Issue No.05 - May (2010 vol.32)
pp: 875-888
Chong-Yung Chi , National Tsing Hua University, Hsinchu
Tsung-Han Chan , National Tsing Hua University, Hsinchu
Fa-Yu Wang , National Tsing Hua University, Hsinchu
Although significant efforts have been made in developing nonnegative blind source separation techniques, accurate separation of positive yet dependent sources remains a challenging task. In this paper, a joint correlation function of multiple signals is proposed to reveal and confirm that the observations after nonnegative mixing would have higher joint correlation than the original unknown sources. Accordingly, a new nonnegative least-correlated component analysis (n{\rm LCA}) method is proposed to design the unmixing matrix by minimizing the joint correlation function among the estimated nonnegative sources. In addition to a closed-form solution for unmixing two mixtures of two sources, the general algorithm of n{\rm LCA} for the multisource case is developed based on an iterative volume maximization (IVM) principle and linear programming. The source identifiability and required conditions are discussed and proven. The proposed n{\rm LCA} algorithm, denoted by n{\rm LCA\hbox{-}IVM}, is evaluated with both simulation data and real biomedical data to demonstrate its superior performance over several existing benchmark methods.
Nonnegative blind source separation, nonnegative least-correlated component analysis, dependent sources, joint correlation function of multiple signals, iterative volume maximization.
Chong-Yung Chi, Tsung-Han Chan, Fa-Yu Wang, "Nonnegative Least-Correlated Component Analysis for Separation of Dependent Sources by Volume Maximization", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.32, no. 5, pp. 875-888, May 2010, doi:10.1109/TPAMI.2009.72
[1] N. Keshava and J. Mustard, "Spectral Unmixing," IEEE Signal Processing Magazine, vol. 19, no. 1, pp. 44-57, Jan. 2002.
[2] M. Mjolsness and D. DeCoste, "Machine Learning for Science: State of the Art and Future Prospects," Science, vol. 293, pp. 2051-2055, 2001.
[3] H.R. Herschman, "Molecular Imaging: Looking at Problems, Seeing Solutions," Science, vol. 302, pp. 605-608, 2003.
[4] D.M. McDonald and P.L. Choyke, "Imaging of Angiogenesis: From Microscope to Clinic," Nature Medicine, vol. 9, no. 6, pp. 713-725, 2003.
[5] M.E. Dickinson, G. Bearman, S. Tilie, R. Lansford, and S.E. Fraser, "Multi-Spectral Imaging and Linear Unmixing Add a Whole New Dimension to Laser Scanning Fluorescence Microscopy," Biotechniques, vol. 31, no. 6, pp. 1272-1278, Dec. 2001.
[6] K. Suhling and D. Stephens, Cell Imaging: Methods Express. Scion Publishing, 2005.
[7] A. Rabinovich, S. Agarwal, S. Krajewski, J.C. Reed, J.H. Price, and S. Belongie, "Accuracy of Unsupervised Spectral Decomposition for Densitometry of Histological Sections," IEEE Trans. Medical Imaging, to appear.
[8] Y. Wang, J. Xuan, R. Srikanchana, and P.L. Choyke, "Modeling and Reconstruction of Mixed Functional and Molecular Patterns," Int'l J. Biomedical Imaging, vol. 2006, pp. 1-9, 2006.
[9] D. Nuzillard and A. Bijaoui, "Blind Source Separation and Analysis of Multispectral Astronomical Images," Astronomy and Astrophysics Supplement Series, vol. 147, pp. 129-138, 2000.
[10] S.A. Astakhov, H. Stogbauer, A. Kraskov, and P. Grassberger, "Monte Carlo Algorithm for Least Dependent Non-Negative Mixture Decomposition," Analytical Chemistry, vol. 78, no. 5, pp. 1620-1627, 2006.
[11] S. Haykin, Neural Networks: A Comprehensive Foundation. second ed. Prentice-Hall, 2005.
[12] A. Hyvarinen, J. Karhunen, and E. Oja, Independent Component Analysis. John Wiley, 2001.
[13] E. Oja and M. Plumbley, "Blind Separation of Positive Sources by Globally Convergent Gradient Search," Neural Computation, vol. 16, pp. 1811-1825, 2004.
[14] D. Lee and H.S. Seung, "Learning the Parts of Objects by Non-Negative Matrix Factorization," Nature, vol. 401, pp. 788-791, Oct. 1999.
[15] J.S. Lee, D.D. Lee, S. Choi, K.S. Park, and D.S. Lee, "Non-Negative Matrix Factorization of Dynamic Images in Nuclear Medicine," IEEE Nuclear Science Symp. Conf. Record, vol. 4, pp. 2027-2030, 2001.
[16] M.W. Berry, M. Browne, A.N. Langville, V.P. Pauca, and R.J. Plemmons, "Algorithms and Applications for Approximate Nonnegative Matrix Factorization," Computational Statistics and Data Analysis, vol. 52, no. 1, pp. 155-173, 2007.
[17] W. Liu, N. Zheng, and X. Lu, "Nonnegative Matrix Factorization for Visual Coding," Proc. IEEE Int'l Conf. Acoustics, Speech, and Signal Processing, pp. 293-296, Apr. 2003.
[18] R. Zdunek and A. Cichocki, "Nonnegative Matrix Factorization with Constrained Second-Order Optimization," Signal Processing, vol. 87, no. 8, pp. 1904-1916, 2007.
[19] P. Hoyer, "Nonnegative Matrix Factorization with Sparseness Constraints," J. Machine Learning Research, vol. 5, pp. 1457-1469, 2004.
[20] H. Laurberg, M.G. Christensen, M.D. Plumbley, L.K. Hansen, and S.H. Jensen, "Theorems on Positive Data: On the Uniqueness of NMF," Computational Intelligence and Neuroscience, vol. 2008, pp. 1-9, 2008.
[21] J. Mansfield, K. Gossage, C. Hoyt, and R. Levenson, "Autofluorescence Removal, Multiplexing, and Automated Analysis Methods for In-Vivo Fluorescence Imaging," J. Biomedical Optics, vol. 10, no. 4, pp. 1-9, 2005.
[22] T.-H. Chan, W.-K. Ma, C.-Y. Chi, and Y. Wang, "A Convex Analysis Framework for Blind Separation of Non-Negative Sources," IEEE Trans. Signal Processing, vol. 56, no. 10, pp. 5120-5134, Oct. 2008.
[23] F.-Y. Wang, C.-Y. Chi, T.-H. Chan, and Y. Wang, "Blind Separation of Positive Dependent Sources by Non-Negative Least-Correlated Component Analysis," Proc. IEEE Int'l Workshop Machine Learning for Signal Processing, pp. 73-78, Sept. 2006.
[24] M.E. Winter, "N-Findr: An Algorithm for Fast Autonomous Spectral End-Member Determination in Hyperspectral Data," Proc. SPIE Conf. Imaging Spectrometry, pp. 266-275, Oct. 1999.
[25] J.M.P. Nascimento and J.M.B. Dias, "Vertex Component Analysis: A Fast Algorithm to Unmix Hyperspectral Data," IEEE Trans. Geoscience and Remote Sensing, vol. 43, no. 4, pp. 898-910, Apr. 2005.
[26] C.I. Chang, C.-C. Wu, W.-M. Liu, and Y.-C. Quyang, "A New Growing Method for Simplex-Based Endmember Extraction Algorithm," IEEE Trans. Geoscience and Remote Sensing, vol. 44, no. 10, pp. 2804-2819, Oct. 2006.
[27] A. Prieto, C.G. Puntonet, and B. Prieto, "A Neural Learning Algorithm for Blind Separation of Sources Based on Geometric Properties," Signal Processing, vol. 64, pp. 315-331, 1998.
[28] P. Santago and H.D. Gage, "Statistical Models of Partial Volume Effect," IEEE Trans. Image Processing, vol. 4, no. 11, pp. 1531-1540, Nov. 1995.
[29] D. Stein, S. Beaven, L. Hoff, E. Winter, A. Schaum, and A. Stocker, "Anomaly Detection from Hyperspectral Imagery," IEEE Signal Processing Magazine, vol. 19, no. 1, pp. 58-69, Jan. 2002.
[30] M.S. Bazaraa, H.D. Sherall, and C.M. Shetty, Nonlinear Programming: Theory and Algorithms, second ed. John Wiley, 1993.
[31] S. Lang, Undergraduate Analysis, second ed. Springer-Verlag, 1983.
[32] S.H. Friedberg, A.J. Insel, and L.E. Spence, Linear Algebra, third ed. Prentice-Hall, 1997.
[33] I.J. Lustig, R.E. Marsten, and D.F. Shanno, "Interior Point Methods for Linear Programming: Computational State of the Art," ORSA J. Computing, vol. 6, no. 1, pp. 1-14, 1994.
[34] C.B. Barber, D.P. Dobkin, and H. Huhdanpaa, "The Quickhull Algorithm for Convex Hulls," ACM Trans. Math. Software, vol. 22, pp. 469-483, Dec. 1996.
[35] T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning. Springer, 2001.
[36] R.A. Horn and C.R. Johnson, Matrix Analysis. Cambridge Univ. Press, 1985.
[37] P. Tichavský and Z. Koldovský, "Optimal Pairing of Signal Components Separated by Blind Techniques," IEEE Signal Processing Letters, vol. 11, no. 2, pp. 119-122, Feb. 2004.
[38] A. Cichocki and S. Amari, Adaptive Blind Signal and Image Processing. John Wiley, 2002.
[39] K. Suzuki, R. Engelmann, H. MacMahon, and K. Doi, "Virtual Dual-Energy Radiography: Improved Chest Radiographs by Means of Rib Suppression Based on a Massive Training Artificial Neural Network (Mtann)," Radiology, vol. 238, http://suzukilab. uchicago.eduresearch.htm , 2006.
[40] A.R. Padhani and J.E. Husband, "Dynamic Contrast-Enhanced MRI Studies in Oncology with an Emphasis on Quantification, Validation and Human Studies," Clinical Radiology, vol. 56, no. 8, pp. 607-620, 2001.
[41] T.F. Coleman and Y. Li, "A Reflective Newton Method for Minimizing a Quadratic Function Subject to Bounds on Some of the Variables," SIAM J. Optimization, vol. 6, no. 4, pp. 1040-1058, 1996.
[42] C.-Y. Chi, C.-C. Feng, C.-H. Chen, and C.-Y. Chen, Blind Equalization and System Identification. Springer-Verlag, 2006.