Abstract：Since traditional compressive sensing reconstruction algorithms have lower reconstruction quality and longer running time,a fast reconstruction algorithm based on separable dictionary training is proposed.Firstly,we choose one class of images as training set and construct their models of generalized low-rank matrix approximation.Then,the alternating direction method is used to solve the model,and we can obtain separable dictionaries.Finally,the separable dictionaries are applied to image reconstruction and realize fast reconstruction of image by simple linear operation.The experimental results show that the proposed algorithm has a better reconstruction performance for training set images compared to traditional reconstruction algorithms.In addition,for other types of images,our algorithm has a good reconstruction quality and a lower reconstruction time.
张长伦, 余沾, 王恒友, 何强. 基于广义低秩矩阵分解的分离字典训练及其快速重建算法[J]. 电子学报, 2018, 46(10): 2400-2409.
ZHANG Chang-lun, YU Zhan, WANG Heng-you, HE Qiang. Separable Dictionary Training and Its Fast Reconstruction Algorithm Based on Generalized Low-Rank Matrix Approximation. Acta Electronica Sinica, 2018, 46(10): 2400-2409.
[1] Candès E,Romberg J,Tao T.Robust uncertainty principles:exact signal reconstruction from highly incomplete frequency information[J].IEEE Transactions on Information Theory,2006,52(2):489-509.
[2] Donoho D.Compressed sensing[J].IEEE Transactions on Information Theory,2006,52(4):1289-1306.
[3] 戴琼海,付长军,季向阳.压缩感知研究[J].计算机学报,2011,34(3):425-434. Dai Qiong-hai,Fu Chang-jun,Ji Xiang-yang.Research on compressed sensing[J].Chinese Journal of Computers,2011,34(3):425-434.(in Chinese)
[4] Figueiredo M,Nowak R,Wright S.Gradient projection for sparse reconstruction:application to compressed sensing and other inverse problems[J].IEEE Journal of Selected Topics in Signal Processing,2008,1(4):586-597.
[5] Tropp J,Gilbert A.Signal recovery from random measurements via orthogonal matching pursuit[J].IEEE Transactions on Information Theory,2007,53(12):4655-4666.
[6] Elad M.A wide-angle view at iterated shrinkage algorithms[J].Proceedings of SPIE-The International Society for Optical Engineering,2007,6701(6701):26-29.
[7] He L,Chen H,Carin L.Tree-structured compressive sensing with variational Bayesian analysis[J].IEEE Signal Processing Letters,2010,17(3):233-236.
[8] Dong W,Shi G,Li X,et al.Compressive sensing via nonlocal low-rank regularization[J].IEEE Transactions on Image Processing,2014,23(8):3618.
[9] 宋云,李雪玉,沈燕飞,等.基于非局部相似块低秩的压缩感知图像重建算法[J].电子学报,2017,45(3):695-703. Song Yun,Li Xue-yu,Shen Yan-fei,et al.Compressed sensing image reconstruction based on low rank of non-local similar patches[J].Acta Electronica Sinica,2017,45(3):695-703.(in Chinese)
[10] 李然,干宗良,朱秀昌.基于最佳线性估计的快速压缩感知图像重建算法[J].电子与信息学报,2012,34(12):3006-3012. Li Ran,Gan Zong-liang,Zhu Xiu-chang.A fast compressed-sensing image reconstruction algorithm based on best linear estimate[J].Journal of Electronics and Information Technology,2012,34(12):3006-3012.(in Chinese)
[11] Chen C,Tramel E,Fowler J.Compressed-sensing recovery of images and video using multihypothesis predictions[J].Signals,Systems and Computers,2011,1193-1198.
[12] 陈江琦,马尽文.基于改进粒子群算法的压缩感知[J].信号处理,2016,33(4):488-495. Chen Jiang-qi,Ma Jin-wen.The improved particle swarm optimization algorithm based compressive sensing[J].Journal of Signal Processing,2016,33(4):488-495.(in Chinese)
[13] Aharon M,Elad M,Bruckstein A.The K-SVD:an algorithm for designing of overcomplete dictionaries for sparse representation[J].IEEE Transactions on Signal Processing,2006,54(11):4311-4322.
[14] Hawe S,Seibert M,Kleinsteuber M.Separable dictionary learning[A].IEEE Conference on Computer Vision and Pattern Recognition[C].IEEE,2013.438-445.
[15] Lu C,Chen H.Compressive image sensing for fast recovery from limited samples:A variation on compressive sensing[J].Information Sciences,2015,325:33-47.
[16] Roohi S,Zonoobi D,Kassim A,et al.Multi-dimensional low rank plus sparse decomposition for reconstruction of under-sampled dynamic MRI[J].Pattern Recognition,2016,63:667-679.
[17] Cai J,Jia X,Gao H,et al.Cine cone beam CT reconstruction using low-rank matrix factorization:algorithm and a proof-of-principle study[J].IEEE Transactions on Medical Imaging,2014,33(8):1581-1591.
[18] Chen S,Liu H,Hu Z,et al.simultaneous reconstruction and segmentation of dynamic PET via low-rank and sparse matrix decomposition[J].IEEE Transactions on Biomedical Engineering,2015,62(7):1784-1795.
[19] Wang L,Sun Y,Gao S,et al.Automatic misalignment correction of seismograms using low-rank matrix recovery[J].IEEE Geoscience and Remote Sensing Letters,2012,10(2):352-356.
[20] Candès E,Li X,Ma Y,et al.Robust principal component analysis[J].Journal of the ACM,2009,58(3):1-37.
[21] 徐森,周天,于化龙,等.一种基于矩阵低秩近似的聚类集成算法[J].电子学报,2013,41(6):1219-1224. Xu Sen,Zhou Tian,Yu Hua-long,et al.Matrix low rank approximation-based cluster ensemble algorithm[J].Acta Electronica Sinica,2013,41(6):1219-1224.(in Chinese)
[22] Ye J.Generalized low rank approximations of matrices[J].Machine Learning,2005,61(1):167-191.
[23] Candès E,Emmanuel J.The restricted isometry property and its implications for compressed sensing[J].Comptes Rendus-Mathématique,2008,346(9):589-592.
[24] 贾正华.广义逆矩阵及其性质[J].巢湖学院学报,2005,7(3):38-39. Jia Zheng-hua.Generalized inverse matrix and its some properties[J].Journal of Chaohu College,2005,7(3):38-39.(in Chinese)
[25] 王秀清,陈北英,于朝霞.关于矩阵的Kronecker积的一些性质[J].山东师范大学学报(自然科学版),2010,25(4):147-149. Wang Xiu-qing,Chen Bei-ying,Yu Zhao-xia.Some properties of Kronecker product of matrices[J].Journal of Shandong Normal University(Natural Science),2010,25(4):147-149.(in Chinese)
[26] 邵逸民.行(列)满秩矩阵的一些性质及应用[J].长春师范学院学报(自然科学版),2008,27(6):22-25. Shao Yi-min.Some properties of row(column)full rank matrices and their applications[J].Journal of Changchun Normal University (Natural Science),2008,27(6):22-25.(in Chinese)
[27] 陈永林,李志林.用Kronecker积与广义逆矩阵求解矩阵方程[J].南京师大学报自然科学版,1991,14(1):9-15. Chen Yong-lin,Li Zhi-lin.Solving matrix equation by using kronecker product and generalized inverse of a matrix[J].Journal of Nanjing Normal University(Natural Science),1991,14(1):9-15.(in Chinese)
[28] 王建宏,谢燕,周星月.矩阵内积的性质[J].高师理科学刊,2010,30(2):12-16. Wang Jian-hong,Xie Yan,Zhou Xing-yue.The properties of inner product properties of matrices[J].Journal of Science of Teachers' College and University,2010,30(2):12-16.(in Chinese)
[29] Chen M,Lin Z,Ma Y,et al.The augmented Lagrange multiplier method for exact recovery of corrupted low-rank matrices[J].Eprint Arxiv,2010,9.
[30] Yuan X,Yang J.Sparse and low-rank matrix decomposition via alternating direction methods[J].Pacific Journal of Optimization,2013,9(1):167-180.
[31] Gower J,Dijksterhuis G.Procrustes Problems[M].Oxford University Press,2004,168(2):xiv,233.
[32] Yuan X,Rao V,Han S,et al.Hierarchical infinite divisibility for multiscale shrinkage[J].IEEE Transactions on Signal Processing,2014,62(17):4363-4374.
[33] Li C,Yin W,Zhang Y.TVAL3:TV Minimization by Augmented Lagrangian and Alternating Direction Algorithm[EB/OL].http://www.caam.rice.edu/~optimization/L1/TVAL3/,2009.
[34] Wang H,Cen Y,He Z,et al.Robust generalized low-rank decomposition of multi-matrices for image recovery[J].IEEE Transactions on Multimedia,2017,19(5),969-983.
2018, Vol. 46 
