Component Analysis Approach to Estimation of Tissue Intensity Distributions of 3D images Arridhana Ciptadi, Cheng Chen and Vitali Zagorodnov Department of Computer Engineering Nanyang Technological University, Singapore carridhana@ntu.edu.sg ccheng@pmail.ntu.edu.sg zvitali@ntu.edu.sg Abstract Many segmentation problems in medical imaging rely on accurate modeling and estimation of tissue intensity probability density functions. Gaussian mixture modeling, currently the most common approach, has several drawbacks, such as reliance on a specific model and iterative optimization. It also does not take advantage of substantially larger amount of data provided by 3D acquisitions, which are becoming standard in clinical environment. We propose a novel completely non-parametric algorithm to estimate the tissue intensity probabilities in 3D images. Instead of relying on traditional framework of iterating between classification and estimation, we pose the problem as an instance of a blind source separation problem, where the unknown distributions are treated as sources and histograms of image subvolumes as mixtures. The new approach performed well on synthetic data and real magnetic resonance (MR) scans, robustly capturing intensity distributions of even small image structures and partial volume voxels. I. . Introduction Many segmentation problems in medical imaging rely on accurate modeling and estimation of tissue intensity probability density functions (pdfs) [20], [24], [30], [14], [21], usually in the context of statistical region-based segmentation. Commonly, tissue intensity probabilities are represented using the finite mixture (FM) model [20], [7], [29], and its special case the finite Gaussian mixture (FGM) [27], [1]. In these models the intensity pdf of each tissue class is represented by a parametric (e.g. Gaussian in the case of FGM) function called the component density while the intensity pdf of the whole image is modeled by a weighted sum of the tissue component densities. The fitting is usually done using Expectation Maximization (EM) algorithm [4], [16], [24], [21], [8], which iterates between soft classification and parameter estimation until a stable state is reached. The main deficiency of FGM models is that the tissue intensity distributions do not always have a Gaussian form. The noise in magnetic resonance (MR) images is known to be Rician rather than Gaussian [9]. Partial volume (PV) voxels represent a mixture of ‘pure’ classes and have non-Gaussian distribution even when the pure classes are Gaussian [12], [14], [26], [24], [28]. Another problem generally associated with the FM+EM framework is the local convergence properties of the iterative EM algorithm, requiring sufficiently close parameter initialization [7], especially for distribution means [6]. The convergence of the EM algorithm to a more meaningful optimum can be improved by including prior information in the classification step, such as pixel correlations [25], MRF priors [15], [28], [30], [18] or probabilistic atlas [15], [18], [21]. However, probabilistic atlases are not available for some applications, as is in the case of segmentation of brain lesions [22] or localization of fMRI activity [23]. Moreover, reliance on prior information can cause bias in estimation [25]. Finally, the FM+EM approach often fails to take advantage of substantially larger amount of data present in 3D images, which are becoming more and more common due to increasing use of MR and CT scanning techniques. We propose a novel non-parametric algorithm to estimate tissue intensity probabilities in 3D images that completely departs from traditional classification-estimation framework. To illustrate the main idea behind our approach, consider the following introductory example. Shown in Figure 1 are the histograms of a 3D T1weighted MR image and two of its 2D slices. The variability in the shape of 2D histograms is due to varying tissue proportions across the slices. While this variability can 1765 2009 IEEE 12th International Conference on Computer Vision (ICCV) 978-1-4244-4419-9/09/$25.00 ©2009 IEEE 3 1.2 Let hi ∈ RL be the L-bin histogram of Vi , normalized to sum to 1. Then x 10 1 0.8 hi = (ΣK j=1 λij fj ) + ei 0.6 0.4 0.2 0 0 50 100 150 200 250 300 200 250 300 (a) 3D Image and its histogram 3 4.5 x 10 4 3.5 3 2.5 2 1.5 1 0.5 0 0 50 100 150 (b) Transverse slice 128 and its histogram (1 ≤ i ≤ N ) (1) where λij is the j-th tissue proportion in the i-th subvolume, Σj λij = 1, and ej is the noise term that reflects the difference between the actual probability distribution and its finite approximation by a histogram. Let H = [h1 h2 . . . hN ]T and Λ = {λij }. Rewriting (1) in a matrix form yields ⎡ ⎤ ⎤ ⎡ ⎤ ⎡ h1 f1 e1 ⎢ h2 ⎥ ⎢ f2 ⎥ ⎢ e2 ⎥ ⎥ ⎥ ⎢ ⎥ ⎢ H =⎢ (2) ⎣ ... ⎦ = Λ⎣ ... ⎦ + ⎣ ... ⎦ hN fK eN 3 9 x 10 8 7 6 5 4 3 2 1 0 0 50 100 150 200 250 300 (c) Transverse slice 152 and its histogram Fig. 1. Histograms of a 3D brain image and several of its slices potentially provide useful information for mixture estimation, it is traditionally discarded by performing estimation directly on the 3D histogram. Instead, the histogram of each 2D slice can be treated as a mixture realization of component densities, with the number of realizations potentially larger than the number of components. This allows stating the unmixing problem as a blind source separation (BSS) problem [3], [5]. To solve the problem we use a framework that is similar to that of independent component analysis (ICA) [11], but without relying on the independence assumption. Instead we use the fact that underlying components must be valid probability distributions with different means, which results in a simple convex optimization problem that guarantees convergence to a global optimum. II. . Problem Statement Let V be a 3D image volume partitioned into a set of N subvolumes V1 , V2 , . . . , VN . We assume the voxel intensities of V can take L distinct values and are drawn from K unknown probability mass functions (pmf) f1 , f2 , . . . , fK ∈ RL . For example, a brain volume can be assumed to have 3 main tissues, white matter (WM), gray matter (GM) and cerebrospinal fluid (CSF), so K = 3. For an 8-bit acquisition, L = 256. Subvolumes can be chosen arbitrary, for example as coronal, sagittal or transverse slices of the 3D volume. which is identical to Blind Source Separation (BSS) formulation with subvolume histograms as mixtures and unknown tissue pmf’s as sources. Our goal is to estimate f1 , f2 , . . . , fK as well as their mixing proportions Λ given the mixture matrix H. Our solution requires several assumptions, most of which are general to a BSS problem: N ≥ K, L > K, and sufficient variability of mixing proportions Λ. These can be easily satisfied with proper choice of partitioning and histogram dimensionality. We also assume that distributions f1 , f2 , . . . , fK have different means and are sufficiently separated from each other, where the meaning of sufficient separation is detailed in Section III. These assumptions are not very restrictive and are generally satisfied for medical images [27], [19]. III. . Proposed Solution BSS problem has been studied extensively in recent years, with several solutions proposed for selected special cases, e.g. factor analysis (FA) [13] for Gaussian sources and independent factor analysis (IFA) [2] or independent component analysis (ICA) [11] for independent non-Gaussian sources. These cannot be extended to our case because our source components are neither Gaussian nor independent but instead are constrained to represent valid probability distribution functions of voxel intensities within approximately uniform image regions. As with the ICA, the first step is to determine the original subspace spanned by tissue pmf’s by applying principal component analysis (PCA) to the mixture matrix H. Assuming that PCA was successful, there will be a linear relationship between f1 , f2 , . . . , fK and p1 , p2 , . . . , pK F = TP (3) where F = [f1 , f2 , . . . , fK ]T and P = [p1 , p2 , . . . , pK ]T . Estimation of T for K = 2 is the topic of Section III-A. Case K > 2 is treated in Section III-B. 1766 A..Estimating Two Unknown Components For K = 2, let p1 ,p2 be the estimated principle components and μ1 ,μ2 be the means of f1 and f2 respectively. Without loss of generality, we assume μ1 < μ2 . Then according to the following lemma, the mean μ of any valid pmf that is a linear combination of p1 and p2 must satisfy μ1 − 1 ≤ μ ≤ μ2 + 2 , where 1 and 2 are usually small and can be ignored. Lemma 1: Let K = 2 and μ1 < μ2 be the means of underlying probability distributions. Let g = τ1 p1 + τ2 p2 satisfy ΣL j=1 g(j) = 1 and g(j) > 0 for 1 ≤ j ≤ L. Then μ1 − 1 (μ2 − μ1 ) ≤ where 1 = ΣL j=1 jg(j) 1 f (i) maxi ( f2 (i) )−1 ≤ μ2 + 2 (μ2 − μ1 ) (4) and 2 = 1 1 . f (i) maxi ( f1 (i) )−1 The 2 equalities are achieved when g = f1 − 1 (f2 − f1 ) and g = f2 + 2 (f2 − f1 ). Proof: From g = τ1 p1 + τ2 p2 and equation (3) it follows that g is also linear combination of fi ’s: g = ζ1 f1 + ζ2 f2 If we ignore 1 and 2 , the equalities in (4) are achieved when g = f1 or g = f2 . Similar statement can be made in case of more than two components, see the following corollary. Corollary 1: Let K > 2 and μ1 < μ2 < . . . < μK be the means of underlying probability distributions. Let L g = ΣK i=1 τi pi satisfy Σj=1 g(j) = 1 and g(j) > 0 for 1 ≤ j ≤ L. Then μ1 ≤ ΣL j=1 jg(j) ≤ μK The equalities holds when g = f1 or g = fK . An important implication of Lemma 1 and its corollary is that one can unmix two components by minimizing or maximizing the mean of a linear combination of principal components, subject to constraints that this linear combination represents a valid pmf. In other words, the coefficients K τ1i of f1 = ΣK i=1 τ1i pi and τKi of fK = Σi=1 τKi pi can be estimated by solving the following Linear Programming optimization problems: L minimize: ΣK i=1 τ1i Σj=1 (jpi (j)) K subject to: Σi=1 τ1i pi (j) ≥ 0 , 1 ≤ j ≤ L L ΣK i=1 τ1i Σj=1 pi (j) = 1 Then L ΣL i=1 g(j) = 1 → Σj=1 ζ1 f1 (j) + ζ2 f2 (j) = ζ1 + ζ2 = 1 (5) and L minimize: −ΣK i=1 τKi Σj=1 (jpi (j)) subject to: ΣK i=1 τKi pi (j) ≥ 0 , 1 ≤ j ≤ L L ΣK i=1 τKi Σj=1 pi (j) = 1 L μ = ΣL j=1 jg(j) = Σj=1 j[ζ1 f1 (j) + ζ2 f2 (j)] L = ζ 1 ΣL j=1 jf1 (j) + ζ2 Σj=1 jf2 (j) = ζ1 μ1 + ζ2 μ2 From (5) we can use the following parametrization: ζ1 = 1 + α and ζ2 = −α. Then, μ = μ1 − α(μ2 − μ1 ) and to minimize μ we should make α as large as possible. However, the largest possible α is controlled by nonnegativity of g: (1 + α)f1 (i) − αf2 (i) f1 (i) α α ≥ 0 ≥ α(f2 (i) − f1 (i)) ≤ f2 (i)1 ≤ −1 1 f (i) maxi ( f2 (i) )−1 f1 (i) 1 1 maxi ( ff21 (i) (i) ) −1 (6) which leads to the right side of inequality (4). In practice, i ’s can be assumed small or even zero. For example, if fi components are Gaussian on unbounded domain, it can be straightforwardly shown that maxi ff21 (i) (i) = maxi ff12 (i) (i) (8) (9) When the number of components is 2, (8) and (9) provide the complete solution. When the number of components is larger than two, (8) and (9) produce components with smallest and largest mean. The next section discusses how remaining components can be estimated. B..Estimating More Than Two Components This leads to the left side of inequality (4). Similarly, using parametrization ζ1 = −α and ζ2 = 1 + α we obtain μ = μ2 + α(μ2 − μ1 ) and hence to maximize μ we need to choose α as large as possible, which results in α≤ (7) First, let’s assume that K = 3, and f1 and f3 have been estimated using (8) and (9). Then the remaining component f2 can be estimated by minimizing its overlap with the first two components, which can be solved using another linear programming problem, as shown in the following lemma. Here notation ·, · stands for inner product between two vectors. 2 2 Lemma 2: Let min(f1 , f3 ) > f1 , f2 +f2 , f3 . If f1 , f3 are known, then the τi coefficients of f2 = ΣK i=1 τi pi are the solution of the following linear programming problem: minimize: Σ3i=1 τi pi , (f1 + f3 ) subject to: Σ3i=1 τi pi (j) ≥ 0, 1 ≤ j ≤ L Σ3i=1 τi ΣL j=1 pi (j) = 1 = ∞ and hence 1 = 2 = 0. 1767 (10) Proof: The sum of overlaps between the unknown component and f1 ,f3 is g, (f1 + f3 ) = Σ3i=1 ζi fi , (f1 + f3 ) = Σ3i=1 ζi fi , (f1 + f3 ) = ζ1 f1 2 + (ζ1 + ζ3 )f1 , f3 + ζ2 (f1 , f2 + f2 , f3 ) + ζ3 f3 2 Let function w(ζi ) = g, (f1 + f3 ), then IV. . Experimental Results wζ 1 (ζi ) = f1 2 + f1 , f3 wζ 3 (ζi ) wζ 2 (ζi ) 2 = f3 + f1 , f3 = f1 , f2 + f2 , f3 A..Implementation Since w(ζi ) is a linear function of ζi ’s, 0 ≤ ζi ≤ 1 and ζ1 + ζ2 + ζ3 = 1, its minimum occurs at ζk = 1, ζj = 0, j = k, where k is the coordinate along which w has the smallest slope, i.e. k = arg min wζ i i According to the lemma conditions, min(f1 2 , f3 2 ) > f1 , f2 +f2 , f3 , hence min(wζ 1 , wζ 3 ) > wζ 2 . Therefore, the minimum value of w(ζi ) is achieved when ζ1 = 0, ζ3 = 0 and ζ2 = 1 and g = Σ3i=1 ζi fi = f2 . The necessary condition in Lemma 2, min(f1 2 , f3 2 ) > f1 , f2 + f2 , f3 , can be interpreted as sufficient separation between the underlying components, since the underlying functions are nonnegative. More specifically, it requires that the sum of overlaps between f1 and f2 and between f2 and f3 is smaller than the norm of f1 or f3 . This translates into a minimum SNR of 1.655 in the case of Gaussian components. This requirement can be easily satisfied for most medical images. In the case of more than three components, starting from the two first estimated components, all other components can be estimated one by one by minimizing their overlap with all previously estimated components, as shown in the following lemma. Lemma 3: Let f1 , . . . , fK be the underlying components, of which the first n are already known. Let min Σnj=1 fi , fj > max Σnj=1 fi , fj i,i≤n i,i>n (11) Then g = ΣK i=1 τi pi , where τi are solutions of the following linear programming problem, will coincide with one of the remaining unknown components. minimize: subject to: n ΣK i=1 τi pi , Σa=1 fa K Σi=1 τi pi (j) ≥ 0, 1 ≤ j ≤ L L ΣK i=1 τi Σj=1 pi (j) = 1 k is the coordinate along which w has the smallest slope. According to (11), the minimum must correspond to one of the unknown components, i.e. k > n. Note that one of the terms on the right side of inequality (11) is equal to the norm of a component, while all other terms on both sides correspond to overlaps between components. Hence condition (11) can be again interpreted as sufficient separation between the underlying components. (12) Proof: Let function w(ζ1 , . . . , ζK ) = g, Σnj=1 fj . Then wζ i = Σnj=1 fi , fj . As mentioned in Lemma 2, the minimum of w occurs when ζk = 1, ζj = 0, j = k, where Our algorithm was implemented fully in Matlab, using built-in functions pcacov and linprog to estimate PCA components and perform linear programming optimization. The volume partition was limited to cuboids of size 5 × 5 × 5, which was chosen empirically and used for all subsequent experiments. During initial testing on simulated data we discovered that the non-negative constraint imposed on estimated components fi was too strict. The histogram noise and errors in estimating the subspace can lead to an infeasible optimization problem or a very narrow search space. To overcome this we relaxed the non-negativity constraint 1 fi ≥ 0 to fi ≥ − 2L , where L is the number of histogram bins. This negative bound was small enough not to cause any visible estimation problems in our experiments. B..Estimating intensity distributions from structural MR data We applied our algorithm on T1-weighted brain images from two publicly available data sets, BrainWeb (http://www.bic.mni.mcgill.ca/brainweb/) and IBSR (http://www.cma.mgh.harvard.edu/ibsr/). The BrainWeb data set contains realistic synthesized brain volumes with varying degrees of noise and intensity nonuniformity, and 1 × 1 × 1mm3 resolution. The IBSR data set contains real MR acquisitions made on a 1.5 T scanner with resolution 1 × 1 × 1.5mm3 . Both data sets contained ground truth for GM and WM. In addition, the BrainWeb data set also contained ground truth for CSF. We further augmented the ground truth to include mix classes of partial volume voxels, namely the GM-WM (for both data sets) and the CSF-GM (for BrainWeb data set only). The partial volume voxels were defined as the voxels located near the boundary between two tissues. Practically, these were identified by performing a one voxel erosion of each tissue with standard 6-neighbor connectivity. All non-brain tissues were removed prior to processing. Our algorithm provided excellent estimates of peak position, shape and proportion of each distribution on BrainWeb data set, when only the main classes were estimated (Fig. 2). Inclusion of mix classes slightly reduced 1768 0.015 0.015 0.01 probability probability Correlation - mean [range] 0.005 0.01 Prop. error - mean [range] 0.005 0 50 100 150 pixel intensity 200 0 250 50 100 150 pixel intensity 200 250 [GM WM] 0.947 [0.758-0.995] 0.072 [0.002-0.172] [GM WM GM-WM] 0.911 [0.704-0.997] 0.099 [0.005-0.241] TABLE I. Correlation and proportion error between estimated and true distributions, averaged over 18 IBSR volumes (a) Noise=3 C.. Estimating distribution of activated voxels from simulated functional MR data 0.012 0.01 0.01 probability probability 0.008 0.008 0.006 0.006 0.004 0.004 0.002 0.002 0 0 50 100 150 pixel intensity 200 250 50 100 150 200 250 100 150 200 250 pixel intensity (b) Noise=5 0.01 0.01 0.009 0.008 0.008 probability probability 0.007 0.006 0.004 0.006 0.005 0.004 0.003 0.002 0.002 0.001 0 50 100 150 pixel intensity 200 0 250 50 pixel intensity (c) Noise=7 Fig. 2. Estimating 3 classes [CSF GM WM] (left) and 3 pure classes [CSF GM WM] + 2 mix classes [CSF-GM GM-WM] (right) on BrainWeb data. Ground truth distributions are shown using dotted lines. the quality of estimation, especially for large noise levels. However, it is remarkable that our algorithm was capable of capturing a two-peak shape of CSF-GM mix class distribution [Fig. 2 a) and b)], which would not be possible with a Gaussian mixture model. To compare our algorithm with several other approaches on the volume with noise=7, compare Fig. 2 c) with Fig. 6 in the reference [6]. 0.01 0.009 0.008 0.008 0.007 0.007 probability probability 0.01 0.009 0.006 0.006 0.005 0.005 0.004 0.004 0.003 0.003 0.002 0.002 0.001 0 0.001 50 100 150 pixel intensity 200 250 0 50 100 150 pixel intensity 200 250 Fig. 3. Estimating 2 classes [GM WM] (left) and 2 pure classes [GM WM] + 1 mix class [GM-WM] (right) on IBSR volume 8. Ground truth distributions are shown using dotted lines. Our algorithm also performed well on the IBSR data set (Fig. 3, Table IV-B). Here by proportion error we meant the mean absolute difference between estimated and true proportions for each class. To estimate the quality of distribution shape estimation we used the average correlation between estimated and true distributions. Activated regions in functional MRI experiments are typically detected using significance threshold testing [17]. This allows controlling for Type I error but not for Type II error. While the knowledge of “activated” distribution would be helpful in determining a more appropriate threshold [10], the small size of activated class makes it challenging to estimate its distribution. To simulate functional MRI data we created a set of synthetic 200 × 200 × 200 resolution images, where activated regions were modeled as uniform intensity cubes of size 3×3×3 voxels on a uniform background. The images were corrupted by Gaussian noise, thus creating two Gaussian distributions for non-activated and activated classes. We then varied the difference between the means of the two distributions and proportion of activated (smaller) class to obtain different samples for our experiments. To provide quantitative performance assessment we used parameter estimates to determine the optimal threshold that minimizes misclassification error (the sum of Type I and II errors). We then recorded the percentage increase in misclassification error when using the found threshold vs. the optimal one, derived from the true distribution parameters, comparing the results with those obtained by the EM algorithm (Figure 4). The EM algorithm was initialized with true parameter values, corresponding to ideal performance that can rarely be achieved in practice, as the parameter values are never known precisely. The performance of our algorithm was not affected for SNR (ratio of the difference between means and standard deviation) range of 2 to 6. In each case, the estimated threshold was practically as good as the optimal threshold as long as the proportion of smaller class was larger than 0.68-1%. Performance of EM-based estimation was significantly worse than that of our approach for SNR=2-4, and is comparable (or slightly better) for SNR=6. However, considering that an imperfect initialization would likely to reduce EM algorithm performance, our approach offers a superior alternative to EM algorithm in this application. V. . Conclusions We developed a novel completely non-parametric algorithm to estimate the tissue intensity probability distributions in 3D images, by treating the problem as an 1769 Proporon of smaller class 40% 60% Ours 80% EM 100% 120% 140% 160% 180% 200% Proporon of smaller class 0% 20% 40% 60% Ours 80% EM 100% 120% 140% 160% 180% 200% (a) (b) Percentage increase in misclassificaon rate 0% 20% Percentage increase in misclassificaon rate Percentage increase in misclassificaon rate Proporon of smaller class 0% 20% 40% 60% Ours 80% EM 100% 120% 140% 160% 180% 200% (c) Fig. 4. Our approach vs. EM-based estimation. Percentage increase in misclassification error as a function of smaller class proportion for a) SNR=2, b) SNR=4, c) SNR=6 instance of blind source separation problem. 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