Convergence analysis for iterative data-driven tight frame construction scheme

Bao, Chenglong; Gao, Hui; Shen, Zuowei

doi:10.1016/j.acha.2014.06.007

Cited by 39 publications

(53 citation statements)

References 14 publications

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“…The trivial case is of course the one of finite systems in finite dimensions. If X = {x n } N n=1 ⊂ C M , then, with respect to the standard orthonormal bases, T X is given by the matrix (1) . .…”

Section: Frames Riesz Sequences and The Duality Principlementioning

confidence: 99%

“…Frames provide large flexibility in designing filters with improved performance in applications. For example, the filters used for image restoration in [1,13,91] are learned from the image, resulting in filters that capture certain features of the image and lead to a transform that gives a better sparse representation. In [75] Gabor frame filter banks are designed to achieve high orientation selectivity that adapts to the geometry of image edges for sparse image approximation.…”

Section: Introductionmentioning

confidence: 99%

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Duality for Frames

Fan

Heinecke

Shen

2015

J Fourier Anal Appl

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The subject of this article is the duality principle, which, well beyond its stand at the heart of Gabor analysis, is a universal principle in frame theory that gives insight into many phenomena. Its fiber matrix formulation for Gabor systems is the driving principle behind seemingly different results. We show how the classical duality identities, operator representations and constructions for dual Gabor frames are in fact aspects of the dual Gramian matrix fiberization and its sole duality principle, giving a unified view to all of them. We show that the same duality principle, via dual Gramian matrix analysis, holds for dual (or bi-) systems in abstract Hilbert spaces. The essence of the duality principle is the unitary equivalence of the frame operator and the Gramian of certain adjoint systems. An immediate consequence is, for example, that, even on this level of generality, dual frames are characterized in terms of biorthogonality relations of adjoint systems. We formulate the duality principle for irregular Gabor systems which have no structure whatsoever to the sampling of the shifts and modulations of the generating window. In case the shifts and modulations are sampled from lattices we show how the abstract matrices can be reduced to the simple structured fiber matrices of shift-invariant systems, thus arriving back in the well understood territory. Moreover, in the arena of multiresolution analysis J Fourier Anal Appl (MRA)-wavelet frames, the mixed unitary extension principle can be viewed as the duality principle in a sequence space. This perspective leads to a construction scheme for dual wavelet frames which is strikingly simple in the sense that it only needs the completion of an invertible constant matrix. Under minimal conditions on the MRA, our construction guarantees the existence and easy constructability of non-separable multivariate dual MRA-wavelet frames. The wavelets have compact support and we show examples for multivariate interpolatory refinable functions. Finally, we generalize the duality principle to the case of transforms that are no longer defined by discrete systems, but may have discrete adjoint systems.

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Section: Frames Riesz Sequences and The Duality Principlementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Duality for Frames

Fan

Heinecke

Shen

2015

J Fourier Anal Appl

Self Cite

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“…The method is shown in Algorithm 1. Recent works have shown convergence of Algorithm 1 or its variants to critical points in the equivalent unconstrained problems [16], [22], [23]. Here, we further prove local linear convergence of the method to the underlying generative (data) model under mild assumptions that depend on properties of the underlying/generating sparse coefficients.…”

Section: Introductionmentioning

confidence: 60%

Analysis of Fast Alternating Minimization for Structured Dictionary Learning

Ravishankar

Needell

2018

2018 Information Theory and Applications Workshop (ITA)

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Methods exploiting sparsity have been popular in imaging and signal processing applications including compression, denoising, and imaging inverse problems. Data-driven approaches such as dictionary learning enable one to discover complex image features from datasets and provide promising performance over analytical models. Alternating minimization algorithms have been particularly popular in dictionary and transform learning. In this work, we study the properties of alternating minimization for structured (unitary) sparsifying operator learning. While the algorithm converges to the stationary points of the non-convex problem in general, we prove local linear convergence to the underlying generative model under mild assumptions. Our experiments show that the unitary operator learning algorithm is robust to initialization.

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“…Meanwhile, the handcrafted framework of the models grants certain interpretability and theoretical foundation to the models. Successful examples include the method of optimal directions [55], the K-SVD [3], learning based PDE design [98], data-driven tight frame [26,8], Ada-frame [141], low-rank models [154,95,27,29,24], piecewise-smooth image models [109,25], and statistical models [74], etc.…”

Section: Image Reconstruction Modelsmentioning

confidence: 99%

A Review on Deep Learning in Medical Image Reconstruction

Zhang

Dong

2020

J. Oper. Res. Soc. China

132

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Medical imaging is crucial in modern clinics to provide guidance to the diagnosis and treatment of diseases. Medical image reconstruction is one of the most fundamental and important components of medical imaging, whose major objective is to acquire high-quality medical images for clinical usage at the minimal cost and risk to the patients. Mathematical models in medical image reconstruction or, more generally, image restoration in computer vision, have been playing a prominent role. Earlier mathematical models are mostly designed by human knowledge or hypothesis on the image to be reconstructed, and we shall call these models handcrafted models. Later, handcrafted plus data-driven modeling started to emerge which still mostly relies on human designs, while part of the model is learned from the observed data. More recently, as more data and computation resources are made available, deep learning based models (or deep models) pushed the data-driven modeling to the extreme where the models are mostly based on learning with minimal human designs. Both handcrafted and data-driven modeling have their own advantages and disadvantages. Typical handcrafted models are well interpretable with solid theoretical supports on the robustness, recoverability, complexity, etc., whereas they may not be flexible and sophisticated enough to fully leverage large data sets. Data-driven models, especially deep models, on the other hand, are generally much more flexible and effective in extracting useful information from large data sets, while they are currently still in lack of theoretical foundations. Therefore, one of the major research trends in medical imaging is to combine handcrafted modeling with deep modeling so that we can enjoy benefits from both approaches. The major part of this article is to provide a conceptual review of some recent works on deep modeling from the unrolling dynamics viewpoint. This viewpoint stimulates new designs of neural network architectures with inspirations from optimization algorithms and numerical differential equations. Given the popularity of deep modeling, there are still vast remaining challenges in the field, as well as opportunities which we shall discuss at the end of this article.

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Convergence analysis for iterative data-driven tight frame construction scheme

Cited by 39 publications

References 14 publications

Duality for Frames

Duality for Frames

Analysis of Fast Alternating Minimization for Structured Dictionary Learning

A Review on Deep Learning in Medical Image Reconstruction

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