APPLICATIONS OF PLANAR SHAPE ANALYSIS TO IMAGE-BASED INFERENCES Anuj Srivastava * , Shantanu Joshi ** , David Kaziska * , and David Wilson *** * Department of Statistics, Florida State University, Tallahassee, FL ** Department of Eletrical Engineering, Florida State University, Tallahassee, FL *** Department of Mathematics, University of Florida, Gainesville, FL ABSTRACT We describe an approach for statistical analysis of shapes of closed curves using tools from differential geometry. This approach uses geodesic paths to define a metric on shape space, that is used to compare shapes, to compute intrin- sic statistics for a set of shapes, and to define probability models on shape spaces. We demonstrate this approach us- ing: (i) interpolation of heart-wall boundaries in echocar- diographic image sequences and (ii) a study of shapes of human silhouettes in infrared surveillance images. 1. INTRODUCTION Detection, extraction and recognition of objects in an image is an important area of research. Objects can be character- ized using a variety of features: textures, edges, boundaries, colors, motion, shapes, locations, etc. Shape often provides an important clue for determining how an object appears in an image. For example, we have displayed the images of three animals in the top panels of Figure 1. The lower panels show the silhouettes of these animals in the corresponding images. It is easy to see that the shapes of these silhou- ettes can help shortlist, or even identify, the animals present in these images. Tools for shape analysis can prove impor- tant in several applications including medical image analy- sis, human surveillance, military target recognition, finger- print analysis, space exploration, and underwater search. A significant part of the past efforts has been restricted to “landmark-based” analysis, where shapes are represented by a coarse, discrete sampling of the object contours [1]. A recent approach [2] considers the shapes of continuous, closed curves in R 2 . In this paper we describe two appli- cations of this idea: First, we look at a problem in ecocar- diographic image analysis where shapes of epicardial and endocardial boundaries are studied to determine the extent and progression of disease in a patient’s heart. We focus on This research was supported in part by the grants NSF (FRG) DMS- 0101429, NSF (ACT) DMS-0345242, and ARO W911NF-04-01-0268. 50 100 150 200 250 300 350 400 450 500 550 50 100 150 200 250 300 350 50 100 150 200 250 300 350 400 450 500 550 50 100 150 200 250 300 350 50 100 150 200 250 300 350 400 450 500 550 50 100 150 200 250 300 350 400 450 500 Fig. 1. Analysis of shapes of objects’ boundaries in images can help in computer vision tasks such as object recognition. the specific problem of interpolating these boundaries in im- age sequences when an expert provides contours for the first and last frames in the sequence. Secondly, we will present an application involving human surveillance with a goal of detecting humans in low-quality night-vision (infrared) im- ages. Our goal here is to build a statistical model to capture human shapes. The rest of this chapter is organized as follows. In Sec- tion 2 we summarize past work on differential-geometric representation of shapes. In the next two sections we de- scribe two applications of this approach. 2. A FRAMEWORK FOR PLANAR SHAPE ANALYSIS The basic idea presented in [2] is to identify a space of closed curves, remove shape-preserving transformations from it, impose a Riemannian structure on it, and treat the result- ing quotient space as the shape space. 1. A Geometric Representation of Shapes: Consider the boundaries or silhouettes of the imaged objects as closed, planar curves in R 2 , parameterized by the arc length. De- note by θ(s) the angle made by the velocity vector with the positive x-axis, as a function of arc length s. Coordinate function α(s) relates to the angle function θ(s) according to ˙ α(s)= e jθ(s) , j = √ −1, with an example in Figure 2. We choose angle functions to represent and analyze shapes. To V - 1037 0-7803-8874-7/05/$20.00 ©2005 IEEE ICASSP 2005 ➠ ➡