FOR BIOMEDICAL IMAGE ANALYSIS Chong-Yung Chi ( ) Institute of - PowerPoint PPT Presentation

NON-NEGATIVE BLIND SOURCE SEPARATION FOR BIOMEDICAL IMAGE ANALYSIS Chong-Yung Chi ( 祁忠勇 ) Institute of Communications Engineering, & Department of Electrical Engineering National Tsing Hua University, Hsinchu, Taiwan 30013 E-mail: cychi@ee.nthu.edu.tw http://www.ee.nthu.edu.tw/cychi/ Joint work with Dr. Tsung-Han Chan, NTHU, Taiwan, Prof. Yue Wang, Virginia Tech., VA, USA, & Prof. Wing-Kin Ma, CUHK, HK 2nd International Symposium on IT Convergence Engineering , POSTECH, Pohang, Korea, August 19-20, 2010 (ISITCE 2010) .

2 Outline  Biomedical Image Analysis and Blind Source Separation (BSS)  Non-negative Blind Source Separation (nBSS): Challenges, Breakthroughs, & Innovations  Experimental Results with Real Biomedical Data  Summary and Future Researches

3 Outline  Biomedical Image Analysis and Blind Source Separation (BSS)  Non-negative Blind Source Separation (nBSS): Challenges, Breakthroughs, & Innovations  Experimental Results with Real Biomedical Data  Summary and Future Researches

Biomedical Image Analysis 4 4 ... Time t=699 sec t=189 sec t=159 sec t=129 sec DCE-MRI time series of breast cancer images captured at different times.  Dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) uses various molecular weight contrast agents to assess tumor vasculature perfusion and permeability.  Is there any “important” bio-information hidden in these observed images?

Biomedical Image Analysis 5 ... Time t=699 sec t=189 sec t=159 sec t=129 sec DCE-MRI time series of breast cancer images captured at different times.  Due to the limited spatial resolution of the imaging device and/or the partial volume effect in the tumor, the signal at each pixel represents a linear mixture of more than one vasculature compartment.  Biomedical image analysis is to effectively extract the information of interest from these images.

Signal Model for Biomedical Images 6 The observed pixel vector (based on pharmacokinetics analysis): : mixing matrix (temporal pattern matrix),  : k th temporal pattern, : source pixel vector  : no. of sources, : no. of observed images, : no. of pixels. 

Blind Source Separation (BSS) 7 The observed pixel vector (based on pharmacokinetics analysis): Goal of BSS : Estimate the K tissue-distribution maps from the given M observed images without the knowledge of temporal patterns

Non-negative BSS (nBSS) 8 Some general assumptions:  (A1) The tissue-distribution maps are non-negative, i.e., .  (A2) The temporal patterns are linearly independent. BSS under the premise of source non-negativity is referred to as non- negative BSS (nBSS).

Convex Sets for nBSS 9  Affine hull of a set of vectors :  Convex hull of a set of vectors  Solid region of a set of vectors :

Geometric Perspective to nBSS 10 (b) From the observations we can construct a polyhedral set, in which the true source (a) Vector space of the signals. vectors must lie. Solid: true sources; dashed: observations. (d) Implemention of CAMNS: (c) Key result of CAMNS: computationally estimate the the true source vectors are at the extreme points; e.g., by LP. extreme points of the polyhedral set.

An Application of nBSS 11

Another Application of nBSS 12 Separated images (164 × 164) Dual-energy X-ray chest images Bone structure nBSS Soft tissue

13 Outline  Biomedical Image Analysis and Blind Source Separation (BSS)  Non-negative Blind Source Separation (nBSS): Challenges, Breakthroughs, & Innovations  Experimental Results with Real Biomedical Data  Summary and Future Researches

Challenges for nBSS in Biomedical Images 14 Tissue-distribution maps are in general statistically correlated.  E.G., correlation coefficient 1 between bone and soft tissue (in page 12) is 0.65.  Most conventional statistical BSS approaches that rely on the source independence assumption , such as independent component analysis (ICA), almost fail completely in biomedical image analysis. 1 The correlation coefficient between two random variables and is defined as follows: where is the expectation operator, and are means of and , respectively, and and are their standard deviations, respectively. Note that . The larger the , the higher the correlation between and .

nBSS Algorithm Design Methodology 15 Extracted tissue Observations Source distribution maps Preprocessing Separation Algorithm x [ n ] s [ n ] Criterion  Preprocessing:  Region/signal of interest selection and outlier pixel filtering.  Dimension/rank/noise reduction (with least information loss) to significantly reduce the complexity of the subsequent source separation processing.  Source separation criterion:  Utilization of various source and/or mixing matrix characteristics and optimization theory to establish/create a separation criterion in a rigorous fashion.  Algorithm:  Algorithm development and implementation to fulfill the source separation criterion.

Breakthroughs in nBSS 16 Apply convex analysis and optimization to nBSS, including problem formulation,  separation criteria establishment, and algorithm development. Our nBSS algorithms never involve any statistical assumptions and their  performances are supported by rigorous mathematical proofs and analyses. Most of our nBSS algorithms can be efficiently implemented by available convex  optimization solvers E.G., CVX (http:www.stanford.edu/~boyd/cvx/) SeDuMi (http://sedumi.mcmaster.ca/) Some real data experimental results have substantiated the effectiveness of our  convex analysis based nBSS algorithms. nBSS using convex optimization turns out to be an interdisciplinary research from  wireless communications to biomedical image and hyperspectral image analysis.

Innovations in nBSS Algorithms 17 Non-negative least correlated component analysis by iterative volume maximization  ( nLCA-IVM ) – IEEE Trans. Pattern Analysis and Machine Intelligence, May 2010. Convex analysis of mixtures of non-negative sources ( CAMNS ) – IEEE Trans. Signal  Processing, Oct. 2008, and a Chapter in the book entitled “ Convex Optimization in Signal Processing and Communications, ” Cambridge University Press, 2010. Minimum-volume enclosing simplex ( MVES ) algorithm – IEEE Trans. Signal Processing,  Nov. 2009. Alternating volume maximization (AVMAX) algorithm – in Proc. First IEEE Workshop  on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing (WHISPERS), Aug. 2009. Robust MVES (RMVES) algorithm – in Proc. IEEE International Conference on Acoustics,  Speech, and Signal Processing (ICASSP), Mar. 2010. Robust AVMAX (RAVMAX) algorithm – in Proc. Second IEEE WHISPERS, June 2010.  Normalized scatterplot clustering - convex analysis of mixtures ( NSC-CAM )  method – to be submitted to IEEE Trans. Medical Imaging. Matlab source codes of nLCA-IVM, CAMNS and MVES have been released at  http://www.ee.nthu.edu.tw/cychi/ due to numerous international requests.

nLCA-IVM 18 Assumptions: (A1) and (A2) (general assumptions as in page 8)   (A3) The sum of all the elements of each row vector of is unity (which holds in MRI due to partial volume effect).  (A4) The elements of are non-negative. Preprocessing: Principal component analysis (PCA) for rank/noise reduction   Find the approximated data matrix from  The optimal , where , and in which denotes the left singular vector associated with the i th principal singular value of The (rank-reduced) observations to be processed are the ones with the  least approximation errors , followed by setting their non-positive entries equal to zero.

nLCA-IVM 19 Source separation criterion: Design of a square demixing matrix such that the volume  of the solid region formed by the extracted non-negative sources is maximized.  Denote the extracted source vector as where is the demixing matrix, and denote the ith extracted source map as  The nLCA is to solve the following volume maximization problem where is the volume of the solid region , and is a vector with all the entries equal to unity.  The optimum has been proven true under (A1) to (A4) and (A5) for each , there exists an (unknown) index such that and for all . (called the existence of pure source samples)

nLCA-IVM 20 Algorithm: Cofactor expansion & alternating linear programming   As it is known that , the above problem can be written as  Consider the cofactor expansion of (w. r. t. the ith row, of ) where is the submatrix of with the i th column and j th row removed.  Update only one row vector of each time, say , while fixing the other rows (i.e., is fixed for all )