Analysis of Planar Shapes Using Geodesic Paths on Shape Spaces Eric Klassen, Anuj Srivastava, Washington Mio, and Shantanu H. Joshi Abstract—For analyzing shapes of planar, closed curves, we propose differential geometric representations of curves using their direction functions and curvature functions. Shapes are represented as elements of infinite-dimensional spaces and their pairwise differences 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. Using the Surrey fish database, we demonstrate some applications of this approach: 1) interpolation and extrapolations of shape changes, 2) clustering of objects according to their shapes, 3) statistics on shape spaces, and 4) Bayesian extraction of shapes in low-quality images. Index Terms—Shape metrics, geodesic paths, shape statistics, intrinsic mean shapes, shape-based clustering, shape interpolation. æ 1 INTRODUCTION S HAPES play a pivotal role in understanding objects in terms of their behavior and characteristics, such as their growth, health, identity, and functionality. Quantitative characterization of shapes is emerging as a major area of research, which will impact diverse applications. Despite a pressing need for analyzing shapes in many problems, the current methods are limited in their scope and perfor- mance. Although shapes are frequently referred to in the literature, consistent mathematical treatments of shapes are relatively limited. Only a limited number of papers provide specific definitions of shapes or shape spaces, or follow it up with a statistical analysis. It is noteworthy that the notion of shape exists in many branches of science with different meanings attached to it. Although the existence of these diverse notions of shape may be a reason behind the absence of formal treatments, a lack of sophisticated mathematical tools is also an important factor. Among the papers that explicitly study shapes, a major limitation in many of them is the use of landmarks to define shapes. Shapes are often encoded by a coarse sampling of the objects’ boundaries, and the outcome and accuracy of the ensuing shape analysis is heavily dependent on the choices made. In addition, it is usually difficult to automate the selection of these landmarks. A more fundamental approach is to represent the continuous boundaries as curves, and then study their shapes. (Of course, any computer implementation will require an eventual discre- tization, but there are distinct advantages to the philosophy of discretizing as late as possible.) However, this approach requires dealing with infinite-dimensional Riemannian manifolds, spaces for which tools such as optimization, random sampling, and hypothesis testing are frequently not available. Our goal is to develop mathematical formula- tions, optimization strategies, and statistical procedures to fundamentally address the outstanding issues in the study of continuous, planar shapes. 1.1 Motivations for Shape Analysis Tools for efficient shape analysis will impact many areas such as computer vision, structural genomics, medical imaging, and computational topology. The issue of representing, analyzing, estimating, and tracking shapes is central to many problems in these applications. Recognition of objects using observed images is also a well publicized problem in computer vision. Images provide information about shapes of the objects and reflectance functions (textures) associated with the objects’ surfaces. Analyzing the shapes of contours can provide important clues about the identities of the objects. For instance, an algorithm for analyzing shapes can help automate recognition of marine animals using contours of their appearances in images. In practice, where shapes are to be inferred from low-quality data, statistical formulations, and inferences become very important. The importance of statistical inferences in general computer vision is well documented, although one needs to emphasize the same need for a comprehensive statistical analysis of shapes. From computing averages and variations of given shapes to testing shape hypotheses from given data, standard tools for statistical analysis need to be extended to formal shape spaces. Also, note that the scope of any shape theory need not be restricted to image analysis. Although image under- standing forms an important application of shape analysis, a theory should be more general in its scope. 1.2 Past Work in Shape Analysis Historically, there have been many exemplary efforts in characterization and quantification of shapes. Efforts by Kendall [14], Bookstein [2], Dryden and Mardia [7], and Kent and Mardia [15] have resulted in an elegant statistical 372 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 26, NO. 3, MARCH 2004 . E. Klassen and W. Mio are with the Department of Mathematics, Florida State University, Tallahassee, FL 32306. E-mail: {klassen, mio}@math.fsu.edu. . A. Srivastava is with the Department of Statistics, Florida State University, Tallahassee, FL 32306. E-mail: anuj@stat.fsu.edu. . S.H. Joshi is with the Department of Electrical Engineering, Florida State University, Tallahassee, FL 32306. E-mail: joshi@eng.fsu.edu. Manuscript received 30 May 2002; revised 3 Mar. 2003; accepted 9 Oct. 2003. 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