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/

2nd International Symposium on IT Convergence Engineering, POSTECH, Pohang, Korea, August 19-20, 2010 (ISITCE 2010). Joint work with Dr. Tsung-Han Chan, NTHU, Taiwan, Prof. Yue Wang, Virginia Tech., VA, USA, & Prof. Wing-Kin Ma, CUHK, HK

 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

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

4  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? 4

DCE-MRI time series of breast cancer images captured at different times.

t=129 sec t=159 sec t=189 sec t=699 sec

...

Time

Biomedical Image Analysis

Biomedical Image Analysis

5  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.

DCE-MRI time series of breast cancer images captured at different times.

t=129 sec t=159 sec t=189 sec t=699 sec

...

Time

Signal Model for Biomedical Images



: mixing matrix (temporal pattern matrix),



: kth temporal pattern, : source pixel vector



: no. of sources, : no. of observed images, : no. of pixels. The observed pixel vector (based on pharmacokinetics analysis):

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Blind Source Separation (BSS)

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Goal of BSS:

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

 (A1) The tissue-distribution maps are non-negative, i.e., .  (A2) The temporal patterns are linearly independent.

Non-negative BSS (nBSS)

8

Some general assumptions:

BSS under the premise of source non-negativity is referred to as non- negative BSS (nBSS).

Convex Sets for nBSS

 Affine hull of a set of vectors :  Convex hull of a set of vectors  Solid region of a set of vectors :

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Geometric Perspective to nBSS

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

An Application of nBSS

11

Another Application of nBSS

12

Dual-energy X-ray chest images Separated images (164×164) Bone structure Soft tissue

nBSS

 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

13 Outline

Challenges for nBSS in Biomedical Images

1The 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 .



Tissue-distribution maps are in general statistically correlated. E.G., correlation coefficient1 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. The larger the , the higher the correlation between and .

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nBSS Algorithm Design Methodology

Observations Extracted tissue distribution maps Source Separation Criterion Algorithm Preprocessing

x[n] s[n]

 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

• ptimization theory to establish/create a separation criterion in a rigorous

fashion.

 Algorithm:

 Algorithm development and implementation to fulfill the source separation

criterion.

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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

• ptimization 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

• n 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



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



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.

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in which denotes the left singular vector associated with the ith principal singular value of

nLCA-IVM



Source separation criterion: Design of a square demixing matrix such that the volume

• f 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)

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nLCA-IVM



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 ith column and jth row removed.

 Update only one row vector of each time, say , while fixing the other rows

(i.e., is fixed for all )

20

nLCA-IVM



Algorithm: Cofactor expansion & alternating linear programming

 The problem (1) can be decomposed into the following two LPs  The optimal solution of (1), denoted by , is chosen as the optimal solution of

(2a) if , otherwise as the optimal solution of (2b).

 The optimum demixing matrix, denoted by , can be obtained in a cyclic row-

by-row manner until convergence.

21

nLCA-IVM



Algorithm: Cofactor expansion & alternating linear programming (LP)

 Move to the boundary of the feasible set such that the volume of

is maximum while fixing , i.e., the volume of is guaranteed to increase maximally at each iteration.

22

A geometric illustration of the nLCA-IVM algorithm for .

 It can be shown that

where can be obtained by

 A closed-form solution: &

where

CAMNS



Assumptions: (A1) – (A3) (the same as in nLCA-IVM) and

 (A5) For each , there exists an (unknown) index such that

. (also called the “local dominance”)



Preprocessing: Affine set fitting for dimension/noise reduction to

23

CAMNS



Source separation criterion: Identification of K extreme points of an observation- constructed polyhedral set (or convex set) via simplex volume maximization

24

 It can be shown that

whose extreme points are exactly the true sources .

 The nBSS problem now becomes finding

all the extreme points of .

 Define

which is the image of . D d

CAMNS



Source separation criterion: Identification of K extreme points of an observation- constructed polyhedral set (or convex set) via simplex volume maximization

 It can be shown that is a simplex,

where satisfies

 Simplex volume maximization for finding all the extreme points of :

25

 Find such that its

simplex volume is maximized, i.e., where is the volume

• f the simplex

CAMNS



Source separation criterion: Identification of K extreme points of an observation- constructed polyhedral set (or convex set) via simplex volume maximization

 It has been shown that, under (A1)-(A3) and (A5), the optimum  The volume of is known as

The above volume maximization problem can be expressed as



Algorithm: Cofactor expansion & alternating linear programming as in nLCA-IVM

26

 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

27 Outline

Experimental Results for DCE-MRI data

28

Fast flow tissue Slow flow tissue Plasma input

100 200 300 400 500 600 700 0.04 0.045 0.05 0.055 0.06 0.065 0.07 time (sec) Slow flow Fast flow Plasma input

Normalized intensity

(a) (b)

 The DCE-MRI data set (of M=19 images) is provided by Prof. Yue

Wang, Virginia Tech., VA.

 The results were obtained by NSC-CAM for K=3: (a) Estimated time

activity curves (columns of ), and (b) spatial distributions of tissues associated with fast/slow perfusion rate, and plasma input.

 It is an advanced tumor case, e.g., “ring” shape (bright peripheral area)

angiogenesis of the tumor (fast flow tissue). A

Experiments with Real Fluorescence Microscopy

 Fluorescence microscopy uses an optical sensor array to produce

multispectral images in which the targets of a specimen are labeled with different fluorescence probes.

 The ability of identifying spectral biomarkers is limited due to the

spectral-overlapped problem among the probes (i.e., information leak- through from one spectral channel to another).

 Separating the fluorescence microscopy into individual maps associated

with specific biomarkers can be formulated as an nBSS problem.

 We investigate the cell division process in the newt lung cell images,

using the real data taken from http://publications.nigms.nih.gov/insidethecell/chapter1.html

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The results were obtained by nLCA-IVM

30

Experiments with Real Dynamic Fluorescence Imaging (DFI) Data

 DFI exploits highly specific and biocompatible fluorescent contrast

agents to interrogate small animals for drug development and disease research.

 Mainly due to the malign effects of light scattering and absorption,

each DFI image is delineated as a linear mixture of anatomical maps associated with different major organs, thus formulated as an nBSS problem.

 Our goal here is to separate different anatomical maps.  Two real DFI data sets (each consisting of M=150 images of rat) are

provided by Cambridge Research & Instrumentation (CRi), Inc., MA.

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Tongue Salivary gland Lymph nodes Lung Heart Liver Small intestine Blood vessels Tongue (6th) Salivary gland (4th) Lymph nodes (5th) Lungs (2nd) Heart (1st) Liver (7th) Small intestine (4th) Blood vessels (3rd) Estimated anatomical map (overlapped with the pseudo-colored estimated sources)

The above results were obtained by CAMNS

Results for DFI Data (Supine Position)

32

Results for DFI Data (Prone Position)

Liver (5th) Kidney (3rd) Blood vessels (4th) Lungs (1st) Brain (2nd) Blood vessels Liver Brain Lung Kidney

Estimated anatomical map (overlapped with the pseudo-colored estimated sources)

33

The above results were obtained by CAMNS

Experiments with Real Dual-energy Computed Tomography Data

 Dual-energy CT provides an effective tool for assessing the skeleton in

• steoporotic conditions. Bone in the presence of volumes of soft tissue

could make substantial errors in determination of cancellous bone.

 Our goal here is to separate bone structure and soft tissue, which can be

formulated as an nBSS problem.

 A real dual-energy CT data set (thorax) is provided by Prof. Yan Kang,

Northeastern University, Shengyang, China.

The results were obtained by nLCA & CAMNS

34

CT data (80KeV) CT data (140KeV)

 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

35 Outline

Summary and Future Researches

 All the presented nBSS algorithms were devised by convex analysis and

• ptimization theory. Their practical implementations (software), together with

advanced algorithm development are under study.

 Potential applications of our nBSS algorithms so far include DCE-MRI image

analysis (for early detection of disease), DFI image analysis (for drug

development and disease research), CT image analysis (for anatomical abnormality diagnosis), and hyperspectral image analysis (for material identification and distribution in remote sensing).

 More biomedical and biological applications are to be investigated.  Extensive real data tests to our nBSS algorithms are currently under way.  Convex optimization theory and software tools (which have been prevalent in

wireless communications) are essential to advances of these interdisciplinary areas in science and engineering.

36

Acknowledgements

37

Thank you for your attention!!!

Our researches are funded by National Science Council of Taiwan and NTHU.

My research team at WCSP Lab., NTHU.