Galen Reeves, Michael Gastpar · arXiv (Cornell University) · 2010
The field of compressed sensing has shown that a sparse but otherwise arbitrary vector can be recovered exactly from a small number of randomly constructed linear projections (or samples). The question addressed in this paper is whether an even smaller number of samples is sufficient when there exists prior knowledge about the distribution of the unknown vector, or when only partial recovery is needed. An information-theoretic lower bound with connections to free probability theory and an upper bound corresponding to a computationally simple thresholding estimator are derived. It is shown that in certain cases (e.g. discrete valued vectors or large distortions) the number of samples can be decreased. Interestingly though, it is also shown that in many cases no reduction is possible.
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2015
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David Donoho, Yaakov Tsaig, David L. Donoho · 2016
We study the notion of Compressed Sensing (CS) as put forward in [14] and related work [20, 3, 4]. The basic idea behind CS is that a signal or image, unknown but supposed to be compressible by a known transform, (eg. wavelet or Fourier), can be subjected to fewer measurements than the nominal number of pixels, and yet be accurately reconstructed. The samples are nonadaptive and measure ‘random ’ linear combinations of the transform coefficients. Approximate reconstruction is obtained by solving for the transform coefficients consistent with measured data and having the smallest possible `1 norm. We perform a series of numerical experiments which validate in general terms the basic idea proposed in [14, 3, 5], in the favorable case where the transform coefficients are sparse in the strong sense that the vast majority are zero. We then consider a range of less-favorable cases, in which the object has all coefficients nonzero, but the coefficients obey an `p bound, for some p ∈ (0, 1]. These experiments show that the basic inequalities behind the CS method seem to involve reasonable constants. We next consider synthetic examples modelling problems in spectroscopy and image pro-
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Jayaraman J. Tiagarajan, Karthikeyan Natesan Ramamurthy, Pavan Turaga et al. · Synthesis lectures on image, video, and multimedia processing · 2014
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Sameer Sonkusale, Maryam Shojaei Baghini, Shuchin Aeron · 2022
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2021
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David Donoho, Yaakov Tsaig, David L. Donoho · 2006
We study the notion of Compressed Sensing (CS) as put forward in [14] and related work [20, 3, 4]. The basic idea behind CS is that a signal or image, unknown but supposed to be compressible by a known transform, (eg. wavelet or Fourier), can be subjected to fewer measurements than the nominal number of pixels, and yet be accurately reconstructed. The samples are nonadaptive and measure ‘random ’ linear combinations of the transform coefficients. Approximate reconstruction is obtained by solving for the transform coefficients consistent with measured data and having the smallest possible `1 norm. We perform a series of numerical experiments which validate in general terms the basic idea proposed in [14, 3, 5], in the favorable case where the transform coefficients are sparse in the strong sense that the vast majority are zero. We then consider a range of less-favorable cases, in which the object has all coefficients nonzero, but the coefficients obey an `p bound, for some p ∈ (0, 1]. These experiments show that the basic inequalities behind the CS method seem to involve reasonable constants. We next consider synthetic examples modelling problems in spectroscopy and image pro-
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Ruhlig, Klaus · Fraunhofer-Publica (Fraunhofer-Gesellschaft) · 2010
S.90
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David L. Donoho · 2015
Abstract—Suppose is an unknown vector in (a digital image or signal); we plan to measure general linear functionals of and then reconstruct. If is known to be compressible by transform coding with a known transform, and we recon-struct via the nonlinear procedure defined here, the number of measurements can be dramatically smaller than the size. Thus, certain natural classes of images with pixels need only
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Jean‐Luc Starck, Fionn Murtagh, Jalal Fadili · Cambridge University Press eBooks · 2010
In this chapter, we provide essential insights on the theory of compressed sensing (CS) that emerged in the work of Candès et al. (2006b), Candès and Tao (2006), and Donoho (2006a). Compressed sensing is also known by the names of compressive sensing and compressed or compressive sampling.
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2014
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Wu, Hui Juan. · DR-NTU (Nanyang Technological University) · 2010
Compressed sensing is a kind of compressive sampling or sparse sampling. It is also \na new technique for acquiring and reconstructing a signal utilizing the prior \nknowledge that it is sparse or compressible. According to the recently developed \nmathematical theory of Compressed-Sensing (CS), images with a sparse representation can be recovered from randomly undersampled k-space data. The sparsity of MR image can be exploited to significantly reduce scan time, or alternatively, improve the resolution of MR image. However, MR image near metallic implants remains an unmet need because of severe artifacts, which mainly \nstem from large metal-induced field inhomogeneities. This work addresses MRI near \nmetallic implants with an innovative imaging technique called "Slice Encoding for \nMetal Artifact Correction" (SEMAC). The SEMAC technique does not require additional hardware, but it can be deployed to the large installed base of whole-body MRI systems. The efficacy of the SEMAC technique in eliminating metal-induced distortions with feasible scan times is validated in phantom and in vivo spine and knee studies. In this project, the study is about a new technique for accelerating SEMAC acquisition by incorporating with compressed sensing in order to greatly reduce scan times, while producing high-quality distortion correction and signal to noise ratio to SEMAC with full sampling.
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Aveechal Prasad · Radiopaedia.org · 2026
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Francisco Chinesta, Elías Cueto, Victor Champaney et al. · Studies in big data · 2025
Very often, manifold learning techniques employ some form of local least-squares fitting of the data. The novelty in compressed sensing techniques relies in the use of the L1-norm instead [1]. The L1 norm constitutes an elegant way of enforcing sparsity. In regression, L2 norm gives an utmost importance to the outliers. This is due to the use of the squared norm, such that these outliers have a tremendous influence in the resulting fitted curve.
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Gitta Kutyniok · Mitteilungen der Deutschen Mathematiker-Vereinigung · 2014
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Ruhlig, K. · Publikationsdatenbank der Fraunhofer-Gesellschaft (Fraunhofer-Gesellschaft) · 2010
Als Compressed Sensing wird eine neue Entwicklung auf dem Gebiet der Sensorsignalverarbeitung bezeichnet. Gegenwärtig werden die bei einem Sensorsystem, wie z.B. einer Digitalkamera, anfallenden Sensordaten häufig nach ihrer Erfassung noch komprimiert, um sie dadurch besser speichern oder auch übertragen zu können. Beim Compressed Sensing hingegen erfolgt die Gewinnung von Sensordaten und deren Komprimierung gewissermaßen gleichzeitig, d.h. die Sensordaten werden in einer bereits komprimierten Form erfasst.
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Hiroshi Inoue, Hiroshi Inoue · 2013
This paper discusses new bounds for restricted isometry property in compressed sensing. In the literature, E.J. Candès has proved that δ2s < √ 2 − 1 is a sufficient condition via l1 optimization having s-sparse vector solution. Later, many researchers have improved the sufficient conditions on δ2s or δs. In this paper, we have improved the sufficient condition to δs < 0.309 and have given the sufficient condition to δk (s < k) using an idea of Q. Mo and S. Li ’ result. Furthermore, we have improved the sufficient conditions to δ2s < 0.593 and δs < 0.472 in special case.
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David L. Donoho · 2006
Suppose x is an unknown vector in R m (depending on context, a digital image or signal); we plan to acquire data and then reconstruct. Nominally this ‘should ’ require m samples. But suppose we know a priori that x is compressible by transform coding with a known transform, and we are allowed to acquire data about x by measuring n general linear functionals – rather than the usual pixels. If the collection of linear functionals is well-chosen, and we allow for a degree of reconstruction error, the size of n can be dramatically smaller than the size m usually considered necessary. Thus, certain natural classes of images with m pixels need only n = O(m 1/4 log 5/2 (m)) nonadaptive nonpixel samples for faithful recovery, as opposed to the usual m pixel samples. Our approach is abstract and general. We suppose that the object has a sparse representation in some orthonormal basis (eg. wavelet, Fourier) or tight frame (eg curvelet, Gabor), meaning that the coefficients belong to an ℓ p ball for 0 < p ≤ 1. This implies that the N most important coefficients in the expansion allow a reconstruction with ℓ 2 error O(N 1/2−1/p). It is possible to design n = O(N log(m)) nonadaptive measurements which
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David L. Donoho · 2006
Suppose is an unknown vector in (a digital image or signal); we plan to measure general linear functionals of and then reconstruct. If is known to be compressible by transform coding with a known transform, and we reconstruct via the nonlinear procedure defined here, the number of measurements can be dramatically smaller than the size. Thus, certain natural classes of images with pixels need only = ( 1 4 log 5 2 ()) nonadaptive nonpixel samples for faithful recovery, as opposed to the usual pixel samples. More specifically, suppose has a sparse representation in some orthonormal basis (e.g., wavelet, Fourier) or tight frame (e.g., curvelet, Gabor)—so the coefficients belong to an ball for 0 1. The most important coefficients in that expansion allow reconstruction with 2 error ( 1 2 1
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Sidhant Misra, Ablo A. Parrilo, Terry P. Orlando et al. · 2011
The central problem of Compressed Sensing is to recover a sparse signal from fewer measurements than its ambient dimension. Recent results by Donoho, and Candes and Tao giving theoretical guarantees that ( 1-minimization succeeds in recovering the signal in a large number of cases have stirred up much interest in this topic. Subse-quent results followed, where prior information was imposed on the sparse signal and algorithms were proposed and analyzed to incorporate this prior information. In[13] Xu suggested the use of weighted fl-minimization in the case where the additional prior information is probabilistic in nature for a relatively simple probabilistic model. In this thesis, we exploit the techniques developed in [13] to extend the analysis to a more general class of probabilistic models, where the probabilities are evaluations of a continuous function at uniformly spaced points in a given interval. For this case, we use weights which have a similar characterization. We demonstrate our techniques through numerical computations for a certain class of weights and compare some of our results with empirical data obtained through simulations.
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