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Compressed sensing MRI using sparsity induced from adjacent slice similarity 利用相邻切片相似度引起的稀疏性压缩感知MRI
2015 International Conference on Sampling Theory and Applications (SampTA) Pub Date : 2015-05-25 DOI: 10.1109/SAMPTA.2015.7148898
A. Hirabayashi, Norihito Inamuro, Kazushi Mimura, Toshiyuki Kurihara, Toshiyuki Homma
{"title":"Compressed sensing MRI using sparsity induced from adjacent slice similarity","authors":"A. Hirabayashi, Norihito Inamuro, Kazushi Mimura, Toshiyuki Kurihara, Toshiyuki Homma","doi":"10.1109/SAMPTA.2015.7148898","DOIUrl":"https://doi.org/10.1109/SAMPTA.2015.7148898","url":null,"abstract":"We propose a fast magnetic resonance imaging (MRI) technique based on compressed sensing. The main idea is to use a combination of full and compressed sensing. Full sensing is conducted for every several slices (F-slice) while compressed sensing with high compression rate is applied to the rest of slices (C-slice). We can perfectly reconstruct F-slice images, which are used to roughly estimate the C-slices. Since the estimate is already of good quality, its difference from the original image is small and sparse. Therefore, the difference can be reconstructed precisely using the standard compressed sensing technique even with high compression rate. Simulation results show that the proposed method outperforms conventional methods with 3.16dB for arm images, 0.26dB for brain images in average for the C-slices with perfect reconstruction for the F-slices.","PeriodicalId":311830,"journal":{"name":"2015 International Conference on Sampling Theory and Applications (SampTA)","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134163100","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 16
Compressed sensing Petrov-Galerkin approximations for parametric PDEs 参数偏微分方程的压缩感知Petrov-Galerkin逼近
2015 International Conference on Sampling Theory and Applications (SampTA) Pub Date : 2015-05-25 DOI: 10.1109/SAMPTA.2015.7148947
Jean-Luc Bouchot, Benjamin Bykowski, H. Rauhut, C. Schwab
{"title":"Compressed sensing Petrov-Galerkin approximations for parametric PDEs","authors":"Jean-Luc Bouchot, Benjamin Bykowski, H. Rauhut, C. Schwab","doi":"10.1109/SAMPTA.2015.7148947","DOIUrl":"https://doi.org/10.1109/SAMPTA.2015.7148947","url":null,"abstract":"We consider the computation of parametric solution families of high-dimensional stochastic and parametric PDEs. We review theoretical results on sparsity of polynomial chaos expansions of parametric solutions, and on compressed sensing based collocation methods for their efficient numerical computation. With high probability, these randomized approximations realize best N-term approximation rates afforded by solution sparsity and are free from the curse of dimensionality, both in terms of accuracy and number of samples evaluations (i.e. PDE solves). Through various examples we illustrate the performance of Compressed Sensing Petrov-Galerkin (CSPG) approximations of parametric PDEs, for the computation of (functionals of) solutions of intregral and differential operators on high-dimensional parameter spaces. The CSPG approximations reduce the number of PDE solves, as compared to Monte-Carlo methods, while being likewise nonintrusive, and being “embarassingly parallel”, unlike dimension-adaptive collocation or Galerkin methods.","PeriodicalId":311830,"journal":{"name":"2015 International Conference on Sampling Theory and Applications (SampTA)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125473340","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 15
Projection retrieval: Theory and algorithms 投影检索:理论与算法
2015 International Conference on Sampling Theory and Applications (SampTA) Pub Date : 2015-05-25 DOI: 10.1109/SAMPTA.2015.7148876
M. Fickus, D. Mixon
{"title":"Projection retrieval: Theory and algorithms","authors":"M. Fickus, D. Mixon","doi":"10.1109/SAMPTA.2015.7148876","DOIUrl":"https://doi.org/10.1109/SAMPTA.2015.7148876","url":null,"abstract":"We consider the fundamental problem of determining a low-rank orthogonal projection operator P from measurements of the form || Px||. First, we leverage a nonembedding result for the complex Grassmannian to establish and analyze a lower bound on the number of measurements necessary to uniquely determine every possible P. Next, we provide a collection of particularly few measurement vectors that uniquely determine almost every P. Finally, we propose manifold-constrained least-squares optimization as a general technique for projection retrieval.","PeriodicalId":311830,"journal":{"name":"2015 International Conference on Sampling Theory and Applications (SampTA)","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125670935","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 10
Numerical solution of underdetermined systems from partial linear circulant measurements 欠定系统的部分线性循环测量数值解
2015 International Conference on Sampling Theory and Applications (SampTA) Pub Date : 2015-05-25 DOI: 10.1109/SAMPTA.2015.7148893
Jean-Luc Bouchot, Lei Cao
{"title":"Numerical solution of underdetermined systems from partial linear circulant measurements","authors":"Jean-Luc Bouchot, Lei Cao","doi":"10.1109/SAMPTA.2015.7148893","DOIUrl":"https://doi.org/10.1109/SAMPTA.2015.7148893","url":null,"abstract":"We consider the traditional compressed sensing problem of recovering a sparse solution from undersampled data. We are in particular interested in the case where the measurements arise from a partial circulant matrix. This is motivated by practical physical setups that are usually implemented through convolutions. We derive a new optimization problem that stems from the traditional ℓ1 minimization under constraints, with the added information that the matrix is taken by selecting rows from a circulant matrix. With this added knowledge it is possible to simulate the full matrix and full measurement vector on which the optimization acts. Moreover, as circulant matrices are well-studied it is known that using Fourier transform allows for fast computations. This paper describes the motivations, formulations, and preliminary results of this novel algorithm, which shows promising results.","PeriodicalId":311830,"journal":{"name":"2015 International Conference on Sampling Theory and Applications (SampTA)","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127347411","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Wavelet coorbit theory in higher dimensions: An overview 高维小波共轨理论综述
2015 International Conference on Sampling Theory and Applications (SampTA) Pub Date : 2015-05-25 DOI: 10.1109/SAMPTA.2015.7148851
H. Führ
{"title":"Wavelet coorbit theory in higher dimensions: An overview","authors":"H. Führ","doi":"10.1109/SAMPTA.2015.7148851","DOIUrl":"https://doi.org/10.1109/SAMPTA.2015.7148851","url":null,"abstract":"The continuous wavelet transform is frequently described as a mathematical microscope. In higher dimensions, there is an increasingly larger choice of such microscopes available, which significantly differ in the way that the wavelet (the “lense” of the microscope) is scaled/rotated/sheared etc. by elements of the dilation group. Summarizing recent results, this note presents a unified and comprehensive approach that allows to study approximation-theoretic aspects of such wavelet systems arising from a rather general class of dilation groups, including the shearlet dilation groups in dimension 2 and higher, using the language and results of coorbit theory. The key feature which this unified approach is based on is the so-called dual action of the dilation group.","PeriodicalId":311830,"journal":{"name":"2015 International Conference on Sampling Theory and Applications (SampTA)","volume":"162 6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129176928","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Unique recovery from edge information 唯一的恢复边缘信息
2015 International Conference on Sampling Theory and Applications (SampTA) Pub Date : 2015-05-25 DOI: 10.1109/SAMPTA.2015.7148903
Benjamin Allen, M. Kon
{"title":"Unique recovery from edge information","authors":"Benjamin Allen, M. Kon","doi":"10.1109/SAMPTA.2015.7148903","DOIUrl":"https://doi.org/10.1109/SAMPTA.2015.7148903","url":null,"abstract":"We study the inverse problem of recovering a function f from the nodes (zeroes) of its wavelet transform. The solution also provides an answer to a generalization of the Marr conjecture in wavelet and mathematical vision theory, regarding whether an image is uniquely determined by its edge information. The question has also other forms, including whether nodes of heat and related equation solutions determine their initial conditions. The general Marr problem reduces in a natural way to the moment problem for reconstructing f, using the moment basis on Rd (Taylor monomials xα), and its dual basis (derivatives δ(α) of of the Dirac delta distribution), expanding the wavelet transform in moments of f. If f has exponential decay and the wavelet's derivatives satisfy generic positions for their zeroes, then f can be uniquely recovered. We show this is the strongest statement of its type. For the original Gaussian wavelet unique recovery reduces to genericity of zeroes of so-called Laplace-Hermite polynomials, which is proved in one dimension.","PeriodicalId":311830,"journal":{"name":"2015 International Conference on Sampling Theory and Applications (SampTA)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129769663","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
Directionally sensitive weight functions 方向敏感的权重函数
2015 International Conference on Sampling Theory and Applications (SampTA) Pub Date : 2015-05-25 DOI: 10.1109/SAMPTA.2015.7148881
R. Aceska, H. H. Giv
{"title":"Directionally sensitive weight functions","authors":"R. Aceska, H. H. Giv","doi":"10.1109/SAMPTA.2015.7148881","DOIUrl":"https://doi.org/10.1109/SAMPTA.2015.7148881","url":null,"abstract":"We study new properties of a recently introduced, directionally sensitive short-time Fourier transform. Its specific relationship to the classical short-time Fourier transform and its quasi shift-invariance property motivate us to introduce customized weight functions with directional sensitivity.","PeriodicalId":311830,"journal":{"name":"2015 International Conference on Sampling Theory and Applications (SampTA)","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126483073","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Spherical designs as a tool for derandomization: The case of PhaseLift 球形设计作为非随机化的工具:PhaseLift的案例
2015 International Conference on Sampling Theory and Applications (SampTA) Pub Date : 2015-05-25 DOI: 10.1109/SAMPTA.2015.7148878
R. Kueng, D. Gross, F. Krahmer
{"title":"Spherical designs as a tool for derandomization: The case of PhaseLift","authors":"R. Kueng, D. Gross, F. Krahmer","doi":"10.1109/SAMPTA.2015.7148878","DOIUrl":"https://doi.org/10.1109/SAMPTA.2015.7148878","url":null,"abstract":"The problem of retrieving phase information from amplitude measurements alone has appeared in many scientific disciplines over the last century. PhaseLift is a recently introduced algorithm for phase recovery that is computationally tractable and numerically stable. However, initial rigorous performance guarantees relied specifically on Gaussian random measurement vectors. To date, it remains unclear which properties of the measurements render the problem well-posed. With this question in mind, we employ the concept of spherical t-designs to achieve a partial derandomziation of PhaseLift. Spherical designs are ensembles of vectors which reproduce the first 2t moments of the uniform distribution on the complex unit sphere. As such, they provide notions of “evenly distributed” sets of vectors, ranging from tight frames (t = 1) to the full sphere, as t approaches infinity. Beyond the specific case of PhaseLift, this result highlights the utility of spherical designs for the derandomization of data recovery schemes.","PeriodicalId":311830,"journal":{"name":"2015 International Conference on Sampling Theory and Applications (SampTA)","volume":"403 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120882322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 11
Construction of orthonormal directional wavelets based on quincunx dilation subsampling 基于五频扩张子采样的正交定向小波构造
2015 International Conference on Sampling Theory and Applications (SampTA) Pub Date : 2015-05-25 DOI: 10.1109/SAMPTA.2015.7148899
Rujie Yin
{"title":"Construction of orthonormal directional wavelets based on quincunx dilation subsampling","authors":"Rujie Yin","doi":"10.1109/SAMPTA.2015.7148899","DOIUrl":"https://doi.org/10.1109/SAMPTA.2015.7148899","url":null,"abstract":"Using quincunx downsampling for bases and standard downsampling for low-redundancy frames, we construct directional wavelet systems that have the same direction selectivity as shearlets in the first frequency dyadic ring; these are the first step towards the construction of efficient shearlet systems.","PeriodicalId":311830,"journal":{"name":"2015 International Conference on Sampling Theory and Applications (SampTA)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115254683","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Sampling and (sparse) stochastic processes: A tale of splines and innovation 抽样和(稀疏)随机过程:样条和创新的故事
2015 International Conference on Sampling Theory and Applications (SampTA) Pub Date : 2015-05-25 DOI: 10.1109/SAMPTA.2015.7148884
M. Unser
{"title":"Sampling and (sparse) stochastic processes: A tale of splines and innovation","authors":"M. Unser","doi":"10.1109/SAMPTA.2015.7148884","DOIUrl":"https://doi.org/10.1109/SAMPTA.2015.7148884","url":null,"abstract":"The commonality between splines and Gaussian or sparse stochastic processes is that they are ruled by the same type of differential equations. Our purpose here is to demonstrate that this has profound implications for the three primary forms of sampling: uniform, nonuniform, and compressed sensing. The connection with classical sampling is that there is a one-to-one correspondence between spline interpolation and the minimum-mean-square-error reconstruction of a Gaussian process from its uniform or nonuniform samples. The caveat, of course, is that the spline type has to be matched to the operator that whitens the process. The connection with compressed sensing is that the non-Gaussian processes that are ruled by linear differential equations generally admit a parsimonious representation in a wavelet-like basis. There is also a construction based on splines that yields a wavelet-like basis that is matched to the underlying differential operator. It has been observed that expansions in such bases provide excellent M-term approximations of sparse processes. This property is backed by recent estimates of the local Besov regularity of sparse processes.","PeriodicalId":311830,"journal":{"name":"2015 International Conference on Sampling Theory and Applications (SampTA)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115588138","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
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