张量稀疏性度量综述
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摘要
稀疏性是现实数据所共有的一般信息表达特性,其含义为数据可由其本质所蕴含的少量基元素进行充分表达。目前,向量(1阶数组)与矩阵(2阶数组)数据均存在较为成熟的稀疏性表达度量,即向量的非零元素个数与矩阵的秩。然而,对于张量(高阶数组)数据的合理稀疏性度量的构造尚未形成统一的解决方案。对张量稀疏性研究的现状进行综合介绍,回顾目前在此方向的研究进展以及所取得的典型应用,并着重介绍本研究小组基于Tucker分解与CANDECOMP/PARAFAC(CP)分解的张量稀疏性内涵理解所构造的一种新型张量稀疏性度量。将此张量稀疏性度量与10个传统方法应用到多光谱图像去噪中进行对比实验,通过数值与视觉上的实验结果说明所提方法的合理性与有效性。
Sparsity is a common representation characteristic underlying practical data,which intrinsically means that a vector/matrix/tensor can be sufficiently represented by a few basis elements underlying the data. Currently,the sparsity measure of vector( one array datum) and matrix( two array datum) have been widely investigated and the researches on them are relatively mature. However,it is still a critical issue to construct a rational sparity measure for a general tensor( multi-array datum). In this study,we will introduce some current sparsity measures raised for tensor representation,and especially review representative research developments and typical applications along this research line. Specifically,we'll introduce a new tensor sparsity measure presented by our research team,as well as its fine applications on hyper-spectral image denoising. We also compare the proposed sparsity measure with 10 traditional methods in multispectral image denoising experiments. The numerical and visual results verify the rationality and effectiveness of the proposed method.
Sparsity is a common representation characteristic underlying practical data,which intrinsically means that a vector/matrix/tensor can be sufficiently represented by a few basis elements underlying the data. Currently,the sparsity measure of vector( one array datum) and matrix( two array datum) have been widely investigated and the researches on them are relatively mature. However,it is still a critical issue to construct a rational sparity measure for a general tensor( multi-array datum). In this study,we will introduce some current sparsity measures raised for tensor representation,and especially review representative research developments and typical applications along this research line. Specifically,we'll introduce a new tensor sparsity measure presented by our research team,as well as its fine applications on hyper-spectral image denoising. We also compare the proposed sparsity measure with 10 traditional methods in multispectral image denoising experiments. The numerical and visual results verify the rationality and effectiveness of the proposed method.
引文
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[25]MAGGIONI M,FOI A.Nonlocal transform-domain denoising of volumetric data with groupwise adaptive variance estimation[C]//International Society for Optical Engineering.[S.l.]:SPIE,2012.
[26]RENARD S B N,BLANC TJ.Denoising and dimensionality reduction using multilinear tools for hyperspectral images[J].IEEE Trans Geoscience and Remote Sensing,2008,5(2):138-142.
[27]LIU X,BOURENNANE S,FOSSATI C.Denoising of hyperspectral images using the parafac model and statistical performance analysis[J].IEEE Trans Geoscience and Remote Sensing,2012,50(10):3717-3724.
[28]PENG Yi,MENG Deyu,XU Zongben,et al.Decomposable nonlocal tensor dictionary learning for multispectral image denoising[C]//IEEE Conference on Computer Vision and Pattern Recognition.[S.l.]:CVPR,2014.
[29]GOLBABAEE M,VANDERGHEYNST P.Joint Trace/TV Norm Minimization:A New Efficient Approach For Spectral Compressive Imaging[C]//IEEE International Conference on Image Processing.New York:IEEE Press,2013.
[2]CANDES E J,WAKIN M B,BOYD S P.Enhancing sparsity by reweighted l1 minimization[J].Fourier analysis and applications,2008,14(5-6):877-905.
[3]de la TORRE F,BLACK M J.A framework for robust subspace learning[J].International Journal of Computer Vision,2003,54(1-3):117-142.
[4]GU Shuhang,ZHANG Lei,ZUO Wangmeng,et d al.Weighted nuclear norm minimization with application to image denoising[C]//IEEE Conference on Computer Vision and Pattern Recognition.[S.l.]:CVPR,2014.
[5]MENG Deyu,TORRE F.Robust matrix factorization with unknown noise[C]//IEEE International Conference on Computer Vision.[S.l.]:ICCV,2013:1337-1344.
[6]PARVARESH F,VIKALO H,MISRA S,et al.Recovering sparse signals using sparse measurement matrices in compressed dna microarrays[J].IEEE Journal of Selected Topics in Signal Processing,2008,2(3):275-285.
[7]WRIGHT J,GANESH A,RAO S,et al.Robust principal component analysis:Exact recovery of corrupted low-rank matrices via convex optimization[C]//Neural information processing systems conference.[S.l.]:NIPS,2009.
[8]ZHAO Qian,MENG Deyu,XU Zongben,et al.Robust principal component analysis with complex noise[C]//International conference on machine learning.[S.l.]:IC-ML,2014:55-63.
[9]KOLDA T G,BADER B W.Tensor decompositions and applications[J].Society for Industrial and Applied Mathematics Review,2009,51(3):455-500.
[10]TUCKER L R.Some mathematical notes on three-mode factor analysis[J].Psychometrika,1966,31(3):279-311.
[11]LIU Ji,MUSIALSKI P,WONKA P,et al.Tensor completion for estimating missing values in visual data[J].IEEETransactions on Pattern Analysis and Machine Intelligence,2013,35(1):208-220.
[12]GANDY S,RECHT B,YAMADA I.Tensor completion and low-n-rank tensor recovery via convex optimization[J].Inverse Problems,2011,27(2):025010.
[13]TOMIOKA R,HAYASHI K,KASHIMA H.Estimation of low-rank tensors via convex optimization[EB/OL].(2010-10-05)[2019-05-28].https://arxiv.org/abs/1010.0789.
[14]SIGNORETTO M,DINH Q T,de LATHAUWER L,et al.Learning with tensors:a framework based on convex optimization and spectral regularization[J].Machine Learning,2014,94(3):303-351.
[15]HUANG Bo,MUCUN,GOLDFARB D,et al.Provable lowrank tensor recovery[J].Optimization-Online,2014,4252(2):455-500.
[16]CAO Wenfei,WANG Yao,YANG Can,et al.Foldedconcave penalization approaches to tensor completion[J].Neurocomputing,2015(152):261-273.
[17]ZHANG Zemin,ELY G,AERON S,et al.Novel methods for multilinear data completion and de-noising based on tensorsvd[C]//IEEE Conference on Computer Vision and Pattern Recognition.[S.l.]:CVPR,2014.
[18]LU Canyi,FENG Jiashi,CHEN Yudong,et al.Tensor robust principal component analysis:Exact recovery of corrupted lowrank tensors via convex optimization[C]//IEEE Conference on Computer Vision and Pattern Recognition.[S.l.]:CVPR,2016.
[19]JIANG B,MA S,ZHANG S.Low-M-Rank Tensor Completion and Robust Tensor PCA[EB/OL].(2015-01-15)[2019-05-28].https://arxiv.org/abs/1501.03689v5.
[20]XIE Qi,ZHAO Qian,MENG Deyu,et al.Multispectral images denoising by intrinsic tensor sparsity regularization[C]//IEEE Conference on Computer Vision and Pattern Recognition.[S.l.]:CVPR,2016.
[21]AHARON M.K-SVD:An algorithm for designing overcomplete dictionaries for sparse representation[J].IEEETrans Signal Processing,2006,54(11):4311-4322.
[22]DABOV K,FOI A,KATKOVNIK V,et al.Image denoising by sparse 3-d transform-domain collaborative filtering[J].IEEE Trans Image Processing,2007,16(8):2080-2095.
[23]ELAD M,AHARON M.Image denoising via sparse and redundant representations over learned dictionaries[J].IEEE Trans Image Processing,2006,15(12):3736-3745.
[24]MANJ'ON J V,COUP'E P,MART'I B L,et al.Adaptive non-local means denoising of mr images with spatially varying noise levels[J].Journal of Magnetic Resonance Imaging,2010,31(1):192-203.
[25]MAGGIONI M,FOI A.Nonlocal transform-domain denoising of volumetric data with groupwise adaptive variance estimation[C]//International Society for Optical Engineering.[S.l.]:SPIE,2012.
[26]RENARD S B N,BLANC TJ.Denoising and dimensionality reduction using multilinear tools for hyperspectral images[J].IEEE Trans Geoscience and Remote Sensing,2008,5(2):138-142.
[27]LIU X,BOURENNANE S,FOSSATI C.Denoising of hyperspectral images using the parafac model and statistical performance analysis[J].IEEE Trans Geoscience and Remote Sensing,2012,50(10):3717-3724.
[28]PENG Yi,MENG Deyu,XU Zongben,et al.Decomposable nonlocal tensor dictionary learning for multispectral image denoising[C]//IEEE Conference on Computer Vision and Pattern Recognition.[S.l.]:CVPR,2014.
[29]GOLBABAEE M,VANDERGHEYNST P.Joint Trace/TV Norm Minimization:A New Efficient Approach For Spectral Compressive Imaging[C]//IEEE International Conference on Image Processing.New York:IEEE Press,2013.