Sparse signal recovery from modulo observations

Abstract We consider the problem of reconstructing a signal from under-determined modulo observations (or measurements). This observation model is inspired by a (relatively) less well-known imaging mechanism called modulo imaging, which can be used to extend the dynamic range of imaging systems; variations of this model have also been studied under the category of phase unwrapping. Signal reconstruction in the under-determined regime with modulo observations is a challenging ill-posed problem, and existing reconstruction methods cannot be used directly. In this paper, we propose a novel approach to solving the inverse problem limited to two modulo periods, inspired by recent advances in algorithms for phase retrieval under sparsity constraints. We show that given a sufficient number of measurements, our algorithm perfectly recovers the underlying signal and provides improved performance over other existing algorithms. We also provide experiments validating our approach on both synthetic and real data to depict its superior performance.

Download Full-text

Compressed Sensing and Its Application in CT and EEG

Medical Imaging ◽

10.4018/978-1-5225-0571-6.ch046 ◽

2017 ◽

pp. 1126-1149

Author(s):

Sajib Saha ◽

Murat Tahtali

Keyword(s):

Compressed Sensing ◽

Applied Mathematics ◽

Signal Reconstruction ◽

Optimization Technique ◽

Linear Measurements ◽

Reconstruction Procedure ◽

Sparsity Constraints ◽

Ill Posed ◽

X Ray Computed ◽

A New Technique

Compressed sensing, also known as compressive sampling is a new technique being rapidly developed over the last few years. The theory states that when some prior information about the signal is available and appropriately incorporated into the signal reconstruction procedure, a signal can be accurately reconstructed even if the Shannon/ Nyquest sampling requirement is violated. The key idea of compressed sensing is to recover a sparse signal from very few non-adaptive, linear measurements by optimization technique. Following the discovery by Donoho in (2006), that sparsity could enable exact solution of ill-posed problems under certain conditions, there has been a tremendous growth on efficient application of sparsity constraints for solving ill-posed problems. The theoretical foundation of compressed sensing has already become a key concept in various areas of applied mathematics, computer science, and electrical engineering. In this chapter we will detail the application of compressed sensing in X-ray computed tomography (CT) and Electroencephalography. Starting from the very basic principles we will provide theoretical justifications on why and how sparsity prior is used in CT and in EEG.

Download Full-text

Tikhonov regularization with ℓ0-term complementing a~convex penalty: ℓ1-convergence under sparsity constraints

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2019-0008 ◽

2019 ◽

Vol 27 (4) ◽

pp. 575-590 ◽

Cited By ~ 1

Author(s):

Wei Wang ◽

Shuai Lu ◽

Bernd Hofmann ◽

Jin Cheng

Keyword(s):

Convergence Rates ◽

Regularization Parameter ◽

A Priori ◽

Convex Functional ◽

Important Special Case ◽

Bounded Linear ◽

Sparsity Constraints ◽

Ill Posed ◽

Sparse Solutions ◽

Special Case

Abstract Measuring the error by an {\ell^{1}} -norm, we analyze under sparsity assumptions an {\ell^{0}} -regularization approach, where the penalty in the Tikhonov functional is complemented by a general stabilizing convex functional. In this context, ill-posed operator equations {Ax=y} with an injective and bounded linear operator A mapping between {\ell^{2}} and a Banach space Y are regularized. For sparse solutions, error estimates as well as linear and sublinear convergence rates are derived based on a variational inequality approach, where the regularization parameter can be chosen either a priori in an appropriate way or a posteriori by the sequential discrepancy principle. To further illustrate the balance between the {\ell^{0}} -term and the complementing convex penalty, the important special case of the {\ell^{2}} -norm square penalty is investigated showing explicit dependence between both terms. Finally, some numerical experiments verify and illustrate the sparsity promoting properties of corresponding regularized solutions.

Download Full-text

Compressed Sensing and Its Application in CT and EEG

Advances in Bioinformatics and Biomedical Engineering - Biomedical Image Analysis and Mining Techniques for Improved Health Outcomes ◽

10.4018/978-1-4666-8811-7.ch006 ◽

2016 ◽

pp. 123-146

Author(s):

Sajib Saha ◽

Murat Tahtali

Keyword(s):

Compressed Sensing ◽

Applied Mathematics ◽

Signal Reconstruction ◽

Optimization Technique ◽

Linear Measurements ◽

Reconstruction Procedure ◽

Sparsity Constraints ◽

Ill Posed ◽

X Ray Computed ◽

A New Technique

Compressed sensing, also known as compressive sampling is a new technique being rapidly developed over the last few years. The theory states that when some prior information about the signal is available and appropriately incorporated into the signal reconstruction procedure, a signal can be accurately reconstructed even if the Shannon/ Nyquest sampling requirement is violated. The key idea of compressed sensing is to recover a sparse signal from very few non-adaptive, linear measurements by optimization technique. Following the discovery by Donoho in (2006), that sparsity could enable exact solution of ill-posed problems under certain conditions, there has been a tremendous growth on efficient application of sparsity constraints for solving ill-posed problems. The theoretical foundation of compressed sensing has already become a key concept in various areas of applied mathematics, computer science, and electrical engineering. In this chapter we will detail the application of compressed sensing in X-ray computed tomography (CT) and Electroencephalography. Starting from the very basic principles we will provide theoretical justifications on why and how sparsity prior is used in CT and in EEG.

Download Full-text

ON TIKHONOV REGULARIZATION AND COMPRESSIVE SENSING FOR SEISMIC SIGNAL PROCESSING

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202511500084 ◽

2012 ◽

Vol 22 (02) ◽

pp. 1150008 ◽

Cited By ~ 10

Author(s):

YANFEI WANG ◽

CHANGCHUN YANG ◽

JINGJIE CAO

Keyword(s):

Compressive Sensing ◽

Tikhonov Regularization ◽

Signal Reconstruction ◽

Gradient Methods ◽

Seismic Signal ◽

Sampling Theorem ◽

Sparse Regularization ◽

Reflection Seismic ◽

Reconstruction Methods ◽

Ill Posed

Using compressive sensing and sparse regularization, one can nearly completely reconstruct the input (sparse) signal using limited numbers of observations. At the same time, the reconstruction methods by compressing sensing and optimizing techniques overcome the obstacle of the number of sampling requirement of the Shannon/Nyquist sampling theorem. It is well known that seismic reflection signal may be sparse, sometimes and the number of sampling is insufficient for seismic surveys. So, the seismic signal reconstruction problem is ill-posed. Considering the ill-posed nature and the sparsity of seismic inverse problems, we study reconstruction of the wavefield and the reflection seismic signal by Tikhonov regularization and the compressive sensing. The l0, l1 and l2 regularization models are studied. Relationship between Tikhonov regularization and the compressive sensing is established. In particular, we introduce a general lp - lq (p, q ≥ 0) regularization model, which overcome the limitation on the assumption of convexity of the objective function. Interior point methods and projected gradient methods are studied. To show the potential for application of the regularized compressive sensing method, we perform both synthetic seismic signal and field data compression and restoration simulations using a proposed piecewise random sub-sampling. Numerical performance indicates that regularized compressive sensing is applicable for practical seismic imaging.

Download Full-text

Information and estimation in image processing

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100125142 ◽

1987 ◽

Vol 45 ◽

pp. 14-17

Author(s):

B. Roy Frieden

Keyword(s):

Image Processing ◽

Optical System ◽

Large Amplitude ◽

Communication Theory ◽

Image Data ◽

Direct Approach ◽

Ill Posed

Despite the skill and determination of electro-optical system designers, the images acquired using their best designs often suffer from blur and noise. The aim of an “image enhancer” such as myself is to improve these poor images, usually by digital means, such that they better resemble the true, “optical object,” input to the system. This problem is notoriously “ill-posed,” i.e. any direct approach at inversion of the image data suffers strongly from the presence of even a small amount of noise in the data. In fact, the fluctuations engendered in neighboring output values tend to be strongly negative-correlated, so that the output spatially oscillates up and down, with large amplitude, about the true object. What can be done about this situation? As we shall see, various concepts taken from statistical communication theory have proven to be of real use in attacking this problem. We offer below a brief summary of these concepts.

Download Full-text

Complex wavefront sensing based on coherent diffraction imaging using vortex modulation

Scientific Reports ◽

10.1038/s41598-021-88523-x ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

Rujia Li ◽

Liangcai Cao

Keyword(s):

Phase Retrieval ◽

Dynamic Range ◽

A Priori ◽

Measured Intensity ◽

Physical Constraints ◽

Coherent Diffraction Imaging ◽

Effective Dynamic ◽

Ill Posed ◽

Retrieval Problem ◽

Diffraction Imaging

AbstractPhase retrieval seeks to reconstruct the phase from the measured intensity, which is an ill-posed problem. A phase retrieval problem can be solved with physical constraints by modulating the investigated complex wavefront. Orbital angular momentum has been recently employed as a type of reliable modulation. The topological charge l is robust during propagation when there is atmospheric turbulence. In this work, topological modulation is used to solve the phase retrieval problem. Topological modulation offers an effective dynamic range of intensity constraints for reconstruction. The maximum intensity value of the spectrum is reduced by a factor of 173 under topological modulation when l is 50. The phase is iteratively reconstructed without a priori knowledge. The stagnation problem during the iteration can be avoided using multiple topological modulations.

Download Full-text

Compressive Sensing in Signal Processing: Algorithms and Transform Domain Formulations

Mathematical Problems in Engineering ◽

10.1155/2016/7616393 ◽

2016 ◽

Vol 2016 ◽

pp. 1-16 ◽

Cited By ~ 19

Author(s):

Irena Orović ◽

Vladan Papić ◽

Cornel Ioana ◽

Xiumei Li ◽

Srdjan Stanković

Keyword(s):

Signal Processing ◽

Fourier Transform ◽

Compressive Sensing ◽

Signal Reconstruction ◽

Signal Acquisition ◽

Transform Domain ◽

Hermite Transform ◽

Time Frequency ◽

Reconstruction Methods ◽

Fourier Transform Domain

Compressive sensing has emerged as an area that opens new perspectives in signal acquisition and processing. It appears as an alternative to the traditional sampling theory, endeavoring to reduce the required number of samples for successful signal reconstruction. In practice, compressive sensing aims to provide saving in sensing resources, transmission, and storage capacities and to facilitate signal processing in the circumstances when certain data are unavailable. To that end, compressive sensing relies on the mathematical algorithms solving the problem of data reconstruction from a greatly reduced number of measurements by exploring the properties of sparsity and incoherence. Therefore, this concept includes the optimization procedures aiming to provide the sparsest solution in a suitable representation domain. This work, therefore, offers a survey of the compressive sensing idea and prerequisites, together with the commonly used reconstruction methods. Moreover, the compressive sensing problem formulation is considered in signal processing applications assuming some of the commonly used transformation domains, namely, the Fourier transform domain, the polynomial Fourier transform domain, Hermite transform domain, and combined time-frequency domain.

Download Full-text