There have been various developments in shape analysis in the last decade. We describe here some relationships of shape analysis with directional statistics. For shape, rotations are to be integrated out or to be optimized over whilst they are the basis for directional statistics. However, various concepts are connected. In particular, certain distributions of directional statistics have emerged in shape analysis, such a distribution is Complex Bingham Distribution. This paper first gives some background to shape analysis and then it goes on to directional distributions and their applications to shape analysis. Note that the idea of using tangent space for analysis is common to both manifold as well. 1 Introduction Consider shapes of configurations of points in Euclidean space. There are various contexts in which k labelled points (or "landmarks") x 1 ; :::; x k in IR m are given and interest is in the shape of (x 1 ; :::; x k ). Example 1 The microscopic fossil Globorotalia truncat...

For analyzing shapes of planar, closed curves, we propose di#erential geometric representations of curves using their direction functions and curvature functions. Shapes are represented as elements of infinite-dimensional spaces and their pairwise di#erences are quantified using the lengths of geodesics connecting them on these spaces. We use a Fourier basis to represent tangents to the shape spaces and then use a gradient-based shooting method to solve for the tangent that connects any two shapes via a geodesic.

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 present a new framework for recognizing planar object classes, which is based on local feature detectors and a probabilistic model of the spatial arrangement of the features. The allowed object deformations are represented through shape statistics, which are learned from examples. Instances of an object in an image are detected by finding the appropriate features in the correct spatial configuration. The algorithm is robust with respect to partial occlusion, detector false alarms, and missed features. A 94% success rate was achieved for the problem of locating quasi-frontal views of faces in cluttered scenes. 1 Introduction Many early pattern recognition algorithms were based on template matching [13], which is optimal for detecting a known signal in white noise. However, since the underlying assumption that "the signal is known exactly" rarely holds true, considerable effort has been devoted to extending this method to handle variability in the target signal. For example, approach...

In this paper, a face localization system is proposed in which local detectors are coupled with a statistical model of the spatial arrangement of facial features to yield robust performance. The outputs from the local detectors are treated as candidate locations and constellations are formed from these. The effects of translation, rotation, and scale are eliminated by mapping to a set of shape variables. The constellations are then ranked according to the likelihood that the shape variables correspond to a face versus an alternative model. Incomplete constellations, which occur when some of the true features are missed, are handled in a principled way. 1 Introduction The problem of face recognition has received considerable attention in the literature [11, 24, 21, 4, 19, 17, 22, 10]; however, in most of these studies, the faces were either embedded in a benign background or were assumed to have been pre-segmented. For any of these recognition algorithms to work in realworld applicati...

A primary goal of statistical shape analysis is to describe the variability of a population of geometric objects. A standard technique for computing such descriptions is principal component analysis. However, principal component analysis is limited in that it only works for data lying in a Euclidean vector space. While this is certainly sufficient for geometric models that are parameterized by a set of landmarks or a dense collection of boundary points, it does not handle more complex representations of shape. We have been developing representations of geometry based on the medial axis description or m-rep. While the medial representation provides a rich language for variability in terms of bending, twisting, and widening, the medial parameters are not elements of a Euclidean vector space. They are in fact elements of a nonlinear Riemannian symmetric space. In this paper, we develop the method of principal geodesic analysis, a generalization of principal component analysis to the manifold setting. We demonstrate its use in describing the variability of medially-defined anatomical objects. Results of applying this framework on a population of hippocampi in a schizophrenia study are presented.

We describe some techniques that can be used to represent and detect deformable shapes in images. The main di#culty with deformable template models is the very large or infinite number of possible non-rigid transformations of the templates. This makes the problem of finding an optimal match of a deformable template to an image incredibly hard. Using a new representation for deformable shapes we show how to e#ciently find a global optimal solution to the non-rigid matching problem. The representation is based on the description of objects using triangulated polygons. Our matching algorithm can minimize a large class of energy functions, making it applicable to a wide range of problems. We present experimental results of detecting shapes in medical images and images of natural scenes. Our method does not depend on initialization and is very robust, yielding good matches even in images with high clutter.

The goal of this paper is to study a mathematical framework of 2D object shape modeling and learning for middle level vision problems, such as image segmentation and perceptual organization. For this purpose, we pursue generic shape models which characterize the most common features of 2D object shapes. In this paper, shape models are learned from observed natural shapes based on a minimax entropy learning theory (Zhu and Mumford 1997, Zhu, Wu and Mumford 1997)[31, 32]. The learned shape models are Gibbs distributions dened on Markov random elds (MRFs). The neighborhood structures of these MRFs correspond to Gestalt laws {co-linearity, co-circularity, proximity, parallelism, and symmetry. Thus both contour-based and region-based features are accounted for. Stochastic Markov chain Monte Carlo (MCMC) algorithms are proposed for learning and model verication. Furthermore, this paper provides a quantitative measure for the so-called non-accidental statistics, and thus justies some empi...

A new method of model registration is proposed using graphical templates. A graph of landmarks is chosen in the template image. All possible candidates for these landmarks are found in the data image using local operators. A dynamic programming algorithm on decomposable subgraphs of the template graph finds the optimal match to a subset of the candidate points in polynomial time. This combination of local operators to describe points of interest/landmarks and a graph to describe their geometric orientation in the plane, yields fast and precise matches of the model to the data, with no initialization required. Key words: Graphical templates, decomposable graphs, model registration, dynamic programming, image matching. Research supported in part by The University of Chicago Block Fund, and ARO DAAL0392 -G-0322. y Research supported in part by National Institutes of Health grant no. R01-GM46800 x1 Introduction In recent years there has been a growing interest in deformable models for ...

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