we demonstrate the significance of general weights, w∈(0,1), empirically through some applications, including the reconstruction of magnetic resonance images. In particular, we show that the general weights attain significance when we do not have fully accurate or fully corrupt information about the support of the signal to be reconstructed from its linear measurements. We conclude the work by discussing a numerical scheme that chooses the partial support and the weights iteratively

On the roles of sparsity and sparsifying transforms in Computerized Tomography

X-ray computed tomography (CT) is one of the most widely used imaging modalities for diagnostic tasks in the clinical application. As X-ray dosage given to the patient has potential to induce undesirable clinical consequences, there is a need for reduction in dosage while maintaining good quality in reconstruction. This report explores the roles of concept of sparsity and sparsifying trans-forms via Frames, Wavelets etc. and their relevance to low-dose tomography. After giving detailed descriptions of basics of Computed Tomography, Frames/Wavelets and Compressive Sensing Theory, the report proposes a TV-norm based method for the reconstruction of Tomography. Finally, the report ends with some discussion on simulation results

Construction of Structured Incoherent Unit Norm Tight Frames

The exact recovery property of Basis pursuit (BP) and Orthogonal Matching Pursuit (OMP) has a relation with the coherence of the underlying frame. A frame with low coherence provides better guarantees for exact recovery. In particular, Incoherent Unit Norm Tight Frames (IUNTFs) play a significant role in sparse representations. IUNTFs with special structure, in particular those given by a union of several orthonormal bases, are known to satisfy better theoretical guarantees for recovering sparse signals. In the present work, we propose to construct structured IUNTFs consisting of large number of orthonormal bases. For a given r,k,m with k being less than or equal to the smallest prime power factor of m and r<k, we construct a CS matrix of size mk×(mk×mr) with coherence at most rk, which consists of mr number of orthonormal bases and with density 1m. We also present numerical results of recovery performance of union of orthonormal bases as against their Gaussian counterparts

On the existence of equivalence class of RIP-compliant matrices

In Compressed Sensing (CS), the matrices that satisfy the Restricted Isometry Property (RIP) play an important role. But it is known that the RIP properties of a matrix Φ and its `weighted matrix' GΦ (G being a non-singular matrix) vary drastically in terms of RIP constant. In this paper, we consider the opposite question: Given a matrix Φ, can we find a non-singular matrix G such that GΦ has compliance with RIP? We show that, under some conditions, a class of non-singular matrices (G) exists such that GΦ has RIP-compliance with better RIP constant. We also provide a relationship between the Unique Representation Property (URP) and Restricted Isometry Property (RIP), and a direct relationship between RIP and sparsest solution of a linear system of equations

Oversampling of Fourier Coefficients: Theory and Applications

The objective behind the proposal studies around oversampling of Fourier coefficients and their applications. I intend to study how oversampling of Fourier coefficients can be used for hiding messages. In the following report oversampling of Fourier coefficients has been discussed as providing room for storing or transmitting hidden information. The scheme aims at transmitting an arbitrary signal and, simultaneously, embedding a hidden code. The most important feature of this scheme is that without the knowledge of exact oversampling parameter the hidden code cannot be retrieved. This parameter provide the key for retrieving the code

Composition of Binary Compressed Sensing Matrices

In the recent past, various methods have been proposed to construct deterministic compressed sensing (CS) matrices. Of interest has been the construction of binary sensing matrices as they are useful for multiplierless and faster dimensionality reduction. In most of these binary constructions, the matrix size depends on primes or their powers. In this study, we propose a composition rule which exploits sparsity and block structure of existing binary CS matrices to construct matrices of general size. We also show that these matrices satisfy optimal theoretical guarantees and have similar density compared to matrices obtained using Kronecker product. Simulation work shows that the synthesized matrices provide comparable results against Gaussian random matrices

Reliable resource-constrained telecardiology via compressive detection of anomalous ECG signals

Telecardiology is envisaged as a supplement to inadequate local cardiac care, especially, in infrastructure deficient communities. Yet the associated infrastructure constraints are often ignored while designing a traditional telecardiology system that simply records and transmits user electrocardiogram (ECG) signals to a professional diagnostic facility. Against this backdrop, we propose a two-tier telecardiology framework, where constraints on resources, such as power and bandwidth, are met by compressively sampling ECG signals, identifying anomalous signals, and transmitting only the anomalous signals. Specifically, we design practical compressive classifiers based on inherent properties of ECG signals, such as self-similarity and periodicity, and illustrate their efficacy by plotting receiver operating characteristics (ROC). Using such classifiers, we realize a resource-constrained telecardiology system, which, for the PhysioNet databases, allows no more than 0.5% undetected patients even at an average downsampling factor of five, reducing the power requirement by 80% and bandwidth requirement by 83.4% compared to traditional telecardiology

IIT Hyderabad model to make ECGs available in remote areas

A team of four IIT-H professors have proposed a two-tier-cardiology framework in which electrocardiogram (ECG) records can be transmitted even when available resources such as power and bandwidth are limited

Nullspace Property for Optimality of Minimum Frame Angle Under Invertible Linear Operators

Frames with a large minimum angle between any two distinct frame vectors are desirable in many present day applications. For a unit norm frame, the absolute value of the cosine of the minimum frame angle is also known as coherence. Two frames are equivalent if one can be obtained from the other via left action of an invertible linear operator. Frame angles can change under the action of a linear operator. Most of the existing works solve different optimization problems to find an optimal linear operator that maximizes the minimal frame angle (in other words, minimizes the coherence). In the present work, nevertheless, we consider the question: Is it always possible to find an equivalent frame with smaller coherence for a given frame?. In this paper, we derive properties of the initial unit norm frame that can ensure an equivalent frame with strictly larger minimal frame angle compared to the initial one. It turns out that the nullspace property of a certain matrix obtained from the initial frame can guarantee such an equivalent frame. We also present the numerical results that support our theoretical claims. © 1994-2012 IEEE

Optimization of Low-Dose Tomography via Binary Sensing Matrices

X-ray computed tomography (CT) is one of the most widely used imaging modalities for diagnostic tasks in the clinical application. As X-ray dosage given to the patient has potential to induce undesirable clinical consequences, there is a need for reduction in dosage while maintaining good quality in reconstruction. The present work attempts to address low-dose tomography via an optimization method. In particular, we formulate the reconstruction problem in the form of a matrix system involving a binary matrix. We then recover the image deploying the ideas from the emerging field of compressed sensing (CS). Further, we study empirically the radial and angular sampling parameters that result in a binary matrix possessing sparse recovery parameters. The experimental results show that the performance of the proposed binary matrix with reconstruction using TV minimization by Augmented Lagrangian and ALternating direction ALgorithms (TVAL3) gives comparably better results than Wavelet based Orthogonal Matching Pursuit (WOMP) and the Least Squares solution