Left Ventricle Endocardium Segmentation for Cardiac CT Volumes Using an Optimal Smooth Surface Yefeng Zheng a, Bogdan Georgescu a, Fernando Vega-Higuera b, and Dorin Comaniciu a a Integrated Data Systems Department, Siemens Corporate Research, Princeton, NJ, USA b Computed Tomography, Siemens Healthcare, Forchheim, Germany ABSTRACT We recently proposed a robust heart chamber segmentation approach based on marginal space learning. 1, 2 In this paper, we focus on improving the LV endocardium segmentation accuracy by searching for an optimal smooth mesh that tightly encloses the whole blood pool. The refinement procedure is formulated as an optimization problem: maximizing the surface smoothness under the tightness constraint. The formulation is a convex quadratic programming problem, therefore has a unique global optimum and can be solved efficiently. Our approach has been validated on the largest cardiac CT dataset (457 volumes from 186 patients) ever reported. Compared to our previous work, it reduces the mean point-to-mesh error from 1.13 mm to 0.84 mm (22% improvement). Additionally, the system has been extensively tested on a dataset with 2000+ volumes without any major failure. Keywords: Heart segmentation, 3D object segmentation, optimal smooth surface 1. INTRODUCTION Recently, we see more and more applications of machine learning to exploit a large expert annotated dataset for non-rigid shape detection and segmentation. Previously, we proposed an efficient learning based approach, marginal space learning (MSL), to detect and segment all four heart chambers from cardiac computed tomography (CT) volumes. 1, 2 In this paper, we focus on improving the accuracy of left ventricle (LV) endocardium segmentation. As shown in Fig. 1, the goal for LV endocardium segmentation is to use a smooth mesh to tightly enclose the whole blood pool, including the papillary muscles and trabeculations. (To measure LV volumes, there are various preferences among cardiologists on whether to include or exclude the papillary muscles from blood pool. Our system can do either way as shown in Section 3.) In our previous system, 1, 2 the mesh smoothness is enforced by projecting the mesh onto a subspace of the statistical shape model defined by a large set of training shapes based on principal component analysis (PCA). 3 The segmentation result is quite good and robust even for volumes with low contrast and severe streak artifacts. For volumes with high contrast, a small error in boundary delineation is observable. Therefore, the accuracy requirement for these volumes should be much higher than those with low contrast (where we cannot see the boundary clearly). Fig. 1b show such an example where the segmented LV endocardium by our previous system may traverse blood pool, instead of enclosing whole blood pool. Though the errors are quite small, they are clearly visible. In this paper, we propose an optimization based approach to search for an optimal smooth surface that tightly encloses the whole blood pool. Given the segmentation results for both LV endocardium and epicardium from our previous system, 1, 2 we extract the blood pool using histogram-based optimal adaptive thresholding (as shown in Fig. 1a), which minimizes the tissue classification error for blood pool and myocardium. We project the endocardium mesh onto the extracted blood pool, resulting in a zigzagged shape that tightly encloses the whole blood pool. Though smoothness can be enforced by PCA subspace projection, 3 the tightness constraint is not satisfied. Another problem is that the PCA shape space (which can only represent a shape as a linear combination of the training set) cannot cover all shape variations encountered in real applications. There are a few generic mesh smoothing techniques in the literature. The Lapalacian smoothing method is an iterative approach where, in each iteration, a vertex is updated with a weighted average of itself and its neighbors. 4 Lapalacian smoothing acts as a low pass filter, therefore it suffers from the shrinkage problem: when Further information: Send correspondence to Yefeng Zheng, yefeng.zheng@siemens.com.
(a) (b) (c) Figure 1. Illustration of left ventricle endocardium (green) segmentation by using a smooth mesh to tightly enclose the whole blood pool. (a) Original data with blood pool labeled (the region enclosed by the red contour). (b) Segmentation result using our previous learning based method. 2 (c) After refinement using the proposed approach. Here, we also show the segmented epicardium (magenta) by our system. the smoothing method is applied iteratively a large number of times, a shape eventually collapses to a point. Recently, Li et al. 5 proposed a graph cut based method to maximize the boundary detection score under a smoothness constraint. However, their objective function lacks an explicit smoothness measurement. Instead, we propose a novel optimization based approach by maximizing an explicit mesh smoothness measurement under the constraint that the mesh should tightly enclose the whole blood pool, as shown in Fig. 1c. We prove that our objective function is a strictly convex quadratic function with a unique global optimal solution. Therefore, a bunch of efficient methods 6, 7 are readily available in the literature to solve the optimization problem. In summary, we make the following contributions. 1. We propose an optimization based approach that explicitly maximizes a smoothness measurement while satisfies the tightness constraint. 2. We show that the optimization problem is a convex quadratic programming problem, therefore has a unique global optimum and can be solved efficiently. 3. We significantly improve LV endocardium segmentation accuracy, reducing the error from 1.13 mm to 0.84 mm (22% improvement). For most volumes, no manual correction is necessary after automatic segmentation. 4. Our system can provide LV volume measurements for both including and excluding papillary muscles. 5. The whole automatic segmentation procedure takes about 1.2 seconds and has been quantitatively validated on a dataset (457 volumes from 186 patients), which is significantly larger than those reported in the literature. The remaining of the paper is organized as following. In Section 2, we briefly review our previous heart chamber segmentation system using marginal space learning. We discuss blood pool extraction using adaptive optimal thresholding in Section 3. Section 4 presents the proposed method to search for an optimal smooth mesh that tightly encloses the extracted blood pool for LV endocardium segmentation. We compare the proposed method with other mesh smoothing techniques in Section 5. A brief conclusion is presented in Section 6. 2. AUTOMATIC HEART CHAMBER SEGMENTATION USING MARGINAL SPACE LEARNING In this section, we give a brief overview of our previous heart chamber segmentation approach using marginal space learning (MSL). Interested readers are referred to our other publications 1, 2 for more details. Our technique
3D Volume Object localization with marginal space learning Position Estimation Position- Orientation Estimation Position- Orientation-Scale Estimation Non-rigid Deformation Estimation Result Figure 2. Diagram of our original heart chamber segmentation system. 2 is based on recent advances in learning discriminative object models and we exploit a large database of annotated CT volumes. We formulate the segmentation as a two-step learning problem: anatomical structure localization and boundary delineation. Object localization (or detection) is required for an automatic segmentation system and discriminative learning approaches have proved to be efficient and robust for solving 2D problems. 8, 9 Recently, we proposed marginal space learning (MSL) to apply machine learning to 3D object detection. The idea of MSL is not to learn a classifier directly in the full similarity transformation parameter space but to incrementally learn classifiers on marginal spaces. As the dimensionality increases, the valid (positive) space region becomes more restricted by previous marginal space classifiers. In our case, we split the estimation into three problems: position estimation, position-orientation estimation, and position-orientation-scale estimation. Fig. 2 shows the system diagram. Besides reducing the searching space significantly, there is another advantage using MSL: we can use different features or learning methods in each step. For example, in the position estimation step, since we treat rotation as an intra-class variation, we can use the efficient 3D Haar features. 10 In the position-orientation and position-orientation-scale estimation steps, we introduce the steerable features. Steerable features constitute a very flexible framework where the idea is to sample a few points from the volume under a special pattern. We extract a few local features for each sampling point, such as voxel intensity and gradient. To evaluate the steerable features under a specified orientation, we only need to steer the sampling pattern and no volume rotation is involved. After the first stage, we get the position, orientation, and scale of the object. We align the mean shape with the estimated transformation to get a rough estimate of the object shape. We then deform the shape to fit the object boundary. Active shape models (ASM) 3 are widely used to deform an initial estimate of a non-rigid shape under the guidance of the image evidence and the shape priori. Non-learning based generic boundary detector in the original ASM 3 does not work in our application due to the complex background and weak edges. We use a learning based method to exploit more image evidences to achieve a robust boundary detection. Fig. 3 shows several examples for heart chamber segmentation. Our approach is robust even under severe streak artifacts as shown by the second example in the figure. The mean point-to-mesh error, E p2m, ranges form 1.13 mm to 1.57 mm for different chambers. MSL provides a generic framework for automatic object detection. It has been successfully applied to many 3D anatomical structure detection problems in medical imaging (e.g., ileocecal valves, 11 polyps, 12 and livers in abdominal CT, 13 brain tissues 14 and heart chambers 15, 16 in ultrasound images). 3. LV BLOOD POOL EXTRACTION Voxel intensity thresholding is commonly used in the previous work 17, 18 to extract blood pool since normally blood pool has a higher intensity than papillary muscles and myocardium. However, a preset threshold does not work well due to variations in the use of contrast agents. An optimal threshold should be tuned for each volume. In this paper, we propose an automatic scheme to determine the optimal threshold based on the initial segmentation of LV endocardium and epicardium using the approach discussed in Section 2. We calculate two histograms of the voxel intensity, one for all voxels enclosed by the endocardium surface (most of them from blood pool with a few from papillary muscles) and the other for all voxels enclosed between endocardium and epicardium surfaces (which is myocardium). As shown by the first example in Fig. 4, for volumes with high contrast, these histograms are well separated with slight overlapping in the middle. The optimal threshold
Figure 3. Examples of heart chamber segmentation in 3D CT volumes with green for LV endocardium surface, magenta for LV epicardium surface, cyan for LA, brown for RV, and blue for RA. Each row represents three orthogonal views of a volume. (which is 212 HU for this volume) can be quickly searched for to minimize the overall voxel classification error. We do observe that histograms of the blood pool and myocardium may have large overlap for volumes with low contrast and the extracted blood pool may contain some errors (see the second example in Fig. 4). Therefore, if the tissue classification error is larger than a threshold, we stick with the initial learning based segmentation without applying any further refinement. After getting the optimal threshold, we can easily classify those voxels enclosed by the endocardium surface as blood pool or papillary muscles. One issue is that the automatic segmentation algorithm is not perfect around the boundary: a portion of blood pool may be segmented as myocardium and vice verse. We can correct such minor segmentation errors by expanding the endocardium mesh a little bit (5 mm) before voxel intensity thresholding. After thresholding, connected component analysis is performed for voxels with intensity larger than
Figure 4. Voxel intensity histograms of blood pool and myocardium. The left column shows the volumes and the right column shows the corresponding histograms together with the optimal threshold to distinguish blood pool and myocardium. the threshold. Only the largest component is preserved as blood pool and small isolated pieces are discarded. Fig. 5 shows the region of the extracted blood pool (red) overlapped with the original segmentation results of LV endocardium (green) and epicardium (magenta). We can see that small segmentation errors are corrected. After that, we project the LV endocardium mesh onto the extracted blood pool. The resulted mesh tightly encloses the whole blood pool, but not smooth. In the next section, we present our optimization based approach to search for an optimal mesh satisfying both the smoothness and tightness constraints. As a by-product of blood pool extraction, we can easily calculate LV volumes for both including and excluding papillary muscles. For the former, we just need to calculate the volume enclosed by LV endocardium surface. For the latter, we count the number of voxels of the extracted blood pool and convert it to a volume measurement easily since we know the CT scanning resolutions. 4. OPTIMAL SMOOTH SURFACE FOR LV ENDOCARDIUM SEGMENTATION In this section, we present our optimization based surface smoothing method. 4.1 Optimization Based Surface Smoothing The smoothness of a surface, S, is often measured by the sum of squares of derivative as SM(S) = S 2 ds. (1)
Figure 5. An example of left ventricle (LV) blood pool extraction. The extracted blood pool (red) is overlapped with the initial segmentation of LV endocardium (green) and epicardium (magenta). Three orthogonal views from the same volume are shown in each row. The top row visualize the blood pool as an unfilled region and the bottom row visualizes the same blood pool as a filled region. A smaller SM represents a smoother surface. In reality, a discrete surface (e.g., a polyhedral surface) is often used to represent the boundary of a 3D object. We represent a polyhedral surface as a graph S = {V, F }, where V is an array of vertices and F is an array of faces. Triangulated surfaces are the most common, where all faces are triangles. For each vertex V i on the surface, we can define a neighborhood N i. Normally, first order neighborhood is used that vertex j is a neighbor of vertex i if they are on the same face. The smoothness around 4, 19 a vertex is often defined as V i = w ij (V j V i ), (2) j N i where the weights w ij are positive numbers that add up to one for each vertex j N i w ij = 1. (3) Therefore, Equation( 2) can be written as V i = V i j N i w ij V j, (4) In this form, the meaning of this smoothness measurement is clear. A vertex on a extremely smooth surface (a plane) can be represented as a weighted average of its neighbors, therefore with zero V i. The weights can be chosen in many different ways taking into consideration of the neighborhoods. The simplest way is to set w ij to be uniform w ij = 1 #N i, (5)
where #N i is the number of neighbors for vertex i. Given Equation (2), the smoothness of the whole surface is SM(S) = i V i 2. (6) We want to adjust the mesh to generate a smooth surface by minimizing Equation (6). Since there is too much freedom to adjust a mesh, similar to the well accepted practice in active contours 20 and ASM, 3 we only allow the adjustment along the normal direction V i = V i + δ i N i, (7) where δ i is a scalar and N i is the surface normal at vertex i. We can further limit the adjustment of each vertex by enforcing the following constraints, l i δ i u i, (8) where l i and u i are the lower and upper bound of the adjustment for vertex i. For example, in the following application on left ventricle endocardium segmentation, we enforce δ i 0 to guarantee that the mesh encloses the whole blood pool. We also need to get a trade-off between smoothness and the amount of adjustment. Our final optimization problem is min F = w ij (V j V i ) 2 + α δ 2 i, (9) i j N i i subject to the bound constrain of Equation ( 8). Here, α 0 is a scalar for the above trade-off. It turns out that our optimization problem is a classical quadratic programming problem. In the following we will prove that the objective function defined in Equation (9) is a strictly convex function for α > 0. Therefore, it has a unique global optimal solution. Theorem 4.1. For α 0, the objective function defined in (9) is a convex function. If α > 0, it is a strictly convex function. Proof. The second term of the objective function, i δ i 2 is a strictly convex function. We only need to prove the first term, F 1, is convex. Substituting (7) into (9) and after re-organizing, we can get F 1 = δ T Qδ + c T δ + c 0, (10) where δ is a vector formed by δ i, Q is a symmetric square matrix, c is a vector, and c 0 is a constant. To prove F 1 is a convex function, we only need to prove all eigenvalues of matrix Q is nonnegative. Q can be represented as Q = W T ΛW, (11) where W is a matrix with each column corresponding to an eigenvector of Q and Λ is a diagonal matrix composed with the eigenvalues of Q. Therefore, F 1 = δ T W T ΛW δ + c T δ + c 0, (12) Suppose Q has a negative eigenvalue, without loss of generality, suppose the first eigenvalue, λ 1, of Q is less than zero. Let e 1 = [1, 0,..., 0] be a vector with the first element being 1 and all the other elements being 0. Let δ = bw 1 e 1, here b is a scalar. Substituting δ into F 1, we get F 1 = λ 1 b 2 + c T W 1 e 1 b + c 0. (13) F 1 is a quadratic function of b and its value is dominated by the first term λ 1 b 2. Since λ 1 < 0, we get F 1 < 0 for a sufficient large b. From Equation (9), we know F 1 is a sum of squares and always no less than zero. Therefore, matrix Q has no negative eigenvalue. It is well-known that all eigenvalues of a real symmetric matrix is real. Therefore, all eigenvalues of Q are real and nonnegative. Therefore, F 1 is a convex function. This completes the proof. For a strictly convex quadratic programming problem, there is a unique global optimal solution and many algorithms have been proposed in the literature. 6 With a bound constraint as in our case, a more efficient and specialized method is available, 7 which takes less than 0.2 seconds in our case for a mesh with 545 points.
Table 1. Mean and variance (in parentheses) of the point-to-mesh error E p2m (in millimeters) for left ventricle endocardium segmentation on 457 volumes based on a four-fold cross validation. Initialization 2 Subspace Projection 3 Generic Smoothing 4 Our Approach E p2m 1.13 (0.55) 1.31 (0.55) 1.39 (0.60) 0.84 (0.47) 4.2 Comparison with Previous Work The active balloon model is a variation of active contours, where an outward force is applied to each point to inflate the evolving contour. 21 If all forces and parameters are properly set, it may result in a smooth mesh enclosing the blood pool. Our approach has several advantages over the active balloon model. First, our approach guarantees to achieve the global optimum. However, the active balloon model only converges to a local optimum. Second, there are many free parameters in the active balloon, especially the balloon force, which is hard to tune. However, there is only one regularization parameter α in our approach. Last, the final surface achieved by the active balloon model generally cannot tightly enclose the blood pool. Recently, Li et al. 5 proposed a graph-cut based approach to search for an optimal surface. Similar to ours, they also search for an optimal adjustment along the surface normal for each point. However, the adjustment in their approach is restricted to discretized positions. Instead, we use continuous optimization and can achieve finer adjustments. Another major drawback of their approach is that their objective function lacks an explicit smoothness measurement. Smoothness is enforced as hard constraints: the difference in the adjustments of neighboring points should be less than a threshold (which is an application dependent parameter). Therefore, their method is limited to handling terrain-like (height-field) and cylindrical surfaces. For a rough object (e.g., LV endocardium in our case), their method does not guarantee to achieve a smooth mesh. However, with an explicit smoothness measurement, we guarantee to achieve the smoothest mesh under the constraints. 5. EXPERIMENTS The proposed method is verified on a dataset with 457 annotated cardiac CT volumes from 186 patients with various cardiovascular disease. The number of patients is significantly larger than those reported in the literature, for example, 10 in Schramm et al., 22 13 in Ecabert et al., 23 and 18 in Jolly. 17 The imaging protocols are heterogeneous with different capture ranges and resolutions. A volume contains 80 to 350 slices and the size of each slice is 512 512 pixels. The resolution inside a slice is isotropic and varies from 0.28 mm to 0.74 mm, while the distance between neighboring slices varies from 0.4 mm to 2.0 mm. A four-fold cross validation is performed to evaluate our algorithm. Special care is taken to prevent volumes from the same patient appear in both the training and test sets. The accuracy of boundary delineation is measured with the symmetric point-to-mesh distance, E p2m. For each point on a mesh, we search for the closest point on the other mesh to calculate the minimum distance. We calculate the point-to-mesh distance from the detected mesh to the ground-truth and vice verse to make the measurement symmetric. The initial segmentation of LV (both endocardium and epicardium) is provided by our learning based system. 2 As shown in Table 1, the initial error is 1.13 mm. Using the proposed approach, we reduce the mean error by 22% to 0.84 mm. We compare our approach with two alternative methods, the PCA based subspace projection 3 and the generic mesh smoothing approach. 4 Starting from the same extracted blood pool, these generic approaches generate a smooth mesh traversing the blood pool and actually increase the error, as shown in Table 1. The segmentation results using different approaches on a volume are shown in Fig. 6. Our approach is fast with an average speed of less than 1.2 seconds per volume for the whole automatic segmentation procedure (including initialization 2 ) on a computer with duo core 3.2 GHz CPUs and 3 GB memory. Our system can calculate the LV volumes for both including and excluding papillary muscles. The volume enclosed by LV endocardium surface is the measurement including the papillary muscles. Using the proposed adaptive intensity thresholding presented in Section 3, we can extract the LV blood pool. After that, we count the number of voxels of the extracted blood pool and convert it to a volume measurement using the volume resolution information. Fig. 7a shows the ground truth of the volume curves (both including and excluding papillary muscles) for a dynamic 3D sequences with 10 frames. Fig. 7b and 7c compare the volume measures
(a) (b) (c) Figure 6. LV segmentation by (a) PCA subspace projection, 3 (b) generic mesh smoothing, 4 and (c) the proposed approach. Three orthogonal views are shown at each row, with green for endocardium and magenta for epicardium. Black boxes indicate regions with noticeable errors. automatically generated by our system against the ground truth for including and excluding papillary muscles, respectively. From the volume curve, we can calculate the ejection fraction (EF) as follows, EF = Volume ED Volume ES Volume ED, (14) where Volume ED and Volume ES are the volume measures of the end-diastolic (ED) and end-systolic (ES) phases, respectively. For the sequence shown in Fig. 7, the errors in EF estimation are quite small (no more than 2%).
120 130 120 110 Groundtruth Including Muscles (EF=50%) Groundtruth Excluding Muscles (EF=51%) 120 Detection Including Muscles (EF=52%) Groundtruth Including Muscles (EF=50%) 110 Detection Excluding Muscles (EF=50%) Groundtruth Excluding Muscles (EF=51%) 100 110 100 Volume (ml) 90 80 Volume (ml) 100 90 80 Volume (ml) 90 80 70 70 70 60 60 60 50 1 2 3 4 5 6 7 8 9 10 Frame 50 1 2 3 4 5 6 7 8 9 10 Frame 50 1 2 3 4 5 6 7 8 9 10 Frame (a) (b) (c) Figure 7. The left ventricle volume-time curves for one dynamic 3D sequence with 10 frames. (a) Ground truth including/excluding papillary muscles from the volume measurements. (b) Detection vs. ground truth by including the papillary muscles. (c) Detection vs. ground truth by excluding the papillary muscles. 6. CONCLUSION In this paper, we proposed an optimization based approach for LV endocardium segmentation by using an optimal smooth mesh tightly enclosing the whole blood pool. The optimization problem is a strictly convex quadratic programming problem with a unique global optimal solution. Our approach can effectively reduce the mean point-to-mesh error from 1.13 mm to 0.84 mm on a dataset with 457 volumes. Additionally, the system has been extensively tested on a dataset with 2000+ volumes without any major failure. REFERENCES 1. Y. Zheng, A. Barbu, B. Georgescu, M. Scheuering, and D. Comaniciu, Fast automatic heart chamber segmentation from 3D CT data using marginal space learning and steerable features, in Proc. Int l Conf. Computer Vision, 2007. 2. Y. Zheng, A. Barbu, B. Georgescu, M. Scheuering, and D. Comaniciu, Four-chamber heart modeling and automatic segmentation for 3D cardiac CT volumes using marginal space learning and steerable features, IEEE Trans. Medical Imaging 27(11), pp. 1668 1681, 2008. 3. T. F. Cootes, C. J. Taylor, D. H. Cooper, and J. Graham, Active shape models Their training and application, Computer Vision and Image Understanding 61(1), pp. 38 59, 1995. 4. G. Taubin, Curve and surface smoothing without shrinkage, in Proc. Int l Conf. Computer Vision, pp. 852 857, 1995. 5. K. Li, X. Wu, D. Z. Chen, and M. Sonka, Optimal surface segmentation in volumetric images A graphtheoretic approach, IEEE Trans. Pattern Anal. Machine Intell. 28(1), pp. 119 134, 2006. 6. G. Goldfarb and A. Idnani, A numerically stable dual method for solving strictly convex quadratic programs, Mathematical Programming 27(1), pp. 1 33, 1983. 7. J. J. Moré and G. Toraldo, On the solutions of large quadratic programming problems with bound constraints, SIAM J. Optimization 1(1), pp. 93 113, 1991. 8. P. Viola and M. Jones, Rapid object detection using a boosted cascade of simple features, in Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 511 518, 2001. 9. B. Georgescu, X. S. Zhou, D. Comaniciu, and A. Gupta, Database-guided segmentation of anatomical structures with complex appearance, in Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 429 436, 2005. 10. Z. Tu, X. S. Zhou, A. Barbu, L. Bogoni, and D. Comaniciu, Probabilistic 3D polyp detection in CT images: The role of sample alignment, in Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 1544 1551, 2006. 11. L. Lu, A. Barbu, M. Wolf, J. Liang, L. Bogoni, M. Salganicoff, and D. Comaniciu, Simultaneous detection and registration for ileo-cecal valve detection in 3D CT colonography, in Proc. European Conf. Computer Vision, 2008.
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