Restricted Isometric Property Research Articles

Oblique Projection Matching Pursuit

Recent theory of compressed sensing (CS) tells us that sparse signals can be reconstructed from a small number of random samples. In reconstruction of sparse signals, greedy algorithms, such as the orthogonal matching pursuit (OMP), have been shown to be computationally efficient. In this paper, the performance of OMP is shown to be dependent on how well information of the underlying signals is preserved in the residual vector. Further, to improve the information preservation, we present a modification of OMP, called oblique projection matching pursuit (ObMP), which updates the residual in a oblique projection manor. Using the restricted isometric property (RIP), we build a solid yet very intuitive grasp of the more accurate phenomenon of ObMP. We also show from empirical experiments that the ObMP achieves improved reconstruction performance over the conventional OMP algorithm in terms of support detection ratio and mean squared error (MSE).

Recovery Error Analysis of Noisy Measurement in Compressed Sensing

With the rapid development of compressed sensing, practical consideration such as the impact of noise in the measurement should be considered. Existing methods for establishing recovery error bound suffer from inherent drawbacks and are only valid for specific types of noise. In this paper, we establish recovery error bound for various types of noise based on restricted isometric property. Our method is constructed from probability perspective. First, we establish the recovery error bound for noiseless signal to lay the foundation for further analysis. Depending on the probabilistic characteristics of different types of noise, we then establish recovery error bound for probabilistically bounded noise as well as for probabilistically unbounded noise. To further enhance the tightness of the bound, we also establish recovery error bound based on Dantzig-selector.

Energy-Efficient Distributed Compressed Sensing Data Aggregation for Cluster-Based Underwater Acoustic Sensor Networks

Energy-efficient data aggregation is important for underwater acoustic sensor networks due to its energy constrained character. In this paper, we propose a kind of energy-efficient data aggregation scheme to reduce communication cost and to prolong network lifetime based on distributed compressed sensing theory. First, we introduce a distributed compressed sensing model for a cluster-based underwater acoustic sensor network in which spatial and temporal correlations are both considered. Second, two schemes, namely, BUTM-DCS (block upper triangular matrix DCS) and BDM-DCS (block diagonal matrix DCS), are proposed based on the design of observation matrix with strictly restricted isometric property. Both schemes take multihop underwater acoustic communication cost into account. Finally, a distributed compressed sensing reconstruction algorithm, DCS-SOMP (Simultaneous Orthogonal Matching Pursuit for DCS), is adopted to recover raw sensor readings at the fusion center. We performed simulation experiments on both the synthesized and real sensor readings. The results demonstrate that the new data aggregation schemes can reduce energy cost by more than 95 percent compared with conventional data aggregation schemes when the cluster number is 20.

Designing sparse sensing matrix for compressive sensing to reconstruct high resolution medical images

Compressive sensing theory enables faithful reconstruction of signals, sparse in domain Ψ, at sampling rate lesser than Nyquist criterion, while using sampling or sensing matrix Φ which satisfies restricted isometric property. The role played by sensing matrix Φ and sparsity matrix Ψ is vital in faithful reconstruction. If the sensing matrix is dense then it takes large storage space and leads to high computational cost. In this paper, effort is made to design sparse sensing matrix with least incurred computational cost while maintaining quality of reconstructed image. The design approach followed is based on sparse block circulant matrix (SBCM) with few modifications. The other used sparse sensing matrix consists of 15 ones in each column. The medical images used are acquired from US, MRI and CT modalities. The image quality measurement parameters are used to compare the performance of reconstructed medical images using various sensing matrices. It is observed that, since Gram matrix of dictionary matrix...

MR image reconstruction with block sparsity and iterative support detection

This work aims to develop a novel magnetic resonance (MR) image reconstruction approach motivated by the recently proposed sampling framework with union-of-subspaces model (SUoS). Based on SUoS, we propose a mathematical formalism that effectively integrates a block sparsity constraint and support information which is estimated in an iterative fashion. The resulting optimization problem consists of a data fidelity term and a support detection based block sparsity (SDBS) promoting term penalizing entries within the complement of the estimated support. We provide optional strategies for block assignment, and we also derive unique and robust recovery conditions in terms of the structured restricted isometric property (RIP), namely the block-RIP. The block-RIP constant we derive is lower than that of the previous structured sparse method, which leads to a reduction of the measurements. Simulation results for reconstructing individual and multiple T1/T2-weighted images demonstrate the consistency with our theoretical claims, and show considerable improvement in comparison with methods using only block sparsity or support information.

Estimates on compressed neural networks regression

When the neural element number n of neural networks is larger than the sample size m, the overfitting problem arises since there are more parameters than actual data (more variable than constraints). In order to overcome the overfitting problem, we propose to reduce the number of neural elements by using compressed projection A which does not need to satisfy the condition of Restricted Isometric Property (RIP). By applying probability inequalities and approximation properties of the feedforward neural networks (FNNs), we prove that solving the FNNs regression learning algorithm in the compressed domain instead of the original domain reduces the sample error at the price of an increased (but controlled) approximation error, where the covering number theory is used to estimate the excess error, and an upper bound of the excess error is given.

Sparsest Random Scheduling for Compressive Data Gathering in Wireless Sensor Networks

Compressive sensing (CS)-based in-network data processing is a promising approach to reduce packet transmission in wireless sensor networks. Existing CS-based data gathering methods require a large number of sensors involved in each CS measurement gathering, leading to the relatively high data transmission cost. In this paper, we propose a sparsest random scheduling for compressive data gathering scheme, which decreases each measurement transmission cost from O(N) to O(log(N)) without increasing the number of CS measurements as well. In our scheme, we present a sparsest measurement matrix, where each row has only one nonzero entry. To satisfy the restricted isometric property, we propose a design method for representation basis, which is properly generated according to the sparsest measurement matrix and sensory data. With extensive experiments over real sensory data of CitySee, we demonstrate that our scheme can recover the real sensory data accurately. Surprisingly, our scheme outperforms the dense measurement matrix with a discrete cosine transformation basis over 5 dB on data recovery quality. Simulation results also show that our scheme reduces almost 10 × energy consumption compared with the dense measurement matrix for CS-based data gathering.

New conditions for uniformly recovering sparse signals via orthogonal matching pursuit

Recently, lots of work has been done on conditions of guaranteeing sparse signal recovery using orthogonal matching pursuit (OMP). However, none of the existing conditions is both necessary and sufficient in terms of the so-called restricted isometric property, coherence, cumulative coherence (Babel function), or other verifiable quantities in the literature. Motivated by this observation, we propose a new measure of a matrix, named as union cumulative coherence, and present both sufficient and necessary conditions under which the OMP algorithm can uniformly recover sparse signals for all sensing matrices. The proposed condition guarantees a uniform recovery of sparse signals using OMP, and reveals the capability of OMP in sparse recovery. We demonstrate by examples that the proposed condition can be used to more effectively determine the recoverable sparse signals via OMP than the conditions existing in the literature. Furthermore, sparse recovery from noisy measurements is also considered in terms of the proposed union cumulative coherence.

Compressive system identification: Sequential methods and entropy bounds

In the first part of this work, a novel Kalman filtering-based method is introduced for estimating the coefficients of sparse, or more broadly, compressible autoregressive models using fewer observations than normally required. By virtue of its (unscented) Kalman filter mechanism, the derived method essentially addresses the main difficulties attributed to the underlying estimation problem. In particular, it facilitates sequential processing of observations and is shown to attain a good recovery performance, particularly under substantial deviations from ideal conditions, those which are assumed to hold true by the theory of compressive sensing. In the remaining part of this paper we derive a few information-theoretic bounds pertaining to the problem at hand. The obtained bounds establish the relation between the complexity of the autoregressive process and the attainable estimation accuracy through the use of a novel measure of complexity. This measure is used in this work as a substitute to the generally incomputable restricted isometric property.

A class of deterministic construction of binary compressed sensing matrices

Compressed Sensing (CS) is an emerging technology in the field of signal processing, which can recover a sparse signal by taking very few samples and solving a linear programming problem. In this paper, we study the application of Low-Density Parity-Check (LDPC) Codes in CS. Firstly, we find a sufficient condition for a binary matrix to satisfy the Restricted Isometric Property (RIP). Then, by employing the LDPC codes based on Berlekamp-Justesen (B-J) codes, we construct two classes of binary structured matrices and show that these matrices satisfy RIP. Thus, the proposed matrices could be used as sensing matrices for CS. Finally, simulation results show that the performance of the proposed matrices can be comparable with the widely used random sensing matrices.