Parametric Shape Modeling Using Deformable ... - IEEE Xplore

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IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 23, NO. 3, MARCH 2004

Parametric Shape Modeling Using Deformable Superellipses for Prostate Segmentation Lixin Gong, Student Member, IEEE, Sayan D. Pathak, Member, IEEE, David R. Haynor, Paul S. Cho, and Yongmin Kim*, Fellow, IEEE

Abstract—Automatic prostate segmentation in ultrasound images is a challenging task due to speckle noise, missing boundary segments, and complex prostate anatomy. One popular approach has been the use of deformable models. For such techniques, prior knowledge of the prostate shape plays an important role in automating model initialization and constraining model evolution. In this paper, we have modeled the prostate shape using deformable superellipses. This model was fitted to 594 manual prostate contours outlined by five experts. We found that the superellipse with simple parametric deformations can efficiently model the prostate shape with the Hausdorff distance error (model versus manual outline) of 1 32 0 62 mm and mean absolute distance error of 0 54 0 20 mm. The variability between the manual outlinings and their corresponding fitted deformable superellipses was significantly less than the variability between human experts with p-value being less than 0.0001. Based on this deformable superellipse model, we have developed an efficient and robust Bayesian segmentation algorithm. This algorithm was applied to 125 prostate ultrasound images collected from 16 patients. The mean error between the computer-generated boundaries and the manual outlinings was 1 36 0 58 mm, which is significantly less than the manual interobserver distances. The algorithm was also shown to be fairly insensitive to the choice of the initial curve. Index Terms—Deformable superellipse, Fourier descriptor, prostate segmentation, shape modeling, transrectal ultrasound.

I. INTRODUCTION

P

ROSTATE cancer is the most commonly occurring cancer next to skin cancer and the second leading cause of cancer-related death among men in the U.S. [19]. Enhanced prostate awareness and increased life expectancy suggest that the number of newly diagnosed cases will continue to increase each year, making prostate cancer a major medical and societal problem that requires significant improvements

Manuscript received April 28, 2003; revised December 8, 2003. This work was supported in part by the National Institutes of Health under Grants AG19275 and CA89061. The Associate Editor responsible for coordinating the review of this paper and recommending its publication was M. A. Viergever. Asterisk indicates corresponding author. L. Gong was with the Department of Electrical Engineering, University of Washington, Seattle, WA 98195 USA. He is now with the Insightful Corporation, Seattle, WA 98109 USA (e-mail: [email protected]). S. D. Pathak is with the Insightful Corporation, Seattle, WA 98109 USA (e-mail: [email protected]). D. R. Haynor is with the Department of Radiology, University of Washington, Seattle, WA 98195 USA (e-mail: [email protected]). P. S. Cho is with the Department of Radiation Oncology, University of Washington, Seattle, WA 98195 USA (e-mail: [email protected]). *Y. Kim is with the Departments of Bioengineering and Electrical Engineering, University of Washington, Seattle, WA 98195 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TMI.2004.824237

in diagnosis and treatment. As an effective method of treating prostate cancer, transperineal interstitial permanent prostate brachytherapy (TIPPB) tries to preserve prostate and sexual function while controlling cancer. It implants 60–120 small radioactive sources (I-125 or Pd-103 seeds) into the prostate gland under transrectal ultrasound (TRUS) guidance with the goal of treating the entire gland to a specific tumoricidal dose [27], [47]. Prostate boundaries are routinely outlined in parallel transverse TRUS images at various stages of TIPPB. The boundaries are used in calculating prostate volume and creating patient-specific anatomical models. These models are used in diagnosis, treatment planning, and follow-up studies [28]. To improve the clinical treatment outcome, the development of intraoperative dosimetry has become a subject of active investigation in the prostate brachytherapy community, where prostate outlining plays a key role. When completed, it will allow the brachytherapist to identify underdosed regions, remedy seed placement, and optimize the dose distribution during the implant procedure, while the patient is still on the operating table [29]. Robust, accurate, and fast boundary delineation is important not only in TIPPB but also in planning other prostate cancer treatments, e.g., high-intensity focused ultrasound, cryotherapy, and external beam radiation therapy [32]. However, the effort required for manual delineation, especially in an intraoperative environment where time pressure is high, is one of the major problems. Automatic prostate segmentation in TRUS images has been a subject of considerable research. Classical segmentation techniques, such as edge detection [1], [2], [22], [24] and region classification [34], [36], have encountered considerable difficulties when applied to TRUS prostate images. These techniques either fail completely or require some kind of postprocessing to remove false boundaries in the segmentation results. To address these difficulties, deformable models have been used with improved results [17], [21], [23], [31]. By constraining extracted boundaries to be smooth, deformable models offer increased robustness to both image noise and boundary gaps by integrating boundary elements into a coherent and consistent mathematical description. Recently, these standard deformable models have been further improved by incorporating additional prior knowledge of the prostate shape to capture the boundary variability and provide more robust and accurate results [3], [12], [45]. The previously-proposed algorithms for automatic prostate boundary identification, while possessing a certain degree of clinical utility, are not in wide clinical use due to their inability to handle speckle noise, poor contrast between the prostate

0278-0062/04$20.00 © 2004 IEEE

GONG et al.: PARAMETRIC SHAPE MODELING USING DEFORMABLE SUPERELLIPSES FOR PROSTATE SEGMENTATION

and surrounding tissues, missing boundary segments (due to ultrasound’s inability to image interfaces that are parallel to the sound beam or hidden by acoustic shadowing caused by calcifications), bowel gas, protein deposit artifacts (corpora amylacea), etc. In addition, the presence of structures (such as the seminal vesicles and bladder neck near the base, urethra, pelvic musculature, and the posterior part of the pubic arch near the apex) makes the automatic boundary delineation difficult [32]. Achieving accurate and robust prostate boundary identification remains a difficult task, and no effective solution has yet been found. In this paper, we focus on two-dimensional (2-D) prostate modeling and segmentation from TRUS images due to the sparse nature of the TRUS image acquisition (transverse slices at 5-mm spacing). Even though more dense acquisition and three-dimensional (3-D) volume acquisition have been suggested [16], [42], most clinics use sparse data for routine clinical tasks. The proposed segmentation method consists of two steps: model building and prostate segmentation with prior model constraints. In the next section, we introduce the parametrically deformable superellipse model and compare its representation compactness and modeling efficiency with the popular Fourier descriptors in prostate model generation. Section III describes a Bayesian framework for prostate segmentation using the deformable superellipse and the results of our prostate segmentation studies. We discuss some future research directions in Section IV followed by conclusion in Section V. II. PROSTATE SHAPE MODELING

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tureless prostate shapes from different cross sections remains to be tested. The relatively smooth outlines of the prostate do not lend themselves well to morphologic procedures that rely on consistently definable landmarks. An alternative approach is to model global shape properties using parametric shape models. This technique does not require the existence of anatomical landmarks. Traditional deformable models are local models, i.e., contours are assumed to be locally smooth. Global properties, such as orientation and size, are not explicitly modeled. Modeling of global shape properties can provide greater robustness to initialization and model evolution. Furthermore, global properties are important in image interpretation because they can be characterized using only a few parameters. Global properties also tend to be much more stable than local ones. If information about the global properties is known a priori, it can improve the segmentation performance. Several methods have been developed for representing shapes [8], [10]. With many parameters controlling local deformations, complex shapes can be modeled, but at the expense of increased computational complexity. On the other hand, abstracting shape into a small number of parameters through global shape modeling leads to more stable and faster numerical schemes. In practice, it is desirable to reduce the number of parameters and the range of each parameter used in a model to obtain a compact representation. Next, we briefly describe the geometry and the construction of the deformable superellipse and compare its modeling efficiency with that of the Fourier descriptors [39]. For both methods, there is no need to identify homologous points in defining the shapes. This removes the bias that could be introduced by the selection of reference points.

A. Shape Models

B. Deformable Superellipses

When segmenting structures in noisy data such as TRUS images using deformable models, it is generally beneficial to introduce some prior information about the structure that is being recovered to constrain the model deformation [25], [26]. In most medical imaging applications, the general shape, location, and orientation of the anatomical structures under examination are known a priori. In TRUS images, prostate boundaries are smooth and generally define closed near-convex contours. This knowledge could be incorporated into the deformable model as prior information in the form of initial conditions, constraints on the model shape parameters, or into the model fitting procedure to achieve higher specificity in detecting shapes from the data. Incorporating prior information about the object shape requires either training or global shape modeling [46]. Training involves manually capturing object shape variability associated with delineations by different users. One of the ways to train is by identifying homologous points or landmarks for setting up a robust correspondence between different specimens. These anatomical points must be capable of being unambiguously and consistently located in different specimens [13]. However, this approach is difficult to apply directly to the prostate boundary identification because there are no consistent landmarks identifiable in a number of clinical cases [3], [45]. Even though a method has been developed recently to build shape models without explicit landmarks [14], its usability for circular fea-

Superellipses are a flexible representation that naturally generalizes ellipses. They can model a large variety of natural shapes, including ellipse, rectangles, parallelograms, and pinched diamonds, by changing a small number of parameters [6], [33]. A centered superellipse can be defined in a parametric form by and

(1)

where the size parameters define the lengths of specifies the the semi axes and the squareness parameter squareness in the 2-D plane. The implicit form for a superellipse can be derived from its parametric form as (2) where can be any positive real number if the two terms on the left side of (2) are first raised to the second power. The function (3) is called the inside-outside function because it provides a simple test whether a given point lies inside or outside the superellipse, which is important for model fitting. The power of the superellipse representation lies not in its ability to model perfect geometric shapes, but in its ability to model deformed geometric shapes through global deformations, such as tapering and bending [7], [38]. Tapering and bending around a given axis are each described by a single parameter.

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Fig. 1. Deformable superellipses of varying global shape parameters. The dotted, dashed and solid curves are corresponding to the first, second and third parameters respectively.

Fig. 2. Deformable superellipse can model a variety of the prostate shapes. The solid contours are manual outlinings and the dotted contour are fitted deformable superellipses.

The mathematical formulation for the global deformations of the model is presented in the Appendix. A superellipse can be . fully characterized by a vector are the pose parameters: The first four , rotation , and scaling . The last four translation define the deformable superellipse shape: squareness , aspect ratio , tapering , and bending . Fig. 1 shows the effects of changing the shape parameters. Several algorithms have been proposed for fitting superellipses [37]. We use the area-minimizing formulation of the squared algebraic distance [38], [20] (4) is the inside-outside function of the dewhere formable superellipse. Biasing the minimized superellipses to have low areas leads to improved fitting performance. A variety of prostate shapes can be modeled using the deformable superellipse model, and Fig. 2 shows some illustrative examples, with varying quality of fit. The manual outlining in Fig. 2(f) shows an atypical shape. This is because the human operator has incidentally included a part of the urinary bladder in the prostate. We have retained such images in our data set for the sake of completeness and representativeness. C. Experimental Methods A total of 125 prostate TRUS images from 16 patients were collected at the Seattle Prostate Institute during routine examinations of the patients undergoing a preimplant study. These series of transverse TRUS slices were acquired at 5-mm intervals using a Siemens Sonoline Prima ultrasound machine (Siemens Medical Systems Ultrasound Group, Issaquah, WA)

with a 5-MHz biplane TRUS probe. The pixel size of the images was 0.19 mm 0.19 mm. The TRUS probe was placed on a mounting apparatus (Precision Stepper Kit, Seed Plan Pro, Seattle, WA), which allows the probe to move along the longitudinal axis and acquire transverse images that are parallel to each other. A graphical user interface was developed in Visual C++ (Microsoft Corp., Redmond, WA), which allows manual tracing of the prostate and editing of contours using simple drawing tools. Five expert observers including two brachytherapists manually traced the prostate boundaries on each of the 125 images, resulting in 594 contours (some experts considered some images as not having prostate contours). Each contour was then interpolated with 100 points by spline fitting to arrive at a smooth prostate contour. Each contour can be treated equally as an anatomically plausible cross-section shape of the prostate for modeling. The error of model fitting and image segmentation could be measured with area-based or distance-based metrics [48]. Comparing areas is not a stringent evaluation criterion since two outlinings might be very different; yet could result in similar area estimates. Hence, in this paper, we use two distance-based metand the mean absolute disrics: the Hausdorff distance [32], where and are defined as the maxtance imum and the mean distance between the closest points over all points on the two contours, respectively. While the Hausdorff distance measures the worst possible disagreement between the two contours, the mean absolute distance estimates the average disagreement between them. D. Model Fitting Results The deformable superellipse model was fitted to the 594 manual prostate outlinings. The ellipse defined by the bounding


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TABLE I STATISTICS OF THE DISTANCES BETWEEN MANUAL OUTLINING AND THEIR CORRESPONDING DEFORMABLE SUPERELLIPSE FITTING

Fig. 3. The boxplots of disagreement between manual outlinings and deformable superellipse fittings measured as the Hausdorff distance and the mean absolute distance compared with the disagreement between different manual outlinings on the same images. The rectangular box in the boxplot has lines at the lower quartile, median, and upper quartile values. The whiskers extend from each end of the box to show the extent of the rest of the data. Outliers are data plotted in “ ” with values beyond the ends of the whiskers.

+

Fig. 4. Analysis of an outlining with the largest Hausdorff distance for deformable superellipse fitting. The solid contours are the five manual outlinings on the same image and the dotted contours are the fitted deformable superellipses. Image (a) had the largest Hausdorff distance between manual outline and fitted superellipse.

box of the corresponding manual outlining was used as the initialization, which was then deformed to fit the manual outlining through least square minimization of the cost function in (4) with respect to the parameter vector . The optimal parameters obtained for each contour were used to generate a set of 100 equidistant points along the deformable superellipse boundary. The goodness-of-fit or residual (the difference between the fitted models and the manual outlinings) was calculated. and measure for the Table I shows the statistics of the manual outlinings and the corresponding fitted superellipses. The confidence intervals for these statistics were estimated using nonparametric bootstrap technique [15]. The mean and ( mm and mm) values of are significantly smaller than the interobserver variation in mm and human expert outlinings ( mm) [32], with p-values being less than 0.0001. The boxplots (Fig. 3) for the two metrics reflect relative small uncertainties associated with the model fitting when compared to manual and for the model outlinings. The 95% percentiles of fittings are 2.49 mm and 0.91 mm, respectively. This indicates

that deformable superellipses can accurately approximate the majority of prostate shapes. We observed that the shape parameters are generally uncorrelated and each one of them can be well represented by a Gaussian distribution. This is desirable as it allows easy and natural incorporation of the shape prior using a multivariate independent Gaussian distribution for prostate segmentation. We also observed that the prostate is better localized in the x direction (horizontal) than in the y direction (vertical). The rotation parameter has a zero mean and a small standard deviation, indicating that the prostate contours are generally situated upright. The scale parameter has a large standard deviation, which is expected because we are modeling prostate cross sections ranging from the mid-gland (where the cross section of the prostate is large) to the base and apex (where the cross section of the prostate is small). distance We further investigated the contours with a large measure to gain insight into the fitting performance. Fig. 4(a) shows the manual outlining with the largest Hausdorff distance to its corresponding deformable superellipse fitting. We can see

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Fig. 5. Illustrates the Fourier descriptors as a stepwise procedure being fitted to a manual outlining of a prostate shape with an increasing number of harmonics ranging from 1 (a) to 8 (h).

Fig. 6.

Boxplot of Hausdorff distance and mean absolute distances versus number of harmonics used to reconstruct the contour.

that the outlining with the largest distance measure is not consistent with the outlinings of the other four experts on the same image, which in contrast can be approximated quite well by the deformable superellipse model. This was the case in mm. The prostate is 90% of the outlinings with walnut-shaped and generally has convex or near-convex closed cross-section contours on 2-D TRUS images. Sharp corners and high curvature segments should generally not occur. This suggests that those manual boundaries with a large distance to their deformable superellipse models might have been due to misinterpretation of the images. Hence, the fitting performance of the deformable superellipse model could have been better with those outliers removed. For comparison, the Fourier descriptors [39] were also used to represent the manual outlinings with an increasing number of harmonics, and Fig. 5 shows the results of fitting the atypical prostate contour using Fourier descriptors as a stepwise pro-

cedure. The fitting results of the whole data set are shown in Fig. 6. When comparing the errors in Fourier descriptor-based fitting to the errors in deformable superellipse fitting (Fig. 3), we found that at least three harmonics (a total of 14 parameters) are needed for the Fourier descriptors to achieve the fitting performance similar to the deformable superellipse. This suggests that the deformable superellipse, which requires eight parameters, provides a more compact representation for the prostate shape than the Fourier descriptors. The apparent superiority of the deformable superellipse as a method for prostate might be due to the fact that the deformable superellipse includes nonlinear terms while the Fourier model is linear. Although the dimensionality of the parameter space in deformable superellipses could be reduced via principal component analysis, the parameters might lose their original physical meanings. Thus, we have chosen the deformable superellipse in its original parameterization as the prior shape model for prostate segmentation.


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Fig. 7. Examples of edge map generated by the robust edge detection algorithm.

III. PROSTATE SEGMENTATION

B. Segmentation Results

A. Bayesian Framework Using the deformable superellipse as the prior shape model for the prostate, we segment the prostate boundaries in TRUS images. The end goal is to find the optimal parameter vector that best describes the prostate in a given unsegmented image. The search can be formulated as a maximum a posterior criterion using the Bayes rule that allows seamless integration of prior shape knowledge. We model the shape prior using a multivariate Gaussian disand the pose prior using a uniform distribution tribution . We also assume that shape is independent from pose based on our modeling study. The edge strength of an image can be used as the likelihood. By using the Bayes rule, the a posterior probability density of the deformed boundary given the input edge strength can be expressed as

(5) The maximization of the a posterior density with respect to can be simplified by taking the logarithm and eliminating , which is the prior probability of the image data that is considered to be equal for all . Thus, it suffices to maximize . This term is also known as the log likelihood function. The log likelihood can be maximized by maximizing the following expression [39]:

(6) is the mean of , the th element of the shape paramHere, eter vector , which is defined to be zero relative to the mean shape configuration. The variance for each of the parameters is is calculated from the training set. is the pixel location of the th point on the model shape defined by . Also, note that is the weighting parameter generally associated with the image noise model [39]. Hence, the a posterior objective is maximized by incorporating a prior bias to likely shapes (first term) and match to the edges in the image by maximizing the sum of the edge strength at the boundary points defined by the vector (second term).

We applied the deformable superellipse-based segmentation algorithm to the set of 125 TRUS images from 16 patients (see Section II-C). The cost function in (6) with set to 0.3 is maximized with respect to p using the simplex algorithm [35]. The average of the five manual outlinings on the same image is treated as the ground truth of the prostate boundary. The prior knowledge of the deformable superellipse parameters representing the prostate shape was obtained using the manual outlinings on images from 15 patients, which was then used in testing the segmentation method on the images from the one remaining patient. This results in 16 such leave-one-out tests. We used the robust edge detection algorithm described in [32] to generate the edge map required for segmentation. This algorithm performs edge detection in three steps. The original image first went through selective contrast enhancement and speckle reduction using the sticks algorithm. The stick length of 15 (corresponding to a line segment of 2.85 mm) was chosen to reduce the speckle while at the same time improving the contrast of the underlying image. Second, the enhanced image was then anisotropically smoothed with the edge-preserving weak membrane algorithm. The penalty associated with a discontinuity and the measure of elasticity of the membrane were chosen to be 0.05 and 8, respectively, to offer a reasonable compromise between preserving the true edges and reducing the noise. Last, edge detection and knowledge-based filtering were performed on the smoothed image using the Canny edge detector with a 9 9 Gaussian kernel with a standard deviation of 2. The high and low values for the hysteresis thresholding and nonmaximal suppression were 0.7 and 0.3, respectively. We elected not to use the edge map generated by the weak membrane algorithm since it generates fragmented edges and misses some of the weaker edges. Instead, the Canny’s edge detector was used on the weak membrane smoothed image. The parameters were selected according to the guideline set forth in [32] and they were fixed for all the images. Fig. 7 shows four prostate ultrasound images with the edge maps obtained using the robust edge detection algorithm described. For deformable model segmentation schemes, an initialization is generally required as the starting point for the model evolution. Since we are modeling prostate cross sections ranging from the mid-gland (where the cross section of the prostate is large) to the base and apex (where the cross section

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Fig. 8. Examples of TRUS prostate segmentation. The solid contours are the ground truth established by averaging five experts’ manual outlinings and the dotted contour are the final results obtained by the proposed Bayesian segmentation scheme.

STATISTICS

OF THE

TABLE II MEAN DISTANCE BETWEEN COMPUTER-GENERATED SEGMENTATION AND THEIR CORRESPONDING MANUAL OUTLINING COMPARED WITH THE MEAN DISTANCE BETWEEN THE FIVE OBSERVERS

Fig. 9. The distances of the computer-generated boundaries to the manual outline on starting the algorithm as a function of the number of points to specify the initial curve.

of the prostate is small), the scale parameter has large variability. Therefore, more than two points specified by a user on the prostate boundary are needed to give the approximate scale for the prostate on each image, and the scaled mean shape can be initialized as the starting point. After the initialization, the segmentation (including robust edge detection and Bayesian model fitting) typically takes less than 5 s for a 256 256 image on a Pentium 4 PC running at 2 GHz. We evaluated our algorithm by calculating the distances between the computer estimate of the prostate boundaries and the manual outlinings on the same image. Fig. 8 shows some illustrative examples when four-point initialization is used, and the quantitative results are presented in Table II. The mean distance between the computer-generated boundaries and the manual outlinings was mm, which is significantly smaller than the average interobserver distance ( mm), with p-value being less than 0.0001. We tested the effect of the number of points used to specify the initial curve. Fig. 9 shows the boxplots of the distances of the

resulting computer-generated boundaries to the manual outlines on starting out with different number of points. We compared the means of these distances using one-way analysis of variance (ANOVA) and found that there is no significant difference in the means of these distances with p-value larger than 0.01. We have also tested the influence of different observers in specifying the initial curve. Starting out with two points specified by five different observers, we computed the distance of the resulting computer-generated boundaries to the average manual outlines. Fig. 10 shows the boxplots of these distances. The means of these distances for the first four observes are not significantly different with large p-values (0.84 for and 0.95 for ) from one-way ANOVA. The outlinings of the fifth observer were significantly different from the other observers. It is interesting to note that this observer is the one who drew the prostate boundary on Fig. 4(a), which is not consistent with the outlinings of the other four experts on the same image. It is apparent that this observer was using a somewhat different criterion for outlining the prostate boundaries than the other observers.


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Fig. 10. Boxplots of distances of the computer-generated boundaries to the average manual outline for prostate boundaries on starting the algorithm with initial curves draw by different observers.

IV. DISCUSSION To develop a fully automated prostate segmentation algorithm, the initial curve has to be generated automatically. A general approach for initialization is to use the mean shape and pose obtained through a training process. In some cases, a single mean shape might not be sufficient to characterize the prostate due to its large shape variability [30]. In [21], two 2-D prostate shape models were used for TRUS prostate segmentation. However, their models were created manually after estimating subjectively the characteristic anatomical variations, which might not be representative. In contrast, by using the deformable superellipse model, we can identify the principal prostate shapes by the fuzzy C-means clustering algorithm from a large set of training outlinings. To realize the idea of fully automatic prostate segmentation in the future, we can fit each of these principal shape models to the image through a hierarchical chamfer matching procedure [9] by equidistant sampling the possible parameters of the similarity transformation (translation, rotation, and scaling) within a reasonable range and choose the one with the highest matching score as the initialization. The range of the similarity transformation can be also obtained in the prostate shape modeling step. We believe that the initial contour generated this way will give the subsequent deformable model evolution process a better starting point than just a simple mean shape. Even though the deformable superellipses can closely model the majority of prostate shape, it can only approximate symmetric shapes. In some rare cases, due to bad image quality and large pathological and individual variations, this model-based segmentation scheme might sometime fail, producing inaccurate results and requiring the intervention of a human operator. In such case, Fourier descriptors can be used for finer refinement with a larger number of harmonics at the cost of longer computation time and possibly increased variability from more user intervention. Based on our clinicians’ feedback, we have developed a graphical user interface with manual editing capability using Fourier descriptors [18].

Given the 3-D nature of the prostate, 3-D modeling and segmentation of the prostate is certainly one of the research directions in the future when the 3-D ultrasound imaging for the prostate finds its way into clinical brachytherapy workspace. This can be carried out with the natural 3-D extension of the deformable superellipse model—deformable superellipsoid model [20]. Due to its inherent symmetry, the deformable superellipsoid model alone might be too limited to model complex 3-D objects precisely. To this end, several researchers have combined deformable superellipsoid with a local deformation scheme, such as splines [40], wavelets [43], spherical harmonics [11], and free-form deformations [5]. These hybrid models have been successfully applied to the shape analysis of the left ventricle [4], [11] and the brain [41], [44], and they can be applied to 3-D prostate segmentation as well. However, the prostate ultrasound studies in prostate brachytherapy currently use multiple 2-D image slices at 5-mm intervals, making the 3-D modeling of the gland challenging. In the future, with the availability of 3-D TRUS systems in brachytherapy, 3-D modeling could provide more accurate anatomic information. With the efficient and robust 2-D modeling and segmentation of the prostate shape provided in this paper, solving more challenging problems, such as prostate segmentation on postimplant TRUS images that have a much higher noise level and multiple missing edges due to shadowing by the implanted seeds, becomes imaginable. V. CONCLUSION We have investigated the deformable superellipse model to characterize the global prostate shape. It was shown that the deformable superellipse model can closely approximate the prostate shape with a fewer parameters compared to Fourier descriptors. Shape analysis of the prostate contributes to our understanding of the introduction of various constraints in the allowed contour shape during the model deformation process. The prior prostate shape information obtained has been integrated into a Bayesian prostate segmentation algorithm and has produced very promising results.

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Fig. 11. Geometry for superellipse circular bending along the y axis (in the patient anterior-posterior direction).

A

APPENDIX Global similarity transformation for superellipses 1) Translation (7) 2) Rotation (8) 3) Scale (9)

B)

Global shape deformation for superellipses 1) Squareness (10) 2) Aspect ratio (11) 3) Linear tapering along

axis (12)

4) Circular bending along the axis ( ) (see Fig. 11) (13) .

where

ACKNOWLEDGMENT The authors wish to thank Dr. P. D. Grimm, Dr. J. C. Blasko, J. Estlund, D. Naidoo, and T. Mills of the Seattle Prostate Institute (Seattle, WA) and Dr. K. Wallner of the University of Washington for participating in the data set outlining and providing valuable feedback and advice on how to improve prostate delineation techniques. REFERENCES [1] R. G. Aarnink, R. J. B. Giesen, A. L. Huynen, J. J. M. C. H. de la Rosette, F. M. J. Debruyne, and H. Wijkstra, “A practical clinical method for contour determination in ultrasonographic prostate images,” Ultrasound Med. Biol., vol. 20, pp. 705–717, 1994. [2] R. G. Aarnink, S. D. Pathak, J. J. M. C. H. de la Rosette, F. M. J. Debruyne, Y. Kim, and H. Wijkstra, “Edge detection in prostatic ultrasound images using integrated edge maps,” Ultrasonics, vol. 36, pp. 635–642, 1998. [3] F. A. Cosio and B. L. Davies, “Automated prostate recognition: A key process for clinically effective robotic prostatectomy,” Med. Biol. Eng. Computing, vol. 37, pp. 236–243, 1999.

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