Abstract. In this paper, we report an active contour algorithm that is capable of using prior shapes. The energy functional of the contour is modified so that the energy depends on the image gradient as well as the prior shape. The model provides the segmentation and the transformation that maps the segmented contour to the prior shape. The active contour is able to find boundaries that are similar in shape to the prior, even when the entire boundary is not visible in the image (i.e., when the boundary has gaps). A level set formulation of the active contour is presented. The existence of the solution to the energy minimization is also established. We also report experimental results of the use of this contour on 2d synthetic images, ultrasound images and fMRI images. Classical active contours cannot be used in many of these images.

This paper proposes a framework for dealing with several problems related to the analysis of shapes. Two related such problems are the definition of the relevant set of shapes and that of defining a metric on it. Following a recent research monograph by Delfour and Zolesio [11], we consider the characteristic functions of the subsets of R² and their distance functions. The L² norm of the difference of characteristic functions, the L # and the W norms of the difference of distance functions define interesting topologies, in particular the well-known Hausdorff distance. Because of practical considerations arising from the fact that we deal with

We address the problem of image segmentation with statistical shape priors in the context of the level set framework. Our paper makes two contributions: Firstly, we propose to generate invariance of the shape prior to certain transformations by intrinsic registration of the evolving level set function. In contrast to existing approaches to invariance in the level set framework, this closed-form solution removes the need to iteratively optimize explicit pose parameters. Moreover, we will argue that the resulting shape gradient is more accurate in that it takes into account the e#ect of boundary variation on the object's pose.

Abstract—We present an approach for comparing two sequences of deforming shapes using both parametric models and nonparametric methods. In our approach, Kendall’s definition of shape is used for feature extraction. Since the shape feature rests on a non-Euclidean manifold, we propose parametric models like the autoregressive model and autoregressive moving average model on the tangent space and demonstrate the ability of these models to capture the nature of shape deformations using experiments on gaitbased human recognition. The nonparametric model is based on Dynamic Time-Warping. We suggest a modification of the Dynamic time-warping algorithm to include the nature of the non-Euclidean space in which the shape deformations take place. We also show the efficacy of this algorithm by its application to gait-based human recognition. We exploit the shape deformations of a person’s silhouette as a discriminating feature and provide recognition results using the nonparametric model. Our analysis leads to some interesting observations on the role of shape and kinematics in automated gait-based person authentication. Index Terms—Shape, shape sequences, shape dynamics, comparison of shape sequences, gait recognition. 1

Using a recently proposed geometric representation of planar shapes, we present algorithmic tools for: (i) hierarchical clustering of imaged objects according to the shapes of their boundaries, (ii) learning of probability models for clustered shapes, and (iii) testing of observed shapes under competing probability models. Clustering at any level of hierarchy is performed using a mimimum dispersion criterion and a Markov search process. Statistical means of clusters provide shapes to be clustered at the next higher level, thus building a hierarchy of shapes.

This paper presents an algorithm for learning the time-varying shape of a non-rigid 3D object from uncalibrated 2D tracking data. We model shape motion as a rigid component (rotation and translation) combined with a non-rigid deformation. Reconstruction is ill-posed if arbitrary deformations are allowed. We constrain the problem by assuming that the object shape at each time instant is drawn from a Gaussian distribution. Based on this assumption, the algorithm simultaneously estimates 3D shape and motion for each time frame, learns the parameters of the Gaussian, and robustly fills-in missing data points. We then extend the algorithm to model temporal smoothness in object shape, thus allowing it to handle severe cases of missing data.

Abstract. All previous geometric active contour models that have been formulated as gradient flows of various energies use the same L 2-type inner product to define the notion of gradient. Recent work has shown that this inner product induces a pathological Riemannian metric on the space of smooth curves. However, there are also undesirable features associated with the gradient flows that this inner product induces. In this paper, we reformulate the generic geometric active contour model by redefining the notion of gradient in accordance with Sobolev-type inner products. We call the resulting flows Sobolev active contours. Sobolev metrics induce favorable regularity properties in their gradient flows. In addition, Sobolev active contours favor global translations, but are not restricted to such motions; they are also less susceptible to certain types of local minima in contrast to traditional active contours. These properties are particularly useful in tracking applications. We demonstrate the general methodology by reformulating some standard edge-based and regionbased active contour models as Sobolev active contours and show the substantial improvements gained in segmentation.

This paper describes methods for recovering time-varying shape and motion of non-rigid 3D objects from uncalibrated 2D point tracks. For example, given a video recording of a talking person, we would like to estimate the 3D shape of the face at each instant, and learn a model of facial deformation. Time-varying shape is modeled as a rigid transformation combined with a non-rigid deformation. Reconstruction is ill-posed if arbitrary deformations are allowed, and thus additional assumptions about deformations are required. We first suggest restricting shapes to lie within a lowdimensional subspace, and describe estimation algorithms. However, this restriction alone is insufficient to constrain reconstruction. To address these problems, we propose a reconstruction method using a Probabilistic Principal Components Analysis (PPCA) shape model, and an estimation algorithm that simultaneously estimates 3D shape and motion for each instant, learns the PPCA model parameters, and robustly fills-in missing data points. We then extend the model to model temporal dynamics in object shape, allowing the algorithm to robustly handle severe cases of missing data.

We present a robust framework for estimating non-rigid 3D shape and motion in video sequences. Given an input video sequence, and a user-specified region to reconstruct, the algorithm automatically solves for the 3D time-varying shape and motion of the object, and estimates which pixels are outliers, while learning all system parameters, including a PDF over non-rigid deformations. There are no user-tuned parameters (other than initialization); all parameters are learned by maximizing the likelihood of the entire image stream. We apply our method to both rigid and non-rigid shape reconstruction, and demonstrate it in challenging cases of occlusion and variable illumination.

We present a novel fully automatic method for high resolution, non-rigid dense 3D point tracking. High quality dense point clouds of non-rigid geometry moving at video speeds are acquired using a phase-shifting structured light ranging technique. To use such data for the temporal study of subtle motions such as those seen in facial expressions, an efficient non-rigid 3D motion tracking algorithm is needed to establish inter-frame correspondences. The novelty of this paper is the development of an algorithmic framework for 3D tracking that unifies tracking of intensity and geometric features, using harmonic maps with added feature correspondence constraints. While the previous uses of harmonic maps provided only global alignment, the proposed introduction of interior feature constraints guarantees that non-rigid deformations will be accurately tracked as well. The harmonic map between two topological disks is a diffeomorphism with minimal stretching energy and bounded angle distortion. The map is stable, insensitive to resolution changes and is robust to noise. Due to the strong implicit and explicit smoothness constraints imposed by the algorithm and the high-resolution data, the resulting registration/deformation field is smooth, continuous and gives dense one-to-one inter-frame correspondences. Our method is validated through a series of experiments demonstrating its accuracy and efficiency. 1. Introduction and Previous