Analysis of data separation and recovery problems using clustered sparsity

King, Emily J.; Kutyniok, Gitta; Zhuang, Xiaosheng

doi:10.1117/12.892723

Cited by 18 publications

(21 citation statements)

References 34 publications

(42 reference statements)

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“…The preliminary results presented in the SPIE Proceedings paper [KKZ11] combined with the theory in this paper provide the first comprehensive analysis of discrete dictionaries inpainting the continuum domain utilizing the novel concept of clustered sparsity, which besides leading to asymptotic error bounds also makes the superior behavior of directional representation systems over wavelets precise. Along the way, our abstract model and analysis lay a common theoretical foundation for data recovery problems when utilizing either analysis-side 1 minimization or thresholding as recovery schemes (Section 2).…”

Section: Introductionmentioning

confidence: 91%

“…Also, since we are only interested in correctly inpainting and not in computing the sparsest expansion, we can circumvent possible problems by solving the inpainting problem by selecting a particular coefficient sequence which expands out to the x, namely the analysis sequence. A similar strategy was pursued in [KKZ11] and [Kut12]. Various inpainting algorithms which are based on the core idea of (Inp) combined with geometric separation are heuristically shown to be successful in [CCS10, DJL + 12, ESQD05].…”

Section: Inpainting Via 1 Minimizationmentioning

confidence: 99%

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Analysis of Inpainting via Clustered Sparsity and Microlocal Analysis

2013

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Recently, compressed sensing techniques in combination with both wavelet and directional representation systems have been very effectively applied to the problem of image inpainting. However, a mathematical analysis of these techniques which reveals the underlying geometrical content is completely missing. In this paper, we provide the first comprehensive analysis in the continuum domain utilizing the novel concept of clustered sparsity, which besides leading to asymptotic error bounds also makes the superior behavior of directional representation systems over wavelets precise. First, we propose an abstract model for problems of data recovery and derive error bounds for two different recovery schemes, namely 1 minimization and thresholding. Second, we set up a particular microlocal model for an image governed by edges inspired by seismic data as well as a particular mask to model the missing data, namely a linear singularity masked by a horizontal strip. Applying the abstract estimate in the case of wavelets and of shearlets we prove that -provided the size of the missing part is asymptotically to the size of the analyzing functions -asymptotically precise inpainting can be obtained for this model. Finally, we show that shearlets can fill strictly larger gaps than wavelets in this model.

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Section: Introductionmentioning

confidence: 91%

Section: Inpainting Via 1 Minimizationmentioning

confidence: 99%

Analysis of Inpainting via Clustered Sparsity and Microlocal Analysis

2013

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“…Inspired by the ideas of compressed sensing (cf. [9,14]), we now present a recovery algorithm which was already used in [25,26]:…”

Section: Inpainting Via ℓ 1 -Minimizationmentioning

confidence: 99%

Asymptotic Analysis of Inpainting via Universal Shearlet Systems

Genzel

Kutyniok

2014

SIAM J. Imaging Sci.

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Recently introduced inpainting algorithms using a combination of applied harmonic analysis and compressed sensing have turned out to be very successful. One key ingredient is a carefully chosen representation system which provides (optimally) sparse approximations of the original image. Due to the common assumption that images are typically governed by anisotropic features, directional representation systems have often been utilized. One prominent example of this class are shearlets, which have the additional benefit to allowing faithful implementations. Numerical results show that shearlets significantly outperform wavelets in inpainting tasks. One of those software packages, www.shearlab.org, even offers the flexibility of using a different parameter for each scale, which is not yet covered by shearlet theory.In this paper, we first introduce universal shearlet systems which are associated with an arbitrary scaling sequence, thereby modeling the previously mentioned flexibility. In addition, this novel construction allows for a smooth transition between wavelets and shearlets and therefore enables us to analyze them in a uniform fashion. For a large class of such scaling sequences, we first prove that the associated universal shearlet systems form band-limited Parseval frames for L 2 (R 2 ) consisting of Schwartz functions. Secondly, we analyze the performance for inpainting of this class of universal shearlet systems within a distributional model situation using an ℓ 1 -analysis minimization algorithm for reconstruction. Our main result in this part states that, provided the scaling sequence is comparable to the size of the (scale-dependent) gap, nearly-perfect inpainting is achieved at sufficiently fine scales.

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“…This sparsity can be used to perform image and data processing tasks including geometric separation of data into components with different fundamental structures [2,3] and image inpainting or, more broadly, data recovery [2,4,5]. These two tasks can be combined.…”

mentioning

confidence: 99%

A theoretical guarantee for data completion via geometric separation

King

Murphy

2017

Proc Appl Math and Mech

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Scientific and commercial data is often incomplete. Recovery of the missing information is an important pre-processing step in data analysis. Real-world data can in many cases be represented as a superposition of two or more different types of structures. For example, images may often be decomposed into texture and cartoon-like components. When incomplete data comes from a distribution well-represented as a mixture of different structures, a sparsity-based method combining concepts from data completion and data separation can successfully recover the missing data. This short note presents a theoretical guarantee for success of the combined separation and completion approach which generalizes proofs from the distinct problems.

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Analysis of data separation and recovery problems using clustered sparsity

Cited by 18 publications

References 34 publications

Analysis of Inpainting via Clustered Sparsity and Microlocal Analysis

Analysis of Inpainting via Clustered Sparsity and Microlocal Analysis

Asymptotic Analysis of Inpainting via Universal Shearlet Systems

A theoretical guarantee for data completion via geometric separation

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