融合数据内部变化信息的丰度估计算法
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摘要
丰度估计(AE)是从高光谱图像中识别地物的关键预处理技术。由于线性模型的可解释性以及数学上的可操作性,基于该模型的线性回归技术CLR(Constrained Linear Regression)在丰度估计中受到了广泛关注。目前,该方法仅仅考虑到了估计数据与被估计数据之间的能量相似性,没有考虑数据内部的变化信息之间的相似性,比如一阶梯度之间的相似性以及二阶梯度之间的相似性。为了提高丰度估计精度,本文提出了融合数据内部变化信息的稀疏低秩丰度估计算法。首先通过增加一阶梯度和二阶梯度的约束项改进传统的丰度估计的数学模型。其次,通过采用范数不等式和优化理论证明了在约束条件下,该模型的有效性及该模型在相关领域的可拓展性。接着,采用辅助变量将改进的数学模型变为增强拉格朗日函数。最后,采用交替双向乘子技术ADMM(Alternating Direction Method of Multipliers)求解该模型并估计高光谱图像的丰度。经仿真实验和实际高光谱图像的实验证明该方法能够改善仿真数据和实际高光谱数据的丰度估计的效果,特别是当端元的丰度存在丰富的变化细节时,丰度估计的精度和抗噪性能均优于当前较流行的丰度估计算法。
Abundance estimation(AE) plays an important role in the processing and analysis of hyperspectral images. Constrained linear regression is usually developed to estimate abundance matrix due to its simplicity and mathematical tractability. However, this approach only focuses on the fitness between the estimated and ground-truth data without considering the internal variability such as the similarity among the first-order gradients and among the second-order gradients. To improve the accuracy of the AE, a novel method of adding internal variability to sparse low-rank AE was proposed.First, first-and second-order gradient constraint terms were used to modify the traditional mathematical model of sparse and low-rank AE. Second, norm inequality and optimization theory were applied to demonstrate the validity of the novel model. The model has been proven applicable to other related fields under constraint conditions. Third, auxiliary variables were utilized to transform the mathematic model to the enhanced Lagrange function(ELF). Finally, the ELF was solved by the alternating direction method of multipliers to estimate the abundance of hyperspectral images. In general, the traditional method of sparse and low-rank AE is the alternating direction sparse and low-rank unmixing(ADSpLRU). In this study, ADSpLRU-FOG refers to the method that adds the first-order gradient to the sparse and lowrank AE, whereas ADSpLRU-FSOG refers to the method that adds first-and second-order gradients to the sparse and low-rank AE.Experiment carried on the USUG library showed that,(1) in the convergent experiment, ADSpLRU-FOG and ADSpLRU-FSOG algorithms converged to a slightly lower NMSE than ADSpLRU. ADSpLRU-FSOG algorithm converged to the lowest NMSE among the three methods.(2) In the robust experiment, ADSpLRU-FOG and ADSpLRU-FSOG algorithms reached higher estimation accuracy than ADSpLRU in terms of SRE under white and colored noises. Among them, ADSpLRU-FSOG achieved remarkably higher SRE value than the other methods.(3) In the visual experiment, ADSpLRU-FOG algorithm could maintain the first-order gradient structure of the data more than ADSpLRU. Meanwhile, the ADSpLRU-FSOG algorithm could preserve the second-order gradient structure of the data better than ADSpLRU-FOG and ADSpLRU algorithms. Experiment based on the Urban and Jasper actual hyperspectral database showed that the accuracy of abundance matrix estimation from ADSpLRU-FSOG was better than those from ADSpLRU and ADSpLRU-FOG.Experimental results suggest that the novel method of adding the internal variability to the abundance matrix estimation can improve convergent behavior, maintain the structure of information of first-order and second-order gradients, obtain comparable estimation accuracy,and enhance robust performance for AE.
Abundance estimation(AE) plays an important role in the processing and analysis of hyperspectral images. Constrained linear regression is usually developed to estimate abundance matrix due to its simplicity and mathematical tractability. However, this approach only focuses on the fitness between the estimated and ground-truth data without considering the internal variability such as the similarity among the first-order gradients and among the second-order gradients. To improve the accuracy of the AE, a novel method of adding internal variability to sparse low-rank AE was proposed.First, first-and second-order gradient constraint terms were used to modify the traditional mathematical model of sparse and low-rank AE. Second, norm inequality and optimization theory were applied to demonstrate the validity of the novel model. The model has been proven applicable to other related fields under constraint conditions. Third, auxiliary variables were utilized to transform the mathematic model to the enhanced Lagrange function(ELF). Finally, the ELF was solved by the alternating direction method of multipliers to estimate the abundance of hyperspectral images. In general, the traditional method of sparse and low-rank AE is the alternating direction sparse and low-rank unmixing(ADSpLRU). In this study, ADSpLRU-FOG refers to the method that adds the first-order gradient to the sparse and lowrank AE, whereas ADSpLRU-FSOG refers to the method that adds first-and second-order gradients to the sparse and low-rank AE.Experiment carried on the USUG library showed that,(1) in the convergent experiment, ADSpLRU-FOG and ADSpLRU-FSOG algorithms converged to a slightly lower NMSE than ADSpLRU. ADSpLRU-FSOG algorithm converged to the lowest NMSE among the three methods.(2) In the robust experiment, ADSpLRU-FOG and ADSpLRU-FSOG algorithms reached higher estimation accuracy than ADSpLRU in terms of SRE under white and colored noises. Among them, ADSpLRU-FSOG achieved remarkably higher SRE value than the other methods.(3) In the visual experiment, ADSpLRU-FOG algorithm could maintain the first-order gradient structure of the data more than ADSpLRU. Meanwhile, the ADSpLRU-FSOG algorithm could preserve the second-order gradient structure of the data better than ADSpLRU-FOG and ADSpLRU algorithms. Experiment based on the Urban and Jasper actual hyperspectral database showed that the accuracy of abundance matrix estimation from ADSpLRU-FSOG was better than those from ADSpLRU and ADSpLRU-FOG.Experimental results suggest that the novel method of adding the internal variability to the abundance matrix estimation can improve convergent behavior, maintain the structure of information of first-order and second-order gradients, obtain comparable estimation accuracy,and enhance robust performance for AE.
引文
Babacan S D,Luessi M,Molina R and Katsaggelos A K.2012.Sparse Bayesian methods for low-rank matrix estimation.IEEE Transactions on Signal Processing,60(8):3964-3977[DOI:10.1109/TSP.2012.2197748]
Bach F R.2008.Consistency of trace norm minimization.The Journal of Machine Learning Research,9:1019-1048
Bioucas-Dias J M,Plaza A,Dobigeon N,Parente M,Du Q,Gader Pand Chanussot J.2012.Hyperspectral unmixing overview:geometrical,statistical,and sparse regression-based approaches.IEEEJournal of Selected Topics in Applied Earth Observations and Remote Sensing,5(2):354-379[DOI:10.1109/JSTARS.2012.2194696]
Boyd S,Parikh N,Chu E,Peleato B and Eckstein J.2011.Distributed optimization and statistical learning via the alternating direction method of multipliers.Foundations and Trends?in Machine Learning,3(1):1-122[DOI:10.1561/2200000016]
Candès E J,Romberg J and Tao T.2006.Robust uncertainty principles:exact signal reconstruction from highly incomplete frequency information.IEEE Transactions on Information Theory,52(2):489-509[DOI:10.1109/TIT.2005.862083]
Chen F and Zhang Y.2013.Sparse hyperspectral unmixing based on constrained Lp-L2 optimization.IEEE Geoscience and Remote Sensing Letters,10(5):1142-1146[DOI:10.1109/LGRS.2012.2232901]
Chen L S and Huang J Z.2012.Sparse reduced-rank regression for simultaneous dimension reduction and variable selection.Journal of the American Statistical Association,107(500):1533-1545[DOI:10.1080/01621459.2012.734178]
Clark R N,Swayze G A,Wise R,Livo E,Hoefen T,Kokaly R F and Sutley S J.2007.USGS digital spectral library splib06a.Reston,VA:U.S.Geological Survey
Feng R Y,Zhong Y F and Zhang L P.2016.Adaptive spatial regularization sparse unmixing strategy based on joint MAP for hyperspectral remote sensing imagery.IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing,9(12):5791-5805[DOI:10.1109/JSTARS.2016.2570947]
Giampouras P V,Themelis K E,Rontogiannis A A and Koutroumbas K D.2016.Simultaneously sparse and low-rank abundance matrix estimation for hyperspectral image unmixing.IEEE Transactions on Geoscience and Remote Sensing,54(8):4775-4789[DOI:10.1109/TGRS.2016.2551327]
Golbabaee M and Vandergheynst P.2012.Compressed sensing of simultaneous low-rank and joint-sparse matrices.eprint arXiv:1211.5058
Iordache M D,Bioucas-Dias J M and Plaza A.2011.Sparse unmixing of hyperspectral data.IEEE Transactions on Geoscience and Remote Sensing,49(6):2014-2039[DOI:10.1109/TGRS.2010.2098413]
Iordache M D,Bioucas-Dias J M and Plaza A.2012.Total variation spatial regularization for sparse hyperspectral unmixing.IEEETransactions on Geoscience and Remote Sensing,50(11):4484-4502[DOI:10.1109/TGRS.2012.2191590]
Iordache M D,Bioucas-Dias J M and Plaza A.2014.Collaborative sparse regression for hyperspectral unmixing.IEEE Transactions on Geoscience and Remote Sensing,52(1):341-354[DOI:10.1109/TGRS.2013.2240001]
Lu T X.1986.Solution of the matrix equation AX-XB=C.Computing,37(4):351-355[DOI:10.1007/BF02251092]
Negahban S and Wainwright M J.2011.Estimation of(near)low-rank matrices with noise and high-dimensional scaling.The Annals of Statistics,39(2):1069-1097[DOI:10.1214/10-AOS850]
Oymak S,Jalali A,Fazel M,Eldar Y C and Hassibi B.2015.Simultaneously structured models with application to sparse and low-rank matrices.IEEE Transactions on Information Theory,61(5):2886-2908[DOI:10.1109/TIT.2015.2401574]
Qian Y T,Jia S,Zhou J and Robles-Kelly A.2011.Hyperspectral unmixing via L1/2 sparsity-constrained nonnegative matrix factorization.IEEE Transactions on Geoscience and Remote Sensing,49(11):4282-4297[DOI:10.1109/TGRS.2011.2144605]
Qu Q,Nasrabadi N M and Tran T D.2014.Abundance estimation for bilinear mixture models via joint sparse and low-rank representation.IEEE Transactions on Geoscience and Remote Sensing,52(7):4404-4423[DOI:10.1109/TGRS.2013.2281981]
Richard E,Obozinski G and Vert J P.2014.Tight convex relaxations for sparse matrix factorization//Advances in Neural Information Processing Systems 27.Montreal,Canada:Curran Associates,Inc.:3284-3292
Richard E,Savalle P A and Vayatis N.2012.Estimation of simultaneously sparse and low rank matrices//Proceedings of the 29th International Conference on International Conference on Machine Learning.Edinburgh,Scotland:ACM:51-58
Rontogiannis A A,Themelis K,Sykioti O and Koutroumbas K.2013.A fast variational Bayes algorithm for sparse semi-supervised unmixing of OMEGA/Mars Express data//2013 5th Workshop on Hyperspectral Image and Signal Processing:Evolution in Remote Sensing.Gainesville,FL,USA:IEEE:1-4[DOI:10.1109/WHIS-PERS.2013.8080749]
Rosin P L.2001.Robust pixel unmixing.IEEE Transactions on Geoscience and Remote Sensing,39(9):1978-1983[DOI:10.1109/36.951088]
Sigurdsson J,Ulfarsson M O and Sveinsson J R.2014.Hyperspectral unmixing with Lq regularization.IEEE Transactions on Geoscience and Remote Sensing,52(11):6793-6806[DOI:10.1109/TGRS.2014.2303155]
Sun L,Jeon B,Zheng Y H and Chen Y J.2016.Hyperspectral unmixing based on L1-L2 sparsity and total variation//2016 IEEE International Conference on Image Processing.Phoenix,AZ,USA:IEEE:4349-4353[DOI:10.1109/lelP.2016.7533181]
Tang W,Shi Z W and Duren Z.2014.Sparse hyperspectral unmixing using an approximate L0 norm.Optik,125(1):31-38[DOI:10.1016/j.ijleo.2013.06.073]
Themelis K E,Rontogiannis A A and Koutroumbas K.2010.Semi-supervised hyperspectral unmixing via the weighted lasso//IEEE International Conference on Acoustics,Speech,and Signal Processing.Dallas,Texas,USA:IEEE:1194-1197[DOI:10.1109/ICASSP.2010.5495385]
Themelis K E,Rontogiannis A A and Koutroumbas K D.2012.A novel hierarchical Bayesian approach for sparse semisupervised hyperspectral unmixing.IEEE Transactions on Signal Processing,60(2):585-599[DOI:10.1109/TSP.2011.2174052]
Yang J X,Zhao Y Q,Chan J C W and Kong S G.2016.Coupled sparse denoising and unmixing with low-rank constraint for hyperspectral image.IEEE Transactions on Geoscience and Remote Sensing,54(3):1818-1833[DOI:10.1109/TGRS.2015.2489218]
Yi C,Zhao Y Q,Yang J X,Chan J C W and Kong S G.2017.Joint hyperspectral superresolution and unmixing with interactive feedback.IEEE Transactions on Geoscience and Remote Sensing,55(7):3823-3834[DOI:10.1109/TGRS.2017.2681721]
Yuan J and Zhang Y J.2017.Application of sparse denoising auto encoder network with gradient difference information for abnormal action detection.Acta Automatica Sinica,43(4):604-610(袁静,章毓晋.2017.融合梯度差信息的稀疏去噪自编码网络在异常行为检测中的应用.自动化学报,43(4):604-610)[DOI:10.16383/j.aas.2017.c150667]
Yuan J,Zhang Y J and Yang D H.2018.Sparse and low-rank abundance estimation with structural information.Journal of Infrared and Millimeter Waves,37(2):144-153(袁静,章毓晋,杨德贺.2018.融入结构信息的稀疏低秩丰度估计在光谱解混中的应用.红外与毫米波学报,37(2):144-153)
Zhang X D.2013.Matrix Analysis and Applications.2nd ed.Beijing:Tsinghua University Press:44(张贤达.2013.矩阵分析与应用.2版.北京:清华大学出版社:44)
Zhu F Y,Wang Y,Fan B,Xiang S M,Meng G F and Pan C H.2014a.Spectral unmixing via data-guided sparsity.IEEE Transactions on Image Processing,23(12):5412-5427[DOI:10.1109/TIP.2014.2363423]
Zhu F Y,Wang Y,Fan B,Meng G F and Pan C H.2014b.Effective spectral unmixing via robust representation and learning-based sparsity.eprint arXiv:1409.0685
Bach F R.2008.Consistency of trace norm minimization.The Journal of Machine Learning Research,9:1019-1048
Bioucas-Dias J M,Plaza A,Dobigeon N,Parente M,Du Q,Gader Pand Chanussot J.2012.Hyperspectral unmixing overview:geometrical,statistical,and sparse regression-based approaches.IEEEJournal of Selected Topics in Applied Earth Observations and Remote Sensing,5(2):354-379[DOI:10.1109/JSTARS.2012.2194696]
Boyd S,Parikh N,Chu E,Peleato B and Eckstein J.2011.Distributed optimization and statistical learning via the alternating direction method of multipliers.Foundations and Trends?in Machine Learning,3(1):1-122[DOI:10.1561/2200000016]
Candès E J,Romberg J and Tao T.2006.Robust uncertainty principles:exact signal reconstruction from highly incomplete frequency information.IEEE Transactions on Information Theory,52(2):489-509[DOI:10.1109/TIT.2005.862083]
Chen F and Zhang Y.2013.Sparse hyperspectral unmixing based on constrained Lp-L2 optimization.IEEE Geoscience and Remote Sensing Letters,10(5):1142-1146[DOI:10.1109/LGRS.2012.2232901]
Chen L S and Huang J Z.2012.Sparse reduced-rank regression for simultaneous dimension reduction and variable selection.Journal of the American Statistical Association,107(500):1533-1545[DOI:10.1080/01621459.2012.734178]
Clark R N,Swayze G A,Wise R,Livo E,Hoefen T,Kokaly R F and Sutley S J.2007.USGS digital spectral library splib06a.Reston,VA:U.S.Geological Survey
Feng R Y,Zhong Y F and Zhang L P.2016.Adaptive spatial regularization sparse unmixing strategy based on joint MAP for hyperspectral remote sensing imagery.IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing,9(12):5791-5805[DOI:10.1109/JSTARS.2016.2570947]
Giampouras P V,Themelis K E,Rontogiannis A A and Koutroumbas K D.2016.Simultaneously sparse and low-rank abundance matrix estimation for hyperspectral image unmixing.IEEE Transactions on Geoscience and Remote Sensing,54(8):4775-4789[DOI:10.1109/TGRS.2016.2551327]
Golbabaee M and Vandergheynst P.2012.Compressed sensing of simultaneous low-rank and joint-sparse matrices.eprint arXiv:1211.5058
Iordache M D,Bioucas-Dias J M and Plaza A.2011.Sparse unmixing of hyperspectral data.IEEE Transactions on Geoscience and Remote Sensing,49(6):2014-2039[DOI:10.1109/TGRS.2010.2098413]
Iordache M D,Bioucas-Dias J M and Plaza A.2012.Total variation spatial regularization for sparse hyperspectral unmixing.IEEETransactions on Geoscience and Remote Sensing,50(11):4484-4502[DOI:10.1109/TGRS.2012.2191590]
Iordache M D,Bioucas-Dias J M and Plaza A.2014.Collaborative sparse regression for hyperspectral unmixing.IEEE Transactions on Geoscience and Remote Sensing,52(1):341-354[DOI:10.1109/TGRS.2013.2240001]
Lu T X.1986.Solution of the matrix equation AX-XB=C.Computing,37(4):351-355[DOI:10.1007/BF02251092]
Negahban S and Wainwright M J.2011.Estimation of(near)low-rank matrices with noise and high-dimensional scaling.The Annals of Statistics,39(2):1069-1097[DOI:10.1214/10-AOS850]
Oymak S,Jalali A,Fazel M,Eldar Y C and Hassibi B.2015.Simultaneously structured models with application to sparse and low-rank matrices.IEEE Transactions on Information Theory,61(5):2886-2908[DOI:10.1109/TIT.2015.2401574]
Qian Y T,Jia S,Zhou J and Robles-Kelly A.2011.Hyperspectral unmixing via L1/2 sparsity-constrained nonnegative matrix factorization.IEEE Transactions on Geoscience and Remote Sensing,49(11):4282-4297[DOI:10.1109/TGRS.2011.2144605]
Qu Q,Nasrabadi N M and Tran T D.2014.Abundance estimation for bilinear mixture models via joint sparse and low-rank representation.IEEE Transactions on Geoscience and Remote Sensing,52(7):4404-4423[DOI:10.1109/TGRS.2013.2281981]
Richard E,Obozinski G and Vert J P.2014.Tight convex relaxations for sparse matrix factorization//Advances in Neural Information Processing Systems 27.Montreal,Canada:Curran Associates,Inc.:3284-3292
Richard E,Savalle P A and Vayatis N.2012.Estimation of simultaneously sparse and low rank matrices//Proceedings of the 29th International Conference on International Conference on Machine Learning.Edinburgh,Scotland:ACM:51-58
Rontogiannis A A,Themelis K,Sykioti O and Koutroumbas K.2013.A fast variational Bayes algorithm for sparse semi-supervised unmixing of OMEGA/Mars Express data//2013 5th Workshop on Hyperspectral Image and Signal Processing:Evolution in Remote Sensing.Gainesville,FL,USA:IEEE:1-4[DOI:10.1109/WHIS-PERS.2013.8080749]
Rosin P L.2001.Robust pixel unmixing.IEEE Transactions on Geoscience and Remote Sensing,39(9):1978-1983[DOI:10.1109/36.951088]
Sigurdsson J,Ulfarsson M O and Sveinsson J R.2014.Hyperspectral unmixing with Lq regularization.IEEE Transactions on Geoscience and Remote Sensing,52(11):6793-6806[DOI:10.1109/TGRS.2014.2303155]
Sun L,Jeon B,Zheng Y H and Chen Y J.2016.Hyperspectral unmixing based on L1-L2 sparsity and total variation//2016 IEEE International Conference on Image Processing.Phoenix,AZ,USA:IEEE:4349-4353[DOI:10.1109/lelP.2016.7533181]
Tang W,Shi Z W and Duren Z.2014.Sparse hyperspectral unmixing using an approximate L0 norm.Optik,125(1):31-38[DOI:10.1016/j.ijleo.2013.06.073]
Themelis K E,Rontogiannis A A and Koutroumbas K.2010.Semi-supervised hyperspectral unmixing via the weighted lasso//IEEE International Conference on Acoustics,Speech,and Signal Processing.Dallas,Texas,USA:IEEE:1194-1197[DOI:10.1109/ICASSP.2010.5495385]
Themelis K E,Rontogiannis A A and Koutroumbas K D.2012.A novel hierarchical Bayesian approach for sparse semisupervised hyperspectral unmixing.IEEE Transactions on Signal Processing,60(2):585-599[DOI:10.1109/TSP.2011.2174052]
Yang J X,Zhao Y Q,Chan J C W and Kong S G.2016.Coupled sparse denoising and unmixing with low-rank constraint for hyperspectral image.IEEE Transactions on Geoscience and Remote Sensing,54(3):1818-1833[DOI:10.1109/TGRS.2015.2489218]
Yi C,Zhao Y Q,Yang J X,Chan J C W and Kong S G.2017.Joint hyperspectral superresolution and unmixing with interactive feedback.IEEE Transactions on Geoscience and Remote Sensing,55(7):3823-3834[DOI:10.1109/TGRS.2017.2681721]
Yuan J and Zhang Y J.2017.Application of sparse denoising auto encoder network with gradient difference information for abnormal action detection.Acta Automatica Sinica,43(4):604-610(袁静,章毓晋.2017.融合梯度差信息的稀疏去噪自编码网络在异常行为检测中的应用.自动化学报,43(4):604-610)[DOI:10.16383/j.aas.2017.c150667]
Yuan J,Zhang Y J and Yang D H.2018.Sparse and low-rank abundance estimation with structural information.Journal of Infrared and Millimeter Waves,37(2):144-153(袁静,章毓晋,杨德贺.2018.融入结构信息的稀疏低秩丰度估计在光谱解混中的应用.红外与毫米波学报,37(2):144-153)
Zhang X D.2013.Matrix Analysis and Applications.2nd ed.Beijing:Tsinghua University Press:44(张贤达.2013.矩阵分析与应用.2版.北京:清华大学出版社:44)
Zhu F Y,Wang Y,Fan B,Xiang S M,Meng G F and Pan C H.2014a.Spectral unmixing via data-guided sparsity.IEEE Transactions on Image Processing,23(12):5412-5427[DOI:10.1109/TIP.2014.2363423]
Zhu F Y,Wang Y,Fan B,Meng G F and Pan C H.2014b.Effective spectral unmixing via robust representation and learning-based sparsity.eprint arXiv:1409.0685