Cholesky分解的逐像元实时高光谱异常探测
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摘要
传统的实时异常探测算法需对高维的背景样本统计矩阵进行求逆运算,数值稳定性差、时间复杂度高。而基于Cholesky分解,将高维矩阵的求逆运算转换为求解下三角线性系统,采用Cholesky分解因子的一阶修正方法快速更新背景统计信息,降低逐像元实时处理的时间复杂度并且保持数值稳定性。由于算法仅涉及下三角矩阵的更新过程,压缩了数据存储空间,适用于机载或星上实时处理。采用3维接收器曲线(3D-ROC)以及计算机实际处理时间对实验结果进行定量化分析,结果表明,该算法在不降低异常探测精度的同时,对当前时刻像元的实时处理时间约缩短为基于QR分解算法的0.4%—0.65%,或减少至基于Woodbury矩阵引理算法的27%—33%,有效提高实时高光谱异常探测器的计算性能,并且保持处理过程中的数值稳定性。
Anomaly detection is one of the most important issues in hyperspectral remote sensing.However,traditional anomaly detection algorithms cannot be used for onboard real-time processing due to heavy computational load caused by the dimensionality curse of hyperspectral data.To implement onboard real-time hyperspectral anomaly detection,the following must be performed:(1) conduct the process in a causal progressive manner without any future data relative to the pixel under test;(2) output the result for each sample right after collecting it;(3) process the data with constant theoretical computational complexity.The present widely used algorithms generally employ QR decomposition or Woodbury’s Identity for real-time anomaly detection.However,their computational load is still extremely high.Also,they suffered serious numerical instabilities led by runoff error in the matrix inverting process.The main objective of this study is to further accelerate the real-time detection process and avoid the matrix inversion module to maintain numerical stabilities.In this work,we proposed a novel real-time sample-wise hyperspectral anomaly detection algorithm based on Cholesky decomposition.The background sample correlation matrix and covariance matrix were symmetric positive definite,which can be factored into a lower triangular matrix and its transpose.We modified the background suppression process into a process that finds the solution to a lower-triangular linear system based on this characteristic.Furthermore,the real-time process is significantly accelerated by virtue of rank-1 updates to the Cholesky factor of the background statistical matrix.Moreover,the numerical stabilities were maintained.Finally,we performed a three dimensional ROC(3D-ROC)analysis to evaluate the performance of real-time anomaly detection in terms of background suppression,detection power,false alarm,and the relationship between each other.An experiment on an actual hyperspectral dataset collected by Field Imaging Spectrometer System(FISS)revealed the following.(1)The proposed algorithm significantly reduced the computing time to process the incoming data.The theoretical computational load demonstrated high efficiency of the technique developed in this work.The time consumption for each incoming sample was reduced to 0.4%—0.65% of the time consumed by traditional QR decomposition-based algorithms,and 27%—33% of the time corresponding to Woodbury’s identity-based techniques.(2) The sample varying background suppression provided an acceptable visual inspection of the anomalies.Real-time processing prevented these weak signals from being suppressed by later detected strong signals.Additionally,3D-ROC analysis could effectively evaluate the detection performance of hyperspectral anomaly detectors.The real-time sample-wise hyperspectral anomaly detector developed in this study is not only computationally efficient but also numerically stable,significantly contributing to onboard implementations.
Anomaly detection is one of the most important issues in hyperspectral remote sensing.However,traditional anomaly detection algorithms cannot be used for onboard real-time processing due to heavy computational load caused by the dimensionality curse of hyperspectral data.To implement onboard real-time hyperspectral anomaly detection,the following must be performed:(1) conduct the process in a causal progressive manner without any future data relative to the pixel under test;(2) output the result for each sample right after collecting it;(3) process the data with constant theoretical computational complexity.The present widely used algorithms generally employ QR decomposition or Woodbury’s Identity for real-time anomaly detection.However,their computational load is still extremely high.Also,they suffered serious numerical instabilities led by runoff error in the matrix inverting process.The main objective of this study is to further accelerate the real-time detection process and avoid the matrix inversion module to maintain numerical stabilities.In this work,we proposed a novel real-time sample-wise hyperspectral anomaly detection algorithm based on Cholesky decomposition.The background sample correlation matrix and covariance matrix were symmetric positive definite,which can be factored into a lower triangular matrix and its transpose.We modified the background suppression process into a process that finds the solution to a lower-triangular linear system based on this characteristic.Furthermore,the real-time process is significantly accelerated by virtue of rank-1 updates to the Cholesky factor of the background statistical matrix.Moreover,the numerical stabilities were maintained.Finally,we performed a three dimensional ROC(3D-ROC)analysis to evaluate the performance of real-time anomaly detection in terms of background suppression,detection power,false alarm,and the relationship between each other.An experiment on an actual hyperspectral dataset collected by Field Imaging Spectrometer System(FISS)revealed the following.(1)The proposed algorithm significantly reduced the computing time to process the incoming data.The theoretical computational load demonstrated high efficiency of the technique developed in this work.The time consumption for each incoming sample was reduced to 0.4%—0.65% of the time consumed by traditional QR decomposition-based algorithms,and 27%—33% of the time corresponding to Woodbury’s identity-based techniques.(2) The sample varying background suppression provided an acceptable visual inspection of the anomalies.Real-time processing prevented these weak signals from being suppressed by later detected strong signals.Additionally,3D-ROC analysis could effectively evaluate the detection performance of hyperspectral anomaly detectors.The real-time sample-wise hyperspectral anomaly detector developed in this study is not only computationally efficient but also numerically stable,significantly contributing to onboard implementations.
引文
Acito N,Matteoli S,Diani M and Corsini G.2011.Complexity-aware algorithm architecture for real-time enhancement of local anomalies in hyperspectral images.Journal of Real-Time Image Processing,8(1):53-68[DOI:10.1007/s11554-011-0205-x]
Bojanczyk A W,Brent R,Van Dooren P and De Hoog F.1987.A note on downdating the Cholesky factorization.SIAM Journal on Scientific and Statistical Computing,8(3):210-221[DOI:10.1137/0908031]
Chang C I.2003.Hyperspectral Imaging:Techniques for Spectral Detection and Classification.New York:Springer[DOI:10.1007/978-1-4419-9170-6]
Chang C I.2013.Hyperspectral Data Processing:Algorithm Design and Analysis.Hoboken,NJ,USA:John Wiley and Sons[DOI:10.1002/9781118269787]
Chang C I.2016.Real-time Progressive Hyperspectral Image Processing:Endmember Finding and Anomaly Detection.New York:Springer[DOI:10.1007/978-1-4419-6187-7]
Chang C I and Chiang S S.2002.Anomaly detection and classification for hyperspectral imagery.IEEE Transactions on Geoscience and Remote Sensing,40(6):1314-1325[DOI:10.1109/TGRS.2002.800280]
Chang C I and Hsueh M.2006.Characterization of anomaly detection in hyperspectral imagery.Sensor Review,26(2):137-146[DOI:10.1108/02602280610652730]
Chang C I,Li H C,Song M P,Liu C H and Zhang L F.2015.Real-time constrained energy minimization for subpixel detection.IEEEJournal of Selected Topics in Applied Earth Observations and Remote Sensing,8(6):2545-2559[DOI:10.1109/jstars.2015.2425417]
Chang C I,Ren H and Chiang S S.2001.Real-time processing algorithms for target detection and classification in hyperspectral imagery.IEEE Transactions on Geoscience and Remote Sensing,39(4):760-768[DOI:10.1109/36.917889]
Chen S Y,Wang Y L,Wu C C,Liu C H and Chang C I.2014.Realtime causal processing of anomaly detection for hyperspectral imagery.IEEE Transactions on Aerospace and Electronic Systems,50(2):1511-1534[DOI:10.1109/taes.2014.130065]
Du B and Zhang L P.2011.Random-selection-based anomaly detector for hyperspectral imagery.IEEE Transactions on Geoscience and Remote Sensing,49(5):1578-1589[DOI:10.1109/TGRS.2010.2081677]
Du Q and Nekovei R.2009.Fast real-time onboard processing of hyperspectral imagery for detection and classification.Journal of Real-Time Image Processing,4(3):273-286[DOI:10.1007/s11554-008-0106-9]
Gill P E,Golub G H,Murray W and Saunders M A.1974.Methods for modifying matrix factorizations.Mathematics of Computation,28(126):505-535[DOI:10.1090/S0025-5718-1974-0343558-6]
Golub G H and Van Loan C F.2012.Matrix Computations.Baltimore:JHU Press[DOI:10.1137/1032141]
Guo W J,Zeng X R,Zhao B W,Ming X,Zhang G F and Lv Q B.2014.Multi-DSP parallel processing technique of hyperspectral RX anomaly detection.Spectroscopy and Spectral Analysis,34(5):1383-1387(郭文记,曾晓茹,赵宝玮,明星,张桂峰,吕群波.2014.高光谱RX异常检测的多DSP并行化处理技术.光谱学与光谱分析,34(5):1383-1387)[DOI:10.3964/j.issn.1000-0593(2014)05-1383-05]
Hsueh M.2007.Reconfigurable Computing for Algorithms in Hyperspectral Image Processing.Baltimore:University of Maryland,Baltimore County
Molero J M,Garzón E M,García I and Plaza A.2013.Analysis and optimizations of global and local versions of the RX algorithm for anomaly detection in hyperspectral data.IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing,6(2):801-814[DOI:10.1109/JSTARS.2013.2238609]
Paz A and Plaza A.2010.Clusters versus GPUs for parallel target and anomaly detection in hyperspectral images.Eurasip Journal on Advances in Signal Processing,2010:915639[DOI:10.1155/2010/915639]
Press W H,Teukolsky S A,Vetterling W T and Flannery B P.2007.Numerical Recipes 3rd Edition:the Art of Scientific Computing.3rd ed.Cambridge:Cambridge University Press[DOI:10.1145/1874391.187410]
Reed I S and Yu X.1990.Adaptive multiple-band cfar detection of an optical pattern with unknown spectral distribution.IEEE Transactions on Acoustics,Speech,and Signal Processing,38(10):1760-1770[DOI:10.1109/29.60107]
Rossi A,Acito N,Diani M and Corsini G.2012.Computationally efficient strategies to perform anomaly detection in hyperspectral images//Proceedings of SPIE 8537,Image and Signal Processing for Remote Sensing XVIII.Edinburgh,United Kingdom:SPIE:85370H[DOI:10.1117/12.973686]
Rossi A,Acito N,Diani M and Corsini G.2014.RX architectures for real-time anomaly detection in hyperspectral images.Journal of Real-Time Image Processing,9(3):503-517[DOI:10.1007/s11554-012-0292-3]
Seeger M.2004.Low rank updates for the Cholesky decomposition.No.EPREPORT-161468.2004.
Stewart G W.1998.Matrix Algorithms:Volume 1:Basic Decompositions.Philadelphia:Society for Industrial and Applied Mathematics[DOI:10-1137/19781611971408]
Zhao B W,Xiang L B,Lv Q B,Zhang G F,Zeng X R and Guo W J.2014.Parallel RX algorithm implementation based on the FPGAand multi-DSP system.Journal of Xidian University,41(3):152-156(赵宝玮,相里斌,吕群波,张桂峰,曾晓茹,郭文记.2014.FPGA和多DSP系统的并行RX探测算法.西安电子科技大学学报(自然科学版),41(3):152-156)[DOI:10.3969/j.issn.1001-2400.2014.03.022]
Zhao C H,Wang Y L and Li X H.2015.A real-time anomaly detection algorithm for hyperspectral imagery based on causal processing.Journal of Infrared and Millimeter Waves,34(1):114-121(赵春晖,王玉磊,李晓慧.2015.一种新型高光谱实时异常检测算法.红外与毫米波学报,34(1):114-121)[DOI:10.3724/sp.j.1010.2015.00114]
Bojanczyk A W,Brent R,Van Dooren P and De Hoog F.1987.A note on downdating the Cholesky factorization.SIAM Journal on Scientific and Statistical Computing,8(3):210-221[DOI:10.1137/0908031]
Chang C I.2003.Hyperspectral Imaging:Techniques for Spectral Detection and Classification.New York:Springer[DOI:10.1007/978-1-4419-9170-6]
Chang C I.2013.Hyperspectral Data Processing:Algorithm Design and Analysis.Hoboken,NJ,USA:John Wiley and Sons[DOI:10.1002/9781118269787]
Chang C I.2016.Real-time Progressive Hyperspectral Image Processing:Endmember Finding and Anomaly Detection.New York:Springer[DOI:10.1007/978-1-4419-6187-7]
Chang C I and Chiang S S.2002.Anomaly detection and classification for hyperspectral imagery.IEEE Transactions on Geoscience and Remote Sensing,40(6):1314-1325[DOI:10.1109/TGRS.2002.800280]
Chang C I and Hsueh M.2006.Characterization of anomaly detection in hyperspectral imagery.Sensor Review,26(2):137-146[DOI:10.1108/02602280610652730]
Chang C I,Li H C,Song M P,Liu C H and Zhang L F.2015.Real-time constrained energy minimization for subpixel detection.IEEEJournal of Selected Topics in Applied Earth Observations and Remote Sensing,8(6):2545-2559[DOI:10.1109/jstars.2015.2425417]
Chang C I,Ren H and Chiang S S.2001.Real-time processing algorithms for target detection and classification in hyperspectral imagery.IEEE Transactions on Geoscience and Remote Sensing,39(4):760-768[DOI:10.1109/36.917889]
Chen S Y,Wang Y L,Wu C C,Liu C H and Chang C I.2014.Realtime causal processing of anomaly detection for hyperspectral imagery.IEEE Transactions on Aerospace and Electronic Systems,50(2):1511-1534[DOI:10.1109/taes.2014.130065]
Du B and Zhang L P.2011.Random-selection-based anomaly detector for hyperspectral imagery.IEEE Transactions on Geoscience and Remote Sensing,49(5):1578-1589[DOI:10.1109/TGRS.2010.2081677]
Du Q and Nekovei R.2009.Fast real-time onboard processing of hyperspectral imagery for detection and classification.Journal of Real-Time Image Processing,4(3):273-286[DOI:10.1007/s11554-008-0106-9]
Gill P E,Golub G H,Murray W and Saunders M A.1974.Methods for modifying matrix factorizations.Mathematics of Computation,28(126):505-535[DOI:10.1090/S0025-5718-1974-0343558-6]
Golub G H and Van Loan C F.2012.Matrix Computations.Baltimore:JHU Press[DOI:10.1137/1032141]
Guo W J,Zeng X R,Zhao B W,Ming X,Zhang G F and Lv Q B.2014.Multi-DSP parallel processing technique of hyperspectral RX anomaly detection.Spectroscopy and Spectral Analysis,34(5):1383-1387(郭文记,曾晓茹,赵宝玮,明星,张桂峰,吕群波.2014.高光谱RX异常检测的多DSP并行化处理技术.光谱学与光谱分析,34(5):1383-1387)[DOI:10.3964/j.issn.1000-0593(2014)05-1383-05]
Hsueh M.2007.Reconfigurable Computing for Algorithms in Hyperspectral Image Processing.Baltimore:University of Maryland,Baltimore County
Molero J M,Garzón E M,García I and Plaza A.2013.Analysis and optimizations of global and local versions of the RX algorithm for anomaly detection in hyperspectral data.IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing,6(2):801-814[DOI:10.1109/JSTARS.2013.2238609]
Paz A and Plaza A.2010.Clusters versus GPUs for parallel target and anomaly detection in hyperspectral images.Eurasip Journal on Advances in Signal Processing,2010:915639[DOI:10.1155/2010/915639]
Press W H,Teukolsky S A,Vetterling W T and Flannery B P.2007.Numerical Recipes 3rd Edition:the Art of Scientific Computing.3rd ed.Cambridge:Cambridge University Press[DOI:10.1145/1874391.187410]
Reed I S and Yu X.1990.Adaptive multiple-band cfar detection of an optical pattern with unknown spectral distribution.IEEE Transactions on Acoustics,Speech,and Signal Processing,38(10):1760-1770[DOI:10.1109/29.60107]
Rossi A,Acito N,Diani M and Corsini G.2012.Computationally efficient strategies to perform anomaly detection in hyperspectral images//Proceedings of SPIE 8537,Image and Signal Processing for Remote Sensing XVIII.Edinburgh,United Kingdom:SPIE:85370H[DOI:10.1117/12.973686]
Rossi A,Acito N,Diani M and Corsini G.2014.RX architectures for real-time anomaly detection in hyperspectral images.Journal of Real-Time Image Processing,9(3):503-517[DOI:10.1007/s11554-012-0292-3]
Seeger M.2004.Low rank updates for the Cholesky decomposition.No.EPREPORT-161468.2004.
Stewart G W.1998.Matrix Algorithms:Volume 1:Basic Decompositions.Philadelphia:Society for Industrial and Applied Mathematics[DOI:10-1137/19781611971408]
Zhao B W,Xiang L B,Lv Q B,Zhang G F,Zeng X R and Guo W J.2014.Parallel RX algorithm implementation based on the FPGAand multi-DSP system.Journal of Xidian University,41(3):152-156(赵宝玮,相里斌,吕群波,张桂峰,曾晓茹,郭文记.2014.FPGA和多DSP系统的并行RX探测算法.西安电子科技大学学报(自然科学版),41(3):152-156)[DOI:10.3969/j.issn.1001-2400.2014.03.022]
Zhao C H,Wang Y L and Li X H.2015.A real-time anomaly detection algorithm for hyperspectral imagery based on causal processing.Journal of Infrared and Millimeter Waves,34(1):114-121(赵春晖,王玉磊,李晓慧.2015.一种新型高光谱实时异常检测算法.红外与毫米波学报,34(1):114-121)[DOI:10.3724/sp.j.1010.2015.00114]