Segmentation of MR Images of a Beating Heart

Transcription

1 Segmentation of MR Images of a Beating Heart Avinash Ravichandran Abstract Heart Arrhythmia is currently treated using invasive procedures. In order use to non invasive procedures accurate imaging modalities and algorithms are required. Magnetic Resonance Imaging (MRI) gives us the ability to visualize the ablations, while not producing any adverse effects such as radiation on the patients. A vital component of such visualization algorithm based on MR images, is the accurate segmentation of the heart from the chest. Since the segmentation would be repeated during the procedure, unsupervised methods would be preferred to methods that would involve any human intervention. 1. Introduction Given a set of ECG gated MR axial heart images, each image consisting of the heart and the chest of the patient,we want to segment the heart and chest from these images. For this segmentation the dynamics of the heart and the chest are exploited. For a given slice (i.e images containing the temporal evolution of a particular spatial location of the heart) the chest is static while the heart is moving. We try to model the heart as a dynamic texture and considering that a dynamic texture lies in a subspace the problem reduces to a subspace clustering problem. Thus in order to segment the subspaces we need to resort to existing algorithms such as Generalized Principal Component Analysis (GPCA) [3] and K-Subspace clustering. Using these algorithms, with some variations we can obtain the subspaces. 1.1 Contribution of the project The project address the following issues. 1. To determine the performance of the various algorithms namely GPCA, K-subspace and spatial GPCA. 2. For the K-subspace algorithm to determine the effect of the number of iterations and the initialization on the performance of the algorithm. 3. Window size effects on the performance of spatial GPCA also needs to be evaluated 2. Intensity Based Segmentation A MR axial image can be considered to be made up of three components. The heart, the chest and the background (which is predominantly noise). A two step approach is used to segment these three groups. We first use an intensity based segmentation on the images and separate the image into two groups, the background and the other group containing both the heart and the chest. Existing algorithms such Polysegment, K-Means and Expectation Maximization (EM) can be used for this step. We assume that the there exists a constant number of intensity groups n throughout the data set and determine this number through experiments.we choose n such that the lowest intensity group among the n groups must contain most of the background and a very small portion of the heart and the chest. If n is very small then along with the background a large part of the heart and the chest will be removed. If n is too large the background will not get removed completely. 3. Dynamics Based Segmentation We propose to model the heart as a dynamic texture. [2] [1] We know that a dynamic texture will lie in a subspace of dimensions R N where N is the order of the system. For a particular slice of the heart, if one observes the temporal evolution, the chest remains static while the heart beats. Thus there are two dynamic textures one of the heart, and one of the chest. This would imply that every dynamic texture would lie in separate subspace of dimensions R N. Hence if one were to consider the temporal evolution of a point in all the frames of the sequence it would lie in one of the two subspaces. Let us consider the number of frames in the sequence to be F, I(t) be vectorised frame at time t 1... F and P be the number of points in each frame. If we were to consider M = [I(1) I(F )] R P F then each row of this matrix represents the temporal evolution of a point. Thus the problem of separating the heart from the chest reduces to a subspace segmentation problem. To achieve the segmentation in this project we use an algebraic algorithm i.e. GPCA and a iterative algorithm namely K-Subspace. For both the algorithms we preprocess the data first by projecting M R P F onto a lower dimensional subspace R d P. This is done by calculating the SVD of M = USV T and then M p R P d is the first d columns of U. 1

2 3.1 Generalized Principal Component Analysis (GPCA) The following section outlines the application of GPCA to M p to segment the data. 1. Since the temporal evolution of each point belongs to either one of the subspaces S h or S c with normal vectors b h and b c, we get v = (b h T x p )(b c T x p ) = 0 (1) where x p R d is the temporal evolution of a point p. In this case the pth row of M p.we see that v will be a homogenous polynomial of degree 2 in d variables 2. b h and b c can be solved from the derivatives of v at points x c and x h Hence we get b h = Dv(x h) Dv(x h ) ; b c = Dv(x c) Dv(x c ) (2) 3. Once b h and b c are known the points can be classified into those belonging to the heart and those belonging to the chest. We propose a slight modification to the GPCA algorithm to incorporate spatial information. In an image we assume that adjacent pixel usually tend to belong to the same object.thus this would help us overcome the problem of isolated pixels in the heart belonging to the chest and vice versa. The spatial version of the algorithm is listed below 1. The derivatives of every point in a frame is calculated. The derivatives are however not normalized. 2. The derivatives of the points in a window Ω of size τ τ is taken. Since the assumption that the points in this window must belong to the same subspace, we want to smooth the derivatives so that the points belong to the same group.the principal component of these derivatives are taken and the derivatives in the window Ω are replaced by it.this process is repeated by sliding the window 3. Once the derivatives are smoothed. The points are segmented subject to k = arg min j=1,2 {bt j D(x)} (3) One can also choose to fit a window function to the window Ω. A popular choice could be a gaussian window. The segmentation is now also a function of s and the window function. 3.2 K-Subspace The iterative algorithm used for the segmentation is the K- subspace. While implementing the algorithm we make the assumption that the dimensions of the subspaces are the same and they are hyperplanes in the ambient space. K- Subspace is usually not restricted by such assumptions but for comparison with GPCA we have imposed the above structure of the subspaces. The K subspaces algorithm is outlined below. 1. The basis for the subpaces S h and S c namely U h, U c R d (d 1) are randomly initialized such that both the matrices are orthogonal 2. The points are then classified based on k = arg min x i U (m) j=1,2 l (U (m) l ) T x i 2 (4) 3. Once the points are classified, the basis U (m+1) h are calculated U (m+1) c U (m+1) h U c (m+1) and = P CA(X (m) h ) (5) = P CA(X (m) c ) (6) To achieve better convergence one can initialize K- subspace with the normals obtained from GPCA. We then obtain the segmentation from of M p using the algorithm. 4 Experimental Results The data set consisted of axial MR images of 9 unique slices of the heart.at each slice 20 frames represented the temporal evolution of the heart.all the experiments in this project perform the segmentation slice by slice. Each image was down sampled from to to accommodate the memory constraints imposed by K-Subspace 4.1 Intensity based segmentation The results of using polysegment and K-means to remove the background from the heart and the chest is show in figure 1. Both the algorithms perform well and they both are obtained by setting n = 15.The pixels thats belong to the background obtained by intensity segmentation are not used in the second stage of the algorithm namely the dynamic segmentation. 4.2 Dynamics based segmentation We first project the data from a subspace of dimension F = 20 to d = 3 and then use the segmentation algorithms to separate the heart and the chest. 2

3 Figure 1. Intensity based segmentation. The figure in the left is the result of the polysegment algorithm while the figure on the right is using kmeans Figure 3. Segmentation using Spatial GPCA. The figure in the left shows one of the frames of the sequence.the middle figure shows the points segmented as the chest and the right figure shows the points segmented as the heart GPCA and Spatial GPCA Figure 2 shows the segmentation for one slice of the heart. One can notice that there is a lot of misclassifications, and the segmentation is very noisy. It is for this purpose we wish to impose additional spatial constraints. Figure 2. Segmentation using GPCA. The figure in the left shows one of the frames of the sequence.the middle figure shows the points segmented as the chest and the right figure shows the points segmented as the heart. Figure 3 shows the results of using the spatial constraints on the segmentation. When compared to the GPCA results one gets really smooth segmentations and the effect of isolated noise pixels is reduced. The size of the window used for these results is s = 3. Different window types where used namely a rectangular window and a gaussian window to scale the normals in Ω. But however the use of these windows does not vary the segmentation much hence we do not continue to explore this path. The window size however makes a difference in the segmentation. Figure 4 shows the effect when we use s = 3 and s = 9, the effect of using a larger window size gives us a better segmentation when comparing the chest alone but then using a larger window groups a large part of the chest with the heart. This is not desirable. Figure 4. Window Size effects. The figures on the left show the chest and the heart segmentation for s = 3 and those on the right are for s = 9 rithm was tested using synthetic data generated to live in two subspaces, the norm error was reached. Figure 5 shows the segmentation of a particular slice of the heart using the K-Subspace algorithm. In this case the K-subspace is randomly initialized K-Subspace In all the implementations of the K-Subspace algorithm the criterion for convergence was that the maximum number of iterations was less than N 0 or the norm of the error was less than ɛ. When the algorithm was run with the heart images never did the norm error reduce below ɛ. Thus N 0 was the only criteria for the convergence. However when the algo- Figure 5. Segmentation using K-Subspace.The figure in the left shows one of the frames of the sequence.the middle figure shows the points segmented as the chest and the right figure shows the points segmented as the heart. The effect of the number of iterations is shown in Figure 6. Looking at these results one can come to the conclusion 3

4 that lesser the number of iterations the better is the segmentation. However this trend does not hold good in all the slices of the heart and hence nothing conclusive can be said about the effect of the number of iterations on the accuracy of the segmentation. Figure 6. Effect of Iterations on K-Subspace. The left figure is for N 0 = 100, the middle one for N 0 = 400 and the right one for N 0 = 600 We also explore the effect of initialization on the performance of K-Subspace algorithm. The figure 7 shows the effect of initialization using GPCA on K-Subspace. The performance marginally improves if K-subspace is manually initialized with the normals from GPCA. The norm error however is not met and the algorithms still runs up to the maximum number of iterations. Figure 7. Effect of Initializations- The right figure shows the K- subspace algorithm initialized using GPCA Finally we compare all the algorithms in Figure 8. It clearly shows us that Spatial GPCA outperforms the other two cases. Figure 9 shows us the performance of the algorithm for all the slices of the heart. Figure 8. Comparison of the Methods- The performance of the GPCA(left), K-Subspace (middle) and Spatial GPCA(right) for a particular slice Figure 9. Spatial GPCA results for the 9 slices of the heart 4

5 5 Summary and Conclusions The accuracy of the three algorithms were tested and it was found that the K-subspace and GPCA perform with similar accuracy. However K-subspace takes about iterations to get the same results, thus we see it terms of speed, GPCA outperforms K-subspace. The spatial GPCA algorithm performs the best among the three. The window size s is a critical factor in getting good results out of the spatial GPCA. Care must be taken to make s optimal such that smoother segmentation is achieved at the same time large parts of the other group do not become a part of the segmentation. Nothing conclusive can be said about the the advantage of initializing K-Subspace with GPCA. In some cases the results are better than randomly initialized basis, but however there is no definite trend. The number of iterations also do not exhibit a trend to say whether more iterations are better or not. We thus see using spatial GPCA is a definite way to achieve good segmentations at least in the axial images. References [1] G. Doretto, A. Chiuso, Y. Wu, and S. Soatto. Dynamic textures. International Journal of Computer Vision, 51(2):91 109, February [2] S. Soatto, G. Doretto, and Y. Wu. Dynamic textures. In IEEE International Conference on Computer Vision, pages , [3] R. Vidal, Y. Ma, and S. Sastry. Generalized principal component analysis (GPCA). In IEEE Conference on Computer Vision and Pattern Recognition, volume I, pages ,