IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 23, NO. 10, OCTOBER Jian Chen and Amir A. Amini*, Senior Member, IEEE

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1 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 23, NO. 10, OCTOBER Quantifying 3-D Vascular Structures in MRA Images Using Hybrid PDE and Geometric Deformable Models Jian Chen and Amir A. Amini*, Senior Member, IEEE Abstract The aim of this paper is to present a hybrid approach to accurate quantification of vascular structures from magnetic resonance angiography (MRA) images using level set methods and deformable geometric models constructed with 3-D Delaunay triangulation. Multiple scale filtering based on the analysis of local intensity structure using the Hessian matrix is used to effectively enhance vessel structures with various diameters. The level set method is then applied to automatically segment vessels enhanced by the filtering with a speed function derived from enhanced MRA images. Since the goal of this paper is to obtain highly accurate vessel borders, suitable for use in fluid flow simulations, in a subsequent step, the vessel surface determined by the level set method is triangulated using 3-D Delaunay triangulation and the resulting surface is used as a parametric deformable model. Energy minimization is then performed within a variational setting with a first-order internal energy; the external energy is derived from 3-D image gradients. Using the proposed method, vessels are accurately segmented from MRA data. Index Terms 3-D Delaunay triangulation, filtering, geometric deformable models, level set methods, multiple scale analysis, vessel segmentation. I. INTRODUCTION ACCURATE quantification of three-dimensional (3-D) vascular structures has become increasingly important for diagnosis and quantification of vascular disease, for vascular surgery planning, and for patient-specific flow simulations. There are a considerable number of approaches in the literatures to segmentation and measurement of 3-D vascular structures [16] [18], [21], [24] [28]. What seems to be lacking, however, are tools to ensure accuracy of segmentation. In addition, most proposed methods are either two-dimensional (2-D) or require user intervention making segmentation somewhat subjective. Automatic and accurate segmentation of vascular structures can, therefore, be highly significant in clinical practice. Motivated by the aforementioned reasons, we present a hybrid approach to automatic and accurate quantification of vascular structures from MRA images. Manuscript received October 30, 2003; revised June 22, This work was supported in part by a grant from BJH foundation and in part by the National Science Foundation (NSF) under Grant IRI The Associate Editor responsible for coordinating the review of this paper and recommending its publication was W. J. Niessen. Asterisk indicates corresponding author. J. Chen is with the Cardiovascular Image Analysis Laboratory, Washington University School of Medicine, Box 8086, 660 S. Euclid Ave., St. Louis, MO USA. *A. Amini is with the Cardiovascular Image Analysis Laboratory, Washington University School of Medicine, Box 8086, 660 S. Euclid Ave., St. Louis, MO USA ( amini@cauchy.wustl.edu). Digital Object Identifier /TMI In order to accurately quantify 3-D vascular structures, first, as a preprocessing step, an appropriate method for enhancement of vascular structures and elimination of other tissues is required; second, a robust method for automatic extraction of the enhanced vessels without user interaction is needed; finally, an appropriate method for correction of the initial segmentation is needed so as to reduce any bias from the preprocessing step and for making the final segmentation fit the true vessel surface. In this paper (a preliminary version appeared in [30]), we use multiple scale filtering for enhancement of vascular structures and for eliminating unwanted image structures. Subsequently, we employ the level set technique for obtaining an initial segmentation. Finally, geometric deformable models are used to correct and refine the segmented vessel surface. Previous work on the segmentation of vascular structures from 3-D MRA images relevant to our work can be classified into the following three categories: Multiple Scale Filtering: Vessel intensity in MRA images varies within a relatively wide range due to blood flow rate and vessel dimensions. Multiple scale filtering was proposed for segmentation of vessels with various diameters on 3-D images [16] [18], [21]. Since the vessels with different diameters can be considered to be line structures with different widths, they can be enhanced by employing multiple scale filtering based on the analysis of eigenvalues of the Hessian matrix at each voxel in the image and segmented by thresholding the enhanced images. Such methods may fail however when vessel diameters change abruptly. In addition, they may also cause bias or artificial vessel structures. Finally, most of the proposed methods are far from being fully automated since they require user interaction for selecting appropriate thresholds. Geometric Deformable Models Minimizing Static Energy: Since Kass et al. first introduced the active contour model or snake [3], a multitude of powerful deformable models for medical image segmentation have been proposed [9]. Klein and Amini [14], Huang and Amini [19], and Frangi et al. [21] proposed to reconstruct 2-D vessel boundaries or 3-D vessel walls using deformable surface models represented by B-spline surfaces. However, it is not possible to employ parameterized geometric models to effectively deal with whole vessel trees, as the models would be required to change topology during evolution. The only exception is the work of McInerney and Terzopoulos who proposed T-snakes for dealing /04$ IEEE

2 1252 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 23, NO. 10, OCTOBER 2004 with topology changes in images [7], [20]. Other papers employing geometric deformable models include papers by Yim et al. who recently proposed a tubular deformable model [26] and deformable isosurface model [28], both of which are based on triangulated meshes for vessel construction. It may be problematic to apply the methods [20], [26], [28] to segmentation of vessels from low-contrast MRA images. Dynamic Implicit Surface Models Level Sets and Geodesic Active Contours Model: Level set methods were first formulated by Osher and Sethian [4], [23], and first introduced to computer vision and image processing community by Malladi et al. [6]. Level set methods have been widely used for image segmentation, providing an elegant solution to change in the initial model s topology as well as model initialization [10], [12], [13], [15], [22], [25], [27], [29]. In the area of 3-D MRA data segmentation, Lorigo et al. used a geodesic active contour model based on the level set method for segmentation of brain vasculatures, and the abdominal aorta from high-contrast MRA and CT images [25]. Bemmel et al.[29] also applied the level set method for segmentation of MRA data using a speed function derived from the multiple scale filtering technique proposed by Frangi et al. [21]. As addressed above, the filtering may cause a bias field in the segmented structures. However, no correction methods for improving the segmentation results were proposed in either paper. Antiga et al. [27] employed multiple steps of level sets evolution for directly segmenting MRA data of vessel of interest instead of whole vascular tree by interactively activating three terms in their evolution equation and refined the resulting models for computational fluid dynamics (CFD) simulations. However, their method only focuses on vessels of interest instead of the entire vascular tree. Multiple scale filtering based on the analysis of the Hessian matrix and its eigenvalues has received a large amount of attention in segmentation of CTA/MRA data[16] [18],[21]. In this paper, we enhance vessel structures by utilizing the eigenvalues of the Hessian matrix and define a classification function associated with the eigenvalues; an effective tool for discriminating vessel structures from nonvessel structures. An initial vessel segmentation is then obtained by applying the level set technique to enhanced vessel structures. The segmented vessel surface is derived as the zero level set of the higher dimensional level set function using the marching cubes algorithm[2]. Although extremely useful is eliminating nonvessel structures, unfortunately, in practice multiple scale filtering often degrades boundaries of vessel structures, and introduces bias in segmentation. Therefore, we need to reduce the bias field andmake the segmentedvessel surfacefit the true vessel surface. A geometric deformable model is, thus, applied to correct and reconstruct the segmented vessel surface. We triangulate the segmented vessel surface using 3-D Delaunay triangulation and use the triangulated surface as the initial deformable model for fine tuning the level-set segmentation. The paper is organized as follows: the method proposed for automatic quantification of vascular structures is described in Section II. In Section III, we address the sensitivity of the proposed method using mathematical simulation of high degree stenoses and in Section IV, experimental results using both in vitro phantom data and three human data are presented. For Fig. 1. Abdominal MRA: the MIP image of the original coronal data volume (left) and two resliced axial image planes at cross section of the abdominal aorta and renal arteries in the axial direction (right) at the 109th and 136th slice location. all cases, we compare automatic segmentations with manually traced boundaries. Finally, in Section V, we summarize the work and provide conclusions. II. METHOD To motivate the need for applying vessel-enhancing filters, Fig. 1 illustrates original image slices from an MRA study and their gradient magnitude images. As shown in the figure, not only vessels but also other tissues such as the left and the right kidneys are enhanced after administration of contrast agent. Therefore, it is difficult to directly segment vessel structures using level set techniques or parametric deformable models without suppressing signal from other tissues. Therefore, in a preprocessing step, application of appropriate filters is necessary in order to enhance vessels and suppress other tissues. In this section, we first introduce the multiple scale filtering method for vessel enhancement, and then show how the level set technique may be used in conjunction to segment the enhanced vessels. Finally, we present a geometric deformable model technique to refine the initial segmentation. A. Vessel Enhancement On MRA images, the vessel intensity may vary within a relatively wide range due to blood flow rate and vessel dimensions. Vessels with various diameters on MRA images can be regarded as 3-D line-structures. Therefore, the problem of vessel enhancement may be cast as a task of multiple scale line analysis. Let be the Hessian matrix of a 3-D function about a point, and with be the eigenvalues of with corresponding eigenvectors given by,, and, respectively. Using the matrix of the eigenvectors, we have (1)

3 CHEN AND AMINI: QUANTIFYING 3-D VASCULAR STRUCTURES IN MRA IMAGES 1253 Equation (1) states that the matrix describes the second-order structure of local intensity variations around each point of, and the local second-order features of can be directly obtained by the eigenvalues,, and which are equivalent to second partial derivatives of with respect to a rotated coordinate system with. The local cross-sectional profile of an ideal 3-D line model is then considered to have a 2-D Gaussian shape where the standard deviation is related to the width of the line,, when is a3 3 rotation matrix, and is the center of the profile. According to (1) and (2), the absolute values of and associated with a point on a line structure should be large. Therefore, we can utilize the following condition to determine if a point is on a line structure or not: (3) and On the other hand, the absolute values of and associated with a point on a plate structure such as left or right kidney (Fig. 1) in abdominal images should be small, but the absolute value of associated with such a point should be large. We can utilize the following condition to eliminate such structures: We use normalized second partial derivatives of the Gaussian function with standard deviation (corresponding to width of the filter) to determine the elements of the Hessian matrix where or denotes,, or. The Hessian matrix can be obtained by (6) where denotes the convolution operator. The eigenvalues of are also given by with. We define to determine if satisfies (3) where, and if and otherwise (8) The derivation of is given Appendix I. As given in (7), if were to satisfy (3), should take a large value. (2) (4) (5) (7) As stated in (4), for a point on a plate-structure, the corresponding should be great and should be small. On the other hand, both and corresponding to a point on a line-structure, should be greater according to (3). Therefore, we can use given in (7) and to measure lineness and generate the line-structure enhanced image for a given from Enhancement of vessels can be achieved by a nonlinear combination of multiple scale filtering. That is where is given by (9). (9) (10) B. Vessel Segmentation In the preprocessing step, vessel structures can be effectively enhanced, and other tissues can be suppressed simultaneously using the proposed multiple scale filtering method. For automatic segmentation of the enhanced vessel structures, a robust method is strongly needed. In this work, we apply the level set method because of its main advantages that include efficient numerical implementation with a fixed discrete grid in spatial domain as well as handling of topological changes. The aim of segmentation is to extract the surface walls of vessels enhanced by the multiple scale filtering method. The problem of segmenting vessel is formulated as evolving a higher dimensional function that contains the embedded motion of implicit vessel surface defined as the zero level set of. Initially, is defined as a function of signed distance from to but is evolved over time with a speed function that can be directly derived from the enhanced vessel image. For given,, and the enhanced vessel image [see (10)], we define the speed function as follows: (11) The speed function is a smooth function and permits stopping the propagation when the propagating vessel surface arrives at the enhanced vessel boundaries. Finally, the level set evolution equation for vessel segmentation is formulated as if is outside if is on if is inside (12) where,,, is an initial vessel surface, and is the distance from to. The numerical discretization of (12) is given in Appendix II. The evolution stops when the difference between the volumes of the vessel regions that are extracted (the level set function taking on a value less than or equal to 0) in iterations and does not change or becomes smaller than a threshold. Since the goal is to extract the zero level set of.wehave implemented (29) using the narrow band technique in order to reduce computational costs. That is, while evolving, we only

4 1254 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 23, NO. 10, OCTOBER 2004 update grid points that are within a fixed appropriate value of signed distance instead of in the whole volume. Moreover, in order to preserve the level set function as a signed distance function during evolution, we use the following to reinitialize it at iteration : if otherwise (13) The numerical discretization of (13) is also given in Appendix III. We used in our experiments. C. Correction of the Segmented Vessel Surface Using the marching cubes algorithm [2], the segmented vessel surface can be extracted as the zero level set of the function [see (12)]. In order to reduce the bias field associated with scale space filtering and so as to make the final vessel surface fit the true vessel surface more accurately, a geometric deformable model is applied. The reasons for applying a geometric deformable model in the correction step instead of an edge-driven level set are as follows. 1) The gradient magnitude of unwanted tissues such as left and right kidneys in our MRA data would be great due to the image contrast in original data. A gradient term in an edgedriven level set evolution equation would attract the propagating vessel surface to all boundaries including both vessels and other unwanted tissues. 2) Although a curvature based smoothing term may suppress unwanted tissues, it may fail where there is a stenosis (due to occurrence of singularity during evolution the wellknown dumbell effect [1]). Therefore, it is difficult to employ the level set method in the correction step. Unlike an implicit level set function, a geometric deformable model is an explicit surface whose topology does not change. Furthermore, since the level-set segmented vessel surface is close to the true vessel boundaries, small deformations would result in the requisite correction. Therefore, a geometric deformable model is more suitable than edge-driven level sets for the correction step. The main idea for the correction step is to use the segmented surface as an explicit initial deformable model, associate an appropriate energy with this model, and proceed to minimize it numerically. 1) Three-Dimensional Delaunay Triangulation: The segmented surface is extracted as a zero level set isosurface from the level set function by the marching cubes algorithm [2] a well-known algorithm for isosurface generation through triangulation. Since the connectivities of vertices on the triangulated isosurface depend on the topological structure of the volumetric data, there would be a problem with triangulation quality once the locations of vertices on the isosurface have changed. On the other hand, even though parametric functions have many advantages for representation of curves and surfaces, their main disadvantage is that it is difficult to represent branching structures such as vascular trees with them. In this paper, we, therefore, adopt 3-D Delaunay triangulation a powerful tool for building topological structures from a given finite set of points. The advantage of a polyhedral surface in this context is that no a priori topological information for constructing Fig. 2. An example of 3-D Delaunay triangulation using incremental-flips algorithm [11]. (a) The existing 3-D Delaunay triangulation. The black point is to be inserted into the point set. (b) The updated 3-D Delaunay triangulation after point insertion. the initial geometric deformable model is required. The 3-D Delaunay triangulation is defined as the unique triangulation that satisfies the 3-D Delaunay criterion a 3-D circumsphere of each tetrahedron contains no other points of the triangulation except for the four points defining the tetrahedron. There are a wide variety of algorithms available for 3-D Delaunay triangulation. Edelsbrunner et al. [5] proposed a well-known method called -Shapes to represent a given finite point set in as multiple level shapes, each of which is derived from 3-D Delaunay triangulation of the point set with a controlling parameter that is similar to the radius of 3-D circumsphere defined in the 3-D Delaunay criterion. A desired level shape can be obtained by selecting an appropriate controlling parameter. In the algorithm of -Shapes, the Delaunay triangulation is first computed for the boundary without a boundary mesh. The points are then inserted incrementally into the triangulation and the topology is updated by splitting and flipping the current triangulation according to the 3-D Delaunay criterion. Fig. 2 demonstrates the incremental insertion algorithm. We adopt -Shapes to triangulate the segmented vessel surface (with ) keeping all vertices lying on it. Subsequently the triangulated surface is used as the initial geometric deformable model. 2) Geometric Deformable Model: Let be the set of vertices lying on the triangulated surface, and represent the connectivity graph derived from 3-D Delaunay triangulation of, the deformable model can be defined as (14) where, is the number of vertices, and. Let and be parameters mapping the vertex of the deformable model in (14) as. The following energy functional is then minimized: (15) where is the weighting parameter. The internal energy function in (15) is a first-order membrane term. The external energy function in (15) is derived from image gradient information in the original unfiltered data. We use first partial derivatives of the

5 CHEN AND AMINI: QUANTIFYING 3-D VASCULAR STRUCTURES IN MRA IMAGES 1255 Gaussian at multiple scales as filters to obtain the gradient information of surface of vessels with different diameters. Hence, is given by and (16) Therefore, the numerical solution to (22) is obtained by iteratively solving (23) over iteration. That is, the set of vertices in (14) is updated by (23). In the numerical implementation of (23), we update the connectivity graph in (14) to maintain the quality of the triangulation. We use the updated to reconstruct the deformable model surface with the -Shapes Delaunay triangulation technique following each iteration. The iteration continues until the average of distance between the vertices at current and the previous iterations is less than a small threshold. (17) where integer, is the convolution operator,,, and denote normalized first partial derivative of the Gaussian (mean 0 and standard deviation ) with respect to,, and, respectively. Our deformable model minimizing the energy functional [see (15)] is given by an evolution equation obtained from the Euler equation of (18) the internal force smoothes the model, and the external potential force pulls the model toward surface boundaries. The internal force is the Laplacian of at node and is defined by [8] III. SENSITIVITY ANALYSIS USING MATHEMATICAL SIMULATION OF VESSELS WITH HIGH DEGREE STENOSES As pointed out in previous sections, due to the enhancement and segmentation steps a bias field may exist around the regions of stenosis, where the diameter of vessel changes abruptly. In order to reveal the influence of the degree of stenosis on the segmentation performance, we performed sensitivity analysis of the proposed method to quantification of vessel structures in images. In particular, we simulated 75%, 80%, and 90% degree stenosed vessels for detailed validation of techniques under a variety of additive noise conditions. An ideal 3-D vessel with degree of stenosis having cross-sectional Gaussian shape can be mathematically represented as (24) where (19) where is the standard deviation of the Gaussian at the cross section where the stenosis exists. We use, 0.80, and 0.90 and in (24) to mathematically synthesize vessels with 75%, 80%, and 90% degree stenoses, respectively. According to (24), the cross section of the synthesized stenosis is located at the axial image plane where, and the highest intensity in the profiles of the synthesized vessels equals 1.0. We mathematically estimate the true region for each synthesized ideal 3-D vessel [see (24)] using the following: (20) and if the vertex vertex otherwise is connected to with (21) The Euler equation of derived in (18) yields the flow that is given by (22) Equation (22) is, thus, discretized as (23) if otherwise (25) We use the estimated as a standard to carry out sensitivity analysis. In order to strictly evaluate the method, we compare the number of voxels in vessel regions obtained from the segmentation or correction method with the number of voxels in the true vessel region obtained from [see (25)]. Let denote the volume of [see (25)] for, 0.80, or 0.90, denote the volume of, and denote the volume of ( ; see Fig. 3). Hence, the rate of correct voxel classification of the method is given by (26)

6 1256 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 23, NO. 10, OCTOBER 2004 TABLE I RESULTS OBTAINED FOR THE SYNTHESIZED VESSELS WITH 75%, 80% and 90% DEGREE STENOSES AND IMPOSED GAUSSIAN NOISE WITH MEAN 0 AND STANDARD DEVIATION =0:0, 0.05, 0.1, 0.15, AND 0.2. NOTE THAT, THE TERM BEFORE CORRECTION INDICATES THE RESULTS OBTAINED FROM THE SEGMENTATION METHOD, AND THE TERM AFTER CORRECTION INDICATES RESULTS OBTAINED BY APPLYING THE CORRECTION METHOD TO THE SEGMENTED RESULTS Fig. 4. (a) The level-set segmented surface (gray) and original true surface (dark). A discrepancy, especially at the stenosis location is clear. (b) Final surface obtained after the application of a geometric deformable model segmentation technique (dark) and original true surface (gray). In this figure, the stenosis degree of the synthesized vessel is 80% area stenosis. Fig. 3. A Venn diagram indicating the true region, as well as artificial, missed, and correctly obtained regions using the segmentation or correction techniques. In addition, the rate of missed voxel classification is obtained as 100%. Let be the artificial region ratio (27) Clearly, a large value for implies a large number of voxels having been misclassified as belonging to true vessel regions. We define the criteria (28) to also evaluate the obtained result. If, i.e.,, the segmentation is perfect and, whereas if,. Therefore,, and becomes larger as ((26)) becomes larger and ((27)) becomes smaller. We applied the method to noise corrupted synthesized 3-D MRA s with stenosis degrees 75%, 80%, and 90% ((24)). The added noise had Gaussian statistics with mean 0 and standard deviation, 0.05, 0.1, 0.15, or 0.2. In these studies, we used and in (23). The obtained results were compared with the estimated ((25)). For each, we calculated ((26)), ((27)), and ((28) for the fifteen data sets. Table I illustrates the results. The findings are summarized in Sections III-A and III-B. A. Sensitivity Analysis of the Segmentation Method The for each case obtained from the segmentation method is relatively large, while is also large. Fig. 4(a) shows an overlapped display of the segmented regions and the estimated true regions for the stenosis data with and [see (24)]. As shown in the figure, the true regions are almost identical to the segmented regions; i.e., is large. On the other hand, artificial regions exist around the location of stenosis; this makes large. This indicates that even though the segmentation method has a good performance for determining vessel regions, it also causes artificial vessel regions due to abrupt diameter changes near the location of a high degree stenosis. B. Sensitivity Analysis of the Correction Method In all cases, obtained from the correction method is less than that obtained from the segmentation method. This indicates the good performance of the correction method in reducing the artificial vessel regions caused by the filtering. Fig. 4(b) shows an overlapped display of the corrected regions and the estimated true regions. As shown, the artificial regions due to the filtering [Fig. 4(a)] are greatly reduced by the correction method. However, due to the internal forces for smoothing the model [see

7 CHEN AND AMINI: QUANTIFYING 3-D VASCULAR STRUCTURES IN MRA IMAGES 1257 Fig. 5. Plots of [see (26)], [see (27)], and [see (28)] as a function of [see (23)] for correction of segmentations obtained for the ideal synthesized vessels [see (24)] in 75% (a), 80% (b), and 90% (c) degree area stenoses. (18)], the regions obtained from the correction method slightly shrink during smoothing. As a result, although obtained from the correction method has decreased in value, obtained from the correction method also decreases in value. This means that there is a trade-off between the internal forces for smoothing the model and the external forces [see (18)] for pulling the model toward the actual boundaries. When, obtained from the correction method is greater than that obtained from the segmentation method. Therefore, the correction method is able to improve the segmentation results when signal-to-noise ratio (SNR) is good. In summary, the correction method is appropriate for refinement of segmented results from MRA images since generally, these images have high SNR. Finally, in order to reveal the influence of [see (23)] on the performance of the correction step, we varied and applied the correction method to three segmented results for the synthesized vessels with 75%, 80%, and 90% stenosis degrees (with ). Fig. 5 plots,, and versus. As illustrated in Fig. 5, when,. In particular, for,,, and. IV. EXPERIMENTAL RESULTS AND VALIDATION We applied the proposed method to MRA data collected from an in vitro vessel phantom with 90% degree stenosis as well as to 3-D abdominal and Carotid arteriography images. In all cases, automatic segmentations were compared to manual segmentations. We used in (7), and in (12), and and in (23). A. Experimental Results 1) Phantom With 90% Stenosis Degree: Fig. 6 displays results of segmentation of Sinc-interpolated, isotropically reconstructed image an MRA image series of a phantom with 90% degree stenosis and The original MRA image series had an in-plane resolution of and axial resolution of 2.4 mm. The reconstructed 3-D isotropic image had a resolution of.as illustrated in Fig. 6(a), the diameter near the location of stenosis changes abruptly because of the high degree of stenosis. As also demonstrated in Fig. 6(b) and (c), comparison of the enhanced and segmented results with the original MRA maximum intensity projection (MIP) reveals that a bias field exists near the location of the As illustrated in Fig. 6(d) (f), the contours obtained from the final corrected segmentation fit the true phantom boundaries more accurately than those obtained by the initial level set technique. In particular, the large bias field resulting from the multiple scale filtering around the location of stenosis was reduced by the proposed correction method. 2) Three Cases of Real MRA Data: We used two abdominal and one carotid MRA cases for validation of the proposed method. The in-plane resolution of the original data in cases 1

8 1258 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 23, NO. 10, OCTOBER 2004 Fig. 6. Results of segmentation of in vitro MRA data collected in a phantom with 90% degree stenosis. (a) MIP of the reconstructed isotropic image of the phantom. (b) MIP of the phantom enhanced by our multiple scale filtering. (c) Overlapping the level-set segmented result (light) and the final result obtained after the application of a geometric deformable model segmentation technique (dark). (d) (f) The intersection contours of a sagittal image slice (d) at slice location 25, a coronal image slice (e) at slice location 31, and axial image slices (f) at slice locations 2, 46, 54 and 89, and the initial level-set segmented (black) and final (white) surfaces, respectively. and 2 (abdominal MRA image series) is. The voxel resolution in the sagittal direction is mm. Both 3-D isotropic images reconstructed from cases 1 and 2 have a resolution of. Case 3 (carotid MRA image series) has an in-plane and axial resolutions of and 0.8 mm, respectively. The isotropic image reconstructed in this case has a resolution of. The filtering step including calculating [see (10)] and [see (16)] with for cases 1, 2, and 3 with sizes,, and consumed most of the computational costs: 557, 874, and 1404 minutes (CPU times), respectively. The processing time for the level set method to segment vessels for cases 1, 2, and 3 were 7, 8, and 19 minutes (CPU times), respectively. Since we retriangulated the deformable model following each iteration, the correction steps for cases 1, 2, and 3 also consumed 69, 52, and 38 minutes (CPU times), respectively. All computations were performed on Sun Blade 100 with 700 Mb of memory. The code was not optimized. Fig. 7 presents the results from segmentation of vessels in case 1. As shown in Fig. 7(a), there is a vessel tree including common hepatic artery, left and right renal arteries, common iliac artery, abdominal aorta, splenic artery, celiac trunk, and superior and inferior mesenteric arteries in the image. In order to demonstrate the usefulness of the proposed multiple scale filtering, Fig. 7(b) shows a result obtained by directly applying the proposed level set method to the original data. As shown in the figure, except for the vessel structures, the unwanted parts of left and right kidneys as well as noise voxels were included in the segmentation. Since the aim of this study is to quantify the entire vascular tree, the filtering step is required. We first applied the multiple scale filtering to images to effectively enhance the vessel tree [Fig. 7(c)], and then automatically segmented the enhanced vessels using the proposed level set method [Fig. 7(d)]. Fig. 7(d) demonstrates the evolution of the propagating implicit surface for segmentation of vessels. In our study, we used a cube as the initial surface, and constructed the initial level set function to embed the initial surface as its zero level set. The level set function is defined by a signed distance function where the sign of the distance from a point to the surface indicates whether the point is located inside or outside the surface [see (12)]. As shown in Fig. 7(d), propagation of the surface moves along the inward normal direction. The arteries in the vessel tree were effectively enhanced and segmented. In addition, other tissues such as the left and right kidneys displayed in the MIP were ef-fectively suppressed by filtering. In order to make surfaces of the level-set segmented vessels fit the true vessel surface more accurately, we corrected and refined the segmented vessel surface with the proposed geometric deformable model [Fig. 7(e) and (f)]. Fig. 7(g) illustrates the effectiveness of the proposed hybrid method in case 1. As shown, both contours obtained from the corrected surfaces fit the actual vessel boundaries on image

9 CHEN AND AMINI: QUANTIFYING 3-D VASCULAR STRUCTURES IN MRA IMAGES 1259 Fig. 7. Results of segmentation of vessels in case 1. (a) Reconstructed images for the abdominal arteriography images with voxel dimensions (b) Labeled result obtained by directly applying the level set method to the original data without using filtering. Besides the vessel structures, unwanted parts of left and right kidneys and noise voxels were part of the segmentation. (c) MIP of enhanced vessel tree. (d) Interim and final results of the evolution of the propagating surface. (e) The final corrected vessel surface. (f) An enlarged view of part of the final vessel surface triangulated with 3-D Delaunay triangulation. (g) The intersection contours of sagittal, coronal, and axial image planes with the level-set segmented result (white) and the final result obtained after the application of a geometric deformable model segmentation technique (gray). slices, while there is a bias in the initial segmented surface. The bias is reduced by the correction of the segmented surface. Fig. 8 presents results from segmentation of vessels in cases 2 and 3 [Fig. 8(a)], respectively, and demonstrates effectiveness of the proposed method in these cases by displaying the intersection contours of image slices with the final corrected surfaces. The final corrected surfaces for these two cases are given in Fig. 8(b). As demonstrated in Fig. 8(c), the intersection contours fit the actual vessel boundaries accurately. However, a few missed regions still remained in the final surface due to the low image contrast in the original images. B. Validation In order to validate the method, we first manually segmented the phantom, the vessels in the two cases of abdominal MRA along the aorta, and the vessel in the case of carotid MRA along the internal carotid artery, and compared the results obtained by the proposed method with the manual segmentation. We calculated [see (26)], [see (27)], and [see (28)] using results obtained before and after correction. Table II displays the calculated,, and for all cases. As indicated in the table, is greater than 87.5% in all cases. In addition, for results obtained by the hybrid method is smaller than for results obtained by level set method alone. This implies that the artificial bias field was reduced by the correction method. In particular, after correction, for the results in the case of the phantom and case 1 decreased substantially. This was also demonstrated in Figs. 6 and 7. It is important to note that after correction became smaller than 6.95% and becomes greater than 88.23% in all cases. Additionally, after correction, for all cases increased in value. V. DISCUSSION AND CONCLUSION We have described a hybrid approach consisting of multiple scale filtering, level set method, and deformable geometric modeling for automatic and accurate quantification of vessel structures from 3-D MRA image data. The method was validated using mathematical simulation of vessels with high degrees of stenoses, with in vitro MRA phantom data, and with in vitro MRA data. We compare the proposed method with previously reported methods as follows. Sato et al. [18] and Frangi et al. [21] independently proposed the eigenvalues of the Hessian matrix based multiple scale filtering for enhancement of vessel structures from 3-D images. In their methods, the enhancement results depends somewhat on the selection of filter parameters. In addition, since the minimal eigenvalue,, is involved as

10 1260 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 23, NO. 10, OCTOBER 2004 Fig. 8. Results from segmentation of vessels in cases 2 and 3. (a) Reconstructed images for the abdominal and carotid arteriography images with voxel dimensions and , respectively. (b) The final corrected vessel surface. (c) The intersection contours of sagittal and coronal image planes and the final vessel surfaces of cases 2 (white) and 3 (black). TABLE II VALIDATION RESULTS FOR REAL MRA DATA one of the dominant factors for enhancing vessel structures, as a result, it could make the filters sensitive to plate-like structures. Our method and their methods share certain similarities in that both utilize the eigenvalues of the Hessian matrix as filters to enhance vessel structures. However, unlike their methods, our method avoids the need to select many filter parameters in advance. In our technique, only is involved, resulting in suppression of nonvessel structures such as the kidney. Moreover, the advantage of the hybrid method is reduction of the bias field due to filtering. Previously, Lorigo et al. applied geodesic active contour models for segmentation of vessel structures from high-contrast MRA and CT images [25]. However, no corrections were applied to the level-set segmentation, especially required for small vessel structures. Yim et al. recently proposed a tubular deformable model [26] and deformable isosurface models [28] both of which are based on triangulated meshes for vessel construction. However, since no pre-processing step for distinguishing vessel structures from other tissues exists, application of the technique to low-contrast MRA images may be difficult. In addition, the difference between our method and the one proposed by Bemmel et al. [29] is that we include a correction step for reducing the bias field that results from multiple scale filtering. Unlike the methods mentioned above, our method consists of three processing steps: enhancement, segmentation, and correction. Both the experimental results from the phantom data and real MRA data demonstrate the effectiveness of the hybrid approach for accurate quantification of vessel structures.

11 CHEN AND AMINI: QUANTIFYING 3-D VASCULAR STRUCTURES IN MRA IMAGES 1261 APPENDIX I DERIVATION OF THE VESSELNESS MEASURE According to (3), we define an ideal reference template set as, where and. Let be any set with. The means for and are given by and, respectively. The normalized correlation between and can be obtained by as in (8). If satisfies (3), then. APPENDIX II NUMERICAL DISCRETIZATION OF (12) We numerically solve (12) by discretizing and on a Cartesian grid and then apply an upwind difference scheme that is first-order accurate to evolve forward in time moving across the grid. That is, (12) is approximated by the following upwind scheme [23]: where The CFL condition for (29) is and,. APPENDIX III NUMERICAL DISCRETIZATION OF (13) (29) (30) (31). Since Equation (13) can be approximated by the following upwind scheme with (30) and (31): if otherwise (32) ACKNOWLEDGMENT The authors would like to thank Dr. K. Bae for providing the MRA data for this project. REFERENCES [1] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geometry, vol. 20, pp , [2] W. E. Lorensen and H. E. Cline, Marching cubes: a high resolution 3D surface construction algorithm, Comput. Graphics, vol. 21, no. 4, pp , [3] M. Kass, A. Witkin, and D. Terzopoulos, Snakes: active contour models, Int. J. Comput. Vis., vol. 1, pp , [4] S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Computational Phys., vol. 79, pp , [5] H. Edelsbrunner and E. Mucke, Three-dimensional alpha shapes, ACM Trans. Graphics, vol. 13, pp , [6] R. Malladi, J. A. Sethian, and B. C. Vemuri, Shape modeling with front propagation: a level set approach, IEEE Trans. Pattern Anal. Machine Intell., vol. 17, pp , Feb [7] T. McInerney and D. Terzopoulos, Topologically adaptable snakes, in Proc. 5th Int. Conf. Computer Vision, 1995, pp [8] G. Taubin, Curve and surface smoothing without shrinkage, in Proc. 5th Int. Conf. Computer Vision, 1995, pp [9] T. McInerney and D. Terzopoulos, Deformable models in medical image analysis: a survey, Med. Image Anal., vol. 1, no. 2, pp , [10] V. Caselles, R. Kimmel, and G. Sapiro, Geodesic active contours, Int. J. Comput. Vis., vol. 22, pp , [11] M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf, Computational Geometry Algorithms and Applications. Berlin, Germany: Springer-Verlag, [12] V. Caselles, R. Kimmel, G. Sapiro, and C. Sbert, Minimal surfaces based object segmentation, IEEE Trans. Pattern Anal. Machine Intell., vol. 19, pp , Apr [13] A. Yezzi, S. Kichenasamy, O. Olver, and A. Tannenbaum, A geometric snake model for segmentation of medical imagery, IEEE Trans. Med. Imag., vol. 16, pp , Apr [14] A. K. Klein, F. Lee, and A. A. Amini, Quantitative coronary angiography with deformable spline models, IEEE Trans. Med. Imag., vol. 16, pp , Oct [15] H. Tek and B. B. Kimia, Volumetric segmentation of medical images by three-dimensional bubbles, Comput. Vis. Image Understanding, vol. 64, no. 2, pp , [16] C. Lorenz, I.-C. Carlsen, T. M. Buzug, C. Fassnacht, and J. Weese, A multi-scale line filter with automatic scale selection based on the hessian matrix for medical image segmentation, in Lecture Notes In Computer Science, 1997, vol. 1252, Proceedings of First International Conference on Scale-Space Theories in Computer Vision, pp [17] K. Krissian, G. Malandain, N. Ayache, R. Vaillant, and Y. Trousset, Model based multiscale detection of 3D vessels, in Proc IEEE Computer Society Conf. Computer Vision and Pattern Recognition, 1998, pp [18] Y. Sato, S. Nakajima, N. Shiraga, H. Atsumi, S. Yoshida, T. Koller, G. Gerig, and R. Kikinis, Three-dimensional multi-scale line filter for segmentation and visualization of curvilinear structures in medical images, Med. Image Anal., vol. 2, no. 2, pp , [19] J. Huang and A. A. Amini, Anatomical object volumes from deformable B-spline surface models, in Proc IEEE Int. Conf. Image Processing, 1998, pp [20] T. McInerney and D. Terzopoulos, Topology adaptive deformable surfaces for medical image volume segmentation, IEEE Trans. Med. Imag., vol. 18, pp , Oct [21] A. F. Frangi, W. J. Niessen, R. M. Hoogeveen, T. van Walsum, and M. A. Viergever, Model-based quantitation of 3-D magnetic resonance angiographic images, IEEE Trans. Med. Imag., vol. 18, pp , Oct [22] X. Zeng, L. H. Staib, R. T. Schultz, and J. S. Duncan, Segmentation and measurement of the cortex from 3-D MR images using coupled-surfaces propagation, IEEE Trans. Med. Imag., vol. 18, pp , Oct

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