A Methodology for Constructing Geometric Priors and Likelihoods for Deformable Shape Models
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Deformable shape models require correspondence across the training population in order to generate a statistical model for use as a future geometric prior. Traditional methods use flxed sampling and assume correspondence, or attempt to induce correspondence by min- imizing variance. In this paper, we deflne a training methodology for sampled medial deformable shape models (m-reps) which generates cor- respondence implicitly via a geometric prior. We present quantitative results of the method applied to real medical images. Automatic segmentation of medical images is a vital step in processing large populations for quantitative study of biological shape variability. In this paper, we present a methodology for statistically training the geometry of deformable model templates that can be used as geometric prior and basis for intensity train- ing for automatic segmentation of gray images. Our method frames the problem as a special case of the general segmentation problem. Given a data set of hu- man or otherwise expertly segmented training cases, we flt models to the labeled data, and then create our template by statistical analysis of the flt population. The fltting is an optimization over model parameters in a Bayesian framework, searching for the model with the highest posterior probability of fltting the data. Our posterior is decomposed into data likelihood and geometric prior terms. The data likelihood accounts for both image match and optional landmark match. The geometric prior encourages models to stay in a legal shape-space. We de- scribe an implementation of the method using m-reps and present results showing that the method is accurate and yields models suitable for statistical analysis. Deformable Models We desire a statistical model for a population of training data. Probabilistic deformable models describe shape variability via a probability distribution on the shape-space. Under the Gaussian model, the distribution of the training data can be modeled by the mean, a point in the space, and several eigenmodes of deformation. This model describes all possible shapes in the training data, and by extension, estimates the actual ambient shape-space from which the training data is drawn. This statistical model can then be used asKeywords:
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We propose an approach for boundary finding where the correspondence of a subset of boundary points to a model is simultaneously determined. Global shape parameters derived from the statistical variation of object boundary points in a training set are used to model the object. A Bayesian formulation, based on this prior knowledge and the edge information of the input image, is employed to find the object boundary with its subset points in correspondence with boundaries in the training set or the mean boundary. We compared the use of a generic smoothness prior and a uniform independent prior with the training set prior in order to demonstrate the power of this statistical information. A number of experiments were performed on both synthetic and real medical images of the brain and heart to evaluate the approach, including the validation of the dependence of the method on image quality, different initialization and prior information.
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A new geometric active contour based level-sets model combining gradient, region and shape knowledge information cues is proposed to robust object detection boundaries in presence of occlusions and cluttered background. The gradient, region and shape knowledge information are incorporated as energy terms. The a priori shape model is based on statistical learning of the training data distribution where the structure of data distribution is approximated by a probability density model. The obtained probability is treated as Kernel Principal Component Analysis (KPC) by allowing the shapes that are close to the training data as energy term and incorporated a prior knowledge about shapes in a more robust manner into evolving equation model to constrain the further segmentation evolution process. We applied successfully the proposed model to synthetic and real MR images. The results drawn by the newer model are compared to expert segmentation and evaluated in terms of F-mesure.
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Purpose: Explicit deformable shape models (DSMs) can be used in a Bayesian statistical framework to provide a priori information for posterior optimization to match the DSM against a target image for automatic segmentation. In this approach a DSM is initialized in the target image and undergoes a series of deformations to closely match the target object. Deformation is driven by optimizing an objective function with terms for geometric typicality (prior) and model‐to‐image match (likelihood). The purpose of this work was to develop strategy, methodology, and tools for constructing the geometric prior and intensity likelihood for a particular form of DSM called m‐reps. Method and Materials: Geometric truth is defined for an object of interest by a statistically significant collection of expert human segmentations of training images. M‐reps are fit to the human drawn contours by minimizing the distance between the surfaces of the m‐rep and the contours under added conditions that lead to positional correspondence across training cases. The geometry of the resulting set of training m‐reps is analyzed in non‐Euclidean space using an approach called principal geodesic analysis (PGA) to yield a set of eigenmodes that define the geometric prior. The intensity likelihood is constructed by registering each training m‐rep with the corresponding gray scale image and collecting regional intensity information that is statistically characterized over all training cases. The intensity information can be in several forms including linear profiles and regional histograms. Results: PGA produces modes that include natural deformations such as local twisting, bending, bulging, and constricting. Unlike analysis in Euclidean space, improper shapes are avoided. The form of the intensity prior can be customized to each object of interest for optimal performance. Conclusion: These methods are powerful, robust and generalizable to other DSMs. Conflict of Interest: The presenting author has a financial interest in Morphormics, Inc.
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Incorporating shape information is essential for the delineation of many organs and anatomical structures in medical images. While previous work has mainly focused on parametric spatial transformations applied on reference template shapes, in this paper, we address the Bayesian inference of parametric shape models for segmenting medical images with the objective to provide interpretable results. The proposed framework defines a likelihood appearance probability and a prior label probability based on a generic shape function through a logistic function. A reference length parameter defined in the sigmoid controls the trade-off between shape and appearance information. The inference of shape parameters is performed within an Expectation-Maximisation approach where a Gauss-Newton optimization stage allows to provide an approximation of the posterior probability of shape parameters. This framework is applied to the segmentation of cochlea structures from clinical CT images constrained by a 10 parameter shape model. It is evaluated on three different datasets, one of which includes more than 200 patient images. The results show performances comparable to supervised methods and better than previously proposed unsupervised ones. It also enables an analysis of parameter distributions and the quantification of segmentation uncertainty including the effect of the shape model.
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In this article, we propose a unified statistical framework for image segmentation with shape prior information. The approach combines an explicitely parameterized point-based probabilistic statistical shape model (SSM) with a segmentation contour which is implicitly represented by the zero level set of a higher dimensional surface. These two aspects are unified in a Maximum a Posteriori (MAP) estimation where the level set is evolved to converge towards the boundary of the organ to be segmented based on the image information while taking into account the prior given by the SSM information. The optimization of the energy functional obtained by the MAP formulation leads to an alternate update of the level set and an update of the fitting of the SSM. We then adapt the probabilistic SSM for multi-shape modeling and extend the approach to multiple-structure segmentation by introducing a level set function for each structure. During segmentation, the evolution of the different level set functions is coupled by the multi-shape SSM. First experimental evaluations indicate that our method is well suited for the segmentation of topologically complex, non spheric and multiple-structure shapes. We demonstrate the effectiveness of the method by experiments on kidney segmentation as well as on hip joint segmentation in CT images.
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Image segmentation, the process of decomposing an image into meaningful regions, is a fundamental problem in image processing and computer vision. Recently, image segmentation techniques based on active contour models with level set implementation have received considerable attention. The objective of this thesis is in the development of advanced active contour-based image segmentation methods that incorporate complex statistical information into the segmentation process, either about the image intensities or about the shapes of the objects to be segmented. To this end, we use nonparametric statistical methods for modeling both the intensity distributions and the shape distributions.
Previous work on active contour-based segmentation considered the class of images in which each region can be distinguished from others by second order statistical features such as the mean or variance of image intensities of that region. This thesis addresses the problem of segmenting a more general class of images in which each region has a distinct arbitrary intensity distribution. To this end, we develop a nonparametric information-theoretic method for image segmentation. In particular, we cast the segmentation problem as the maximization of the mutual information between the region labels and the image pixel intensities. The resulting curve evolution equation is given in terms of nonparametric density estimates of intensity distributions, and the segmentation method can deal with a variety of intensity distributions in an unsupervised fashion.
The second component of this thesis addresses the problem of estimating shape densities from training shapes and incorporating such shape prior densities into the image segmentation process. To this end, we propose nonparametric density estimation methods in the space of curves and the space of signed distance functions. We then derive a corresponding curve evolution equation for shape-based image segmentation. Finally, we consider the case in which the shape density is estimated from training shapes that form multiple clusters. This case leads to the construction of complex, potentially multi-modal prior densities for shapes. As compared to existing methods, our shape priors can: (a) model more complex shape distributions; (b) deal with shape variability in a more principled way; and (c) represent more complex shapes. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.)
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