Abstract. A crucial problem in statistical shape analysis is establishing the correspondence of shape features across a population. While many solutions are easy to express using boundary representations, this has been a considerable challenge for medial representations. This paper uses a new 3-D medial model that allows continuous interpolation of the medial manifold and provides a map back and forth between it and the boundary. A measure defined on the medial surface then allows one to write integrals over the boundary and the object interior in medial coordinates, enabling the expression of important object properties in an object-relative coordinate system. We use these integrals to optimize correspondence during model construction, reducing variability due to the model parameterization that could potentially mask true shape change effects. Discrimination and hypothesis testing of populations of shapes are expected to benefit, potentially resulting in improved significance of shape differences between populations even with a smaller sample size. 1

Abstract. A fundamental problem when computing statistical shape models is the determination of correspondences between the instances of the associated data set. Often, homologies between points that represent the surfaces are assumed which might lead to imprecise mean shape and variability results. We propose an approach where exact correspondences are replaced by evolving correspondence probabilities. These are the basis for a novel algorithm that computes a generative statistical shape model. We developed an unified MAP framework to compute the model parameters (’mean shape ’ and ’modes of variation’) and the nuisance parameters which leads to an optimal adaption of the model to the set of observations. The registration of the model on the instances is solved using the Expectation Maximization- Iterative Closest Point algorithm which is based on probabilistic correspondences and proved to be robust and fast. The alternated optimization of the MAP explanation with respect to the observation and the generative model parameters leads to very efficient and closed-form solutions for (almost) all parameters. Experimental results on brain structure data sets demonstrate the efficiency and well-posedness of the approach. The algorithm is then extended to an automatic classification method using the k-means clustering and applied to synthetic data as well as brain structure classification problems. 1

This paper presents a new method of constructing compact statistical point-based models of populations of human cortical surfaces with functions of spatial locations driving the correspondence optimization. The proposed method is to establish a tradeoff between an even sampling of the surfaces (a low surface entropy) and the similarity of corresponding points across the population (a low ensemble entropy). The similarity metric, however, isn’t constrained to be just spatial proximity, but can be any function of spatial location, thus allowing the integration of local cortical geometry as well as DTI connectivity maps and vasculature information from MRA images. This method does not require a spherical parameterization or fine tuning of parameters. Experimental results are also presented, showing lower local variability for both sulcal depth and cortical thickness measurements, compared to other commonly used methods such as FreeSurfer.

Abstract. For many years we have been developing a variety of methods that together would allow segmentation of 3D objects from medical images in a way reflecting knowledge of both the population of anatomic geometries sought and the population of images consistent with that geometry. To support the probability estimation methods we use to reflect this knowledge, the methods use a medial description, the m-rep, as the object representation and regional intensity quantile functions as the representation of image information in regions relative to the m-rep. Using manually segmented images to which m-reps have been fit and which contain information to allow alignment, our methods use principal geodesic analysis to estimate prior probability density, on the anatomic geometry, and they use principal component analysis to estimate a likelihood density, on the regional intensity quantile functions. They then segment automatically via posterior optimization over principal geodesic coefficients, after initialization via bones or a few contours. Each component of this methodology is briefly reviewed. Pelvic organs from multi-day populations from individual patients were segmented from CT by training a prior and a likelihood density by the methods indicated. The results are compared to human segmentations. The resulting measurements indicate that in a significant majority of cases, maximizing the log posterior objective function provides segmentations in as good or better agreement with experts than they agree with each other. Similar results are reported for other organs, other image types, and between-patient variation. 1

Abstract In this paper, we present a method to compute a statistical shape model based on shapes which are represented by unstructured point sets with arbitrary point numbers. A fundamental problem when computing statistical shape models is the determination of correspondences between the points of the shape observations of the training data set. In the absence of landmarks, exact correspondences can only be determined between continuous surfaces, not between unstructured point sets. To overcome this problem, we introduce correspondence probabilities instead of exact correspondences. The correspondence probabilities are found by aligning the observation shapes with the affine Expectation Maximization- Iterative Closest Points registration algorithm. In a second step, the correspondence probabilities are used as input to compute a mean shape (represented once again by an unstructured point set). Both steps are unified in a single optimization criterion which depends on the two parameters ’ registration transformation ’ and ’mean shape’. In a last step, a variability model which best represent the variability in the training data set is computed. Experiments on synthetic data sets and real brain structure data sets are then designed to evaluate the performance of our algorithm. The method is compared to a statistical shape model built on exact correspondences. Results regarding the established measures ”generalization ability ” and ”specificity ” show the relevance of our approach. 1

In this paper, we present a method to compute a statistical shape model based on shapes which are represented by unstructured point sets with arbitrary point numbers. A fundamental problem when computing statistical shape models is the determination of correspondences between the observations of the associated data set. Often, homologies between points that represent the surfaces are assumed. When working merely with point clouds, this might lead to imprecise mean shape and variability results. To overcome this problem, we propose an approach where exact correspondences are replaced by evolving correspondence probabilities. These are the basis for a novel algorithm that computes a generative statistical shape model. We developed a unified Maximum A Posteriori (MAP) framework to compute the model parameters (’mean shape ’ and ’modes of variation’) and the nuisance parameters which leads to an optimal adaption of the model to the set of observations. The registration of the model on the observations is solved using the Expectation Maximization- Iterative Closest Point algorithm which is based on probabilistic correspondences and proved to be robust and fast. The alternated optimization of the MAP explanation with respect to the observation and the generative model parameters leads to very efficient and closed-form solutions for nearly all parameters. A comparison with a statistical shape model which is built using the Iterative Closest Point (ICP) registration algorithm and a Principal Component Analysis (PCA) shows that our approach leads to better SSM quality measures.