Alexander M. Bronstein, Michael M.Bronstein Department of Computer Science Technion - Israel Institute of Technology Haifa 32000, Israel Michael Zibulevsky, Yehoshua Y. Zeevi Department of Electrical Engineering Technion - Israel Institute of Technology Haifa 32000, Israel ABSTRACT We pose the problem of tissue classification in MRI as a Blind Source Separation (BSS) problem and solve it by means of Sparse Component Analysis (SCA). Assuming that most MR images can be sparsely represented, we consider their optimal sparse represen- tation. Sparse components define a physically-meaningful feature space for classification. We demonstrate our approach on simu- lated and real multi-contrast MRI data. The proposed framework is general in that it is applicable to other modalities of medical imaging as well, whenever the linear mixing model is applicable. 1. INTRODUCTION Tissue classification for diagnosis has been widely addressed from the viewpoint of machine learning and image processing [1]. Mag- netic resonance imaging (MRI) is especially useful for this task owing to its ability to image tissues characterized by their mag- netic properties (spin-lattice relaxation time T1 , spin-spin relax- ation time T2 and proton density PD). By appropriately choosing pulse sequence parameters (echo time TE and repetition time TR), tissue properties can be emphasized, producing a set of images with different contrast. Roughly speaking, brain tissues, for example, can be thought of as consisting of water and fat in different proportions. These substances have different spin properties, and hence contribute dif- ferently to the resulting MR image when different contrasts are used. The underlying physical model in MRI suggests that such ”mixing” is linear. This principle is used in the 2-point Dixon method [2] for fat suppression. The Dixon method requires the images to be acquired exactly in-phase and out-of-phase, such that one component can be removed by simple averaging of the two im- ages. This exact phase relation cannot always be easily achieved, e.g. due to inhomogeneities of the field. In this study, we consider a more general blind source sep- aration (BSS) framework, which can be used on generic multi- contrast MR data. We solve this problem by means of Sparse Component Analysis (SCA). This approach is based on the as- sumption that typical MR images, like other natural images, can be sparsely represented by an appropriate transformation. We address the problem of finding optimal sparse representation for such im- ages. SCA produces a physically-meaningful feature space, wherein tissue classification can be carried out using simple linear methods. This research has been supported by the HASSIP Research Network Program HPRN-CT-2002-00285, sponsored by the European Commission and by the Ollendorff Minerva Center for Vision and Image Sciences. 2. THE LINEAR MIXTURE MODEL Let S1 and S2 denote two Nx × Ny source images, representing concentration of the basic two components (fat and water) of the brain tissue. In multi-contrast MRI, we produce a set of M mix- tures Xi , given by linear combinations X m = a m1 S 1 + a m2 S 2 , m =1...M, (1) and possibly contaminated by noise (accounting also for the pres- ence of other substances with properties different from those of water and fat). In matrix form, (1) can be rewritten as X = A · S, (2) where A is a M × 2 mixing matrix, S is 2 × Nx Ny matrix consist- ing of source images parsed into row vectors, and X is M ×Nx N y matrix of mixtures constructed similarly. The mixing matrix repre- sents the relative response of water and fat at each contrast. Note that unlike the Dixon method, here we assume arbitrary chosen contrasts. We assume no a priori knowledge of A, except that M ≥ 2 and rank(A)=2. In addition, we assume without loss of gen- erality that the sources have zero mean. When M> 2, X is a redundant representation of S (at least in the zero-noise case) with M − 2 linearly dependent combinations of S1 and S2. This re- dundancy can be removed by reducing the dimension of X to 2 for example by using PCA. As a result, one obtains a 2 × Nx N y matrix Y = Φ · X = A ′ · S, (3) whose rows are the first two principal components. The matrix Φ denotes the PCA projection matrix. 3. SPARSE COMPONENT ANALYSIS Our goal is to estimate the sources S, representing water and fat concentrations, given Y = A ′ · S, where A ′ is a 2 × 2 unknown in- vertible matrix. This problem is usually referred to as blind source separation and is often solved by means of Independent Compo- nent Analysis (ICA). The assumption of statistical independence of sources can be relaxed and replaced by the assumption of their sparseness. This gives rise to the Sparse Component Analysis (SCA), introduced in [3]. We use the quasi-maximum likelihood algorithm [4, 5] for SCA. “UNMIXING” TISSUES: SPARSE COMPONENT ANALYSIS IN MULTI-CONTRAST MRI 0-7803-9134-9/05/$20.00 ©2005 IEEE