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Recovery of piecewise smooth images from few fourier samples

Greg Ongie, Mathews Jacob
<span title="">2015</span> <i title="IEEE"> 2015 International Conference on Sampling Theory and Applications (SampTA) </i> &nbsp;
We introduce a Prony-like method to recover a continuous domain 2-D piecewise smooth image from few of its Fourier samples.  ...  Finally, we demonstrate our algorithm on the recovery of MRI phantoms from few low-resolution Fourier samples.  ...  A version of this approach was investigated in an earlier work for the super-resolution recovery of MR signals from few Fourier samples [9] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/sampta.2015.7148950">doi:10.1109/sampta.2015.7148950</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/wf3dhfkhynhk3kc4drbehlhcly">fatcat:wf3dhfkhynhk3kc4drbehlhcly</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170810104747/http://research.engineering.uiowa.edu/cbig/sites/research.engineering.uiowa.edu.cbig/files/files/sampta15.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/9d/08/9d08c944aac60b4103ca5240a7fe26fd896969bc.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/sampta.2015.7148950"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> ieee.com </button> </a>

Off-the-Grid Recovery of Piecewise Constant Images from Few Fourier Samples [article]

Greg Ongie, Mathews Jacob
<span title="2016-04-21">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We introduce a method to recover a continuous domain representation of a piecewise constant two-dimensional image from few low-pass Fourier samples.  ...  We present necessary and sufficient conditions for unique recovery of the image from finite low-pass Fourier samples using the annihilation relation.  ...  Parts of this work were presented at the IEEE International Symposium on Biomedical Imaging (ISBI '15) held April 16-19, 2015 in Brooklyn, New York, and at the International Conference on Sampling Theory  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1510.00384v2">arXiv:1510.00384v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6ispuyxjpvb3jln6xj4li7633a">fatcat:6ispuyxjpvb3jln6xj4li7633a</a> </span>
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Off-the-Grid Recovery of Piecewise Constant Images from Few Fourier Samples

Greg Ongie, Mathews Jacob
<span title="">2016</span> <i title="Society for Industrial &amp; Applied Mathematics (SIAM)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/amdwhvj4srb33cknicgjbgvsze" style="color: black;">SIAM Journal of Imaging Sciences</a> </i> &nbsp;
We introduce a method to recover a continuous domain representation of a piecewise constant twodimensional image from few low-pass Fourier samples.  ...  functions, from few Fourier samples.  ...  Edge set aware recovery of the image In this section we establish conditions for the unique recovery of a piecewise constant image from few of its low-pass Fourier samples with knowledge of the minimal  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1137/15m1042280">doi:10.1137/15m1042280</a> <a target="_blank" rel="external noopener" href="https://www.ncbi.nlm.nih.gov/pubmed/29973971">pmid:29973971</a> <a target="_blank" rel="external noopener" href="https://pubmed.ncbi.nlm.nih.gov/PMC6028195/">pmcid:PMC6028195</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/guwnrj5urfe27gqj6eeukvab6i">fatcat:guwnrj5urfe27gqj6eeukvab6i</a> </span>
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Structured Low-Rank Algorithms: Theory, MR Applications, and Links to Machine Learning [article]

Mathews Jacob, Merry P. Mani, Jong Chul Ye
<span title="2019-10-27">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this survey, we provide a detailed review of recent advances in the recovery of continuous domain multidimensional signals from their few non-uniform (multichannel) measurements using structured low-rank  ...  We demonstrate the utility of the framework in a wide range of MR imaging (MRI) applications, including highly accelerated imaging, calibration-free acquisition, MR artifact correction, and ungated dynamic  ...  recovery of surfaces or points living on surfaces from a few noisy samples.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1910.12162v1">arXiv:1910.12162v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/e7x2qzy52fhlbh2rrtasxkc5fi">fatcat:e7x2qzy52fhlbh2rrtasxkc5fi</a> </span>
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Structured low-rank recovery of piecewise constant signals with performance guarantees [article]

Greg Ongie, Sampurna Biswas, Mathews Jacob
<span title="2016-04-17">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We derive theoretical guarantees for the exact recovery of piecewise constant two-dimensional images from a minimal number of non-uniform Fourier samples using a convex matrix completion algorithm.  ...  The recovery algorithm arranges the known Fourier samples into the structured matrix then attempts recovery of the missing Fourier data by minimizing the nuclear norm subject to structure and data constraints  ...  DISCUSSION AND CONCLUSION We derived a performance guarantee for the recovery of piecewise constant images from non-uniform Fourier samples by a structured matrix completion.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1604.04888v1">arXiv:1604.04888v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5vr23tea7jgxfn5bkswjx5getq">fatcat:5vr23tea7jgxfn5bkswjx5getq</a> </span>
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Signal Representations on Graphs: Tools and Applications [article]

Siheng Chen and Rohan Varma and Aarti Singh and Jelena Kovačević
<span title="2015-12-16">2015</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Based on this framework, we study three typical classes of graph signals: smooth graph signals, piecewise-constant graph signals, and piecewise-smooth graph signals.  ...  We then study how such graph dictionary works in two standard tasks: approximation and sampling followed with recovery, both from theoretical as well as algorithmic perspectives.  ...  and Recovery: The goal of sampling and recovery is to collect a few samples from a graph signal, and then to recover the original graph signal from those samples either exactly or approximately.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1512.05406v1">arXiv:1512.05406v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/hzwsycryqrdzhiu2om6rnrnxfu">fatcat:hzwsycryqrdzhiu2om6rnrnxfu</a> </span>
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Novel Structured Low-rank algorithm to recover spatially smooth exponential image time series [article]

Arvind Balachandrasekaran, Mathews Jacob
<span title="2017-03-29">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We propose a structured low rank matrix completion algorithm to recover a time series of images consisting of linear combination of exponential parameters at every pixel, from under-sampled Fourier measurements  ...  This annihilation relation translates into a structured low rank matrix formed from the k-t samples.  ...  INTRODUCTION The recovery of linear combination of damped exponentials from few of their measurements is a classical problem in signal processing, starting with the seminal work of Prony [1] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1703.09880v1">arXiv:1703.09880v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/bu4yimg6dfdptcvunrh7zig3sy">fatcat:bu4yimg6dfdptcvunrh7zig3sy</a> </span>
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A Fast Algorithm for Convolutional Structured Low-Rank Matrix Recovery [article]

Greg Ongie, Mathews Jacob
<span title="2017-06-25">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
., Toeplitz, Hankel, or their multi-level generalizations, built from Fourier data of the image should be low-rank.  ...  Our experiments on the recovery of images from undersampled Fourier measurements show that the resulting algorithm is considerably faster than previously proposed algorithms, and can accommodate much larger  ...  , p = 1, 0, for recovery of Fourier coefficients of r Diracs, as in(1), from non-uniform random Fourier samples.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1609.07429v3">arXiv:1609.07429v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/pvn62si76rh2ldxz3y4phvk7k4">fatcat:pvn62si76rh2ldxz3y4phvk7k4</a> </span>
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A Fast Algorithm for Convolutional Structured Low-Rank Matrix Recovery

Gregory Ongie, Mathews Jacob
<span title="">2017</span> <i title="Institute of Electrical and Electronics Engineers (IEEE)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/qwupf4tqybcbpesmxen2dy6wyi" style="color: black;">IEEE Transactions on Computational Imaging</a> </i> &nbsp;
., Toeplitz, Hankel, or their multi-level generalizations, built from Fourier data of the image should be low-rank.  ...  Our experiments on the recovery of images from undersampled Fourier measurements show that the resulting algorithm is considerably faster than previously proposed algorithms, and can accommodate much larger  ...  Ongie and Jacob Page 44 Ongie and Jacob Page 45 TABLE I Convergence time of algorithms for structured low-rank recovery of piecewise constant images from noise-free random fourier samples.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/tci.2017.2721819">doi:10.1109/tci.2017.2721819</a> <a target="_blank" rel="external noopener" href="https://www.ncbi.nlm.nih.gov/pubmed/29911129">pmid:29911129</a> <a target="_blank" rel="external noopener" href="https://pubmed.ncbi.nlm.nih.gov/PMC5999344/">pmcid:PMC5999344</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/y3rmdw7qxbfwxcrm36kc6m3qpu">fatcat:y3rmdw7qxbfwxcrm36kc6m3qpu</a> </span>
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Experiments with Compressively Sampled Images and a New Debluring-Denoising Algorithm

Sina Jafarpour, Ali Pezeshki, Robert Calderbank
<span title="">2008</span> <i title="IEEE"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/hvpvmtth5ndrjdu4ahfgq3tizm" style="color: black;">2008 Tenth IEEE International Symposium on Multimedia</a> </i> &nbsp;
In this paper we will examine the effect of different parameters in the quality of real compressively sampled images in the compressed sensing framework.  ...  We will select a variety of different real images of different types and test the quality of the recovered images, the recovery time, and required resources when different measurement methods with different  ...  l1-magic package [22] which was used in the image recovery process.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/ism.2008.119">doi:10.1109/ism.2008.119</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/ism/JafarpourPC08.html">dblp:conf/ism/JafarpourPC08</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/kn7hrzmsqrhvvoz2q3sgiveqam">fatcat:kn7hrzmsqrhvvoz2q3sgiveqam</a> </span>
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Image Reconstruction from Undersampled Fourier Data Using the Polynomial Annihilation Transform

Rick Archibald, Anne Gelb, Rodrigo B. Platte
<span title="2015-09-05">2015</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/dqvu4s4i3jflxoqpe3v2njc6y4" style="color: black;">Journal of Scientific Computing</a> </i> &nbsp;
In so doing, we achieve a more accurate image reconstruction method from under-sampled and noisy Fourier data.  ...  Fourier samples are collected in a variety of applications including magnetic resonance imaging (MRI) and synthetic aperture radar (SAR). The data are typically under-sampled and noisy.  ...  Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program. Accordingly, the U.S.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s10915-015-0088-2">doi:10.1007/s10915-015-0088-2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7tz6b37ybrgdph7qjmltx2uw3m">fatcat:7tz6b37ybrgdph7qjmltx2uw3m</a> </span>
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Hybrid Compressive Sampling via a New Total Variation TVL1 [chapter]

Xianbiao Shu, Narendra Ahuja
<span title="">2010</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Compressive sampling (CS) is aimed at acquiring a signal or image from data which is deemed insufficient by Nyquist/Shannon sampling theorem.  ...  First, by minimizing the 1-norm of partial gradients, it can achieve greater accuracy than the widely-used TV 1 2 based CS.  ...  Conclusion In this paper, we propose a hybrid compressive sampling method using a new TV measure T V 1 , for recovering a piecewise smooth image containing all possible sharper edges from limited measurements  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-15567-3_29">doi:10.1007/978-3-642-15567-3_29</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5ogjeqo4gzck3nprmakvg5nfzy">fatcat:5ogjeqo4gzck3nprmakvg5nfzy</a> </span>
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Two-stage Geometric Information Guided Image Reconstruction [article]

Jing Qin, Weihong Guo
<span title="2021-02-16">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Existing methods mostly work well on piecewise constant images but not so well on piecewise smooth images such as natural images, medical images that contain a lot of details.  ...  In compressive sensing, it is challenging to reconstruct image of high quality from very few noisy linear projections.  ...  Acknowledgement The authors would like to thank the partial support from the Case Western Reserve University ACES+ advance opportunity funds.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1409.7450v2">arXiv:1409.7450v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qqlgwddv2vggbc3aq5kgevu23a">fatcat:qqlgwddv2vggbc3aq5kgevu23a</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20191012094901/https://arxiv.org/pdf/1409.7450v1.pdf" title="fulltext PDF download [not primary version]" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <span style="color: #f43e3e;">&#10033;</span> <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/53/06/53069bae1c5060445b0c230d2b6adf6722675a75.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1409.7450v2" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

A method to reduce the Gibbs ringing artifact in MRI scans while keeping tissue boundary integrity

R. Archibald, A. Gelb
<span title="">2002</span> <i title="Institute of Electrical and Electronics Engineers (IEEE)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/yhnt2pif75h2bmnnmdg7m6nlc4" style="color: black;">IEEE Transactions on Medical Imaging</a> </i> &nbsp;
Gibbs ringing is a well known artifact that effects reconstruction of images having discontinuities.  ...  This paper presents the application of the Gegenbauer reconstruction method to neuro-imaging.  ...  Recovery of images from noisy MRI data was addressed in [1] , which required the expansion of techniques and theory described here.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/tmi.2002.1000255">doi:10.1109/tmi.2002.1000255</a> <a target="_blank" rel="external noopener" href="https://www.ncbi.nlm.nih.gov/pubmed/12022619">pmid:12022619</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/jypzfrq5argjvpn26cy266iddi">fatcat:jypzfrq5argjvpn26cy266iddi</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20060503203021/http://math.la.asu.edu:80/~ag/ArchGelbMed.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/03/66/0366c823ccb3f958fcf908907d9349ecf1f84175.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/tmi.2002.1000255"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> ieee.com </button> </a>

Fourier reconstruction of univariate piecewise-smooth functions from non-uniform spectral data with exponential convergence rates

Rodrigo B. Platte, Alexander J. Gutierrez, Anne Gelb
<span title="">2015</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/d6gsqr7e2vhqfcooye42dqftf4" style="color: black;">Applied and Computational Harmonic Analysis</a> </i> &nbsp;
Reconstruction of piecewise smooth functions from non-uniform Fourier data arises in sensing applications such as magnetic resonance imaging (MRI).  ...  This paper presents a new method that uses edge information to recover the Fourier transform of a piecewise smooth function from data that is sparsely sampled at high frequencies.  ...  Many of the numerical results presented were obtained with the aid of the Chebfun System [21, 23, 24] . We also thank the two anonymous reviewers for their valuable suggestions and comments.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.acha.2014.10.002">doi:10.1016/j.acha.2014.10.002</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/x4depn7bcfapphg24kll3vitoq">fatcat:x4depn7bcfapphg24kll3vitoq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170705183749/https://math.la.asu.edu/%7Eplatte/pub/Driver_ACHArevised.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/8e/b5/8eb58682e7f31dcebad86e4c4e853ddbdd61cae9.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.acha.2014.10.002"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>
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