Comment

We enjoyed reading this article on human recognition based on gait using shape space theory. Having attended the recently held SAMSI workshop on statistical analysis of shapes, we appreciate the need to bring together mathematicians, statisticians, and computer vision researchers. Before we begin a discussion of the article, we would like to provide a brief history of shape analysis in the image analysis and computer vision community. Analysis of planar shapes has a long history in image analysis and computer vision dating back to the early 1960s, when moments were introduced as shape descriptors (Hu 1962). In the early 1970s, Fourier descriptors of several types (Persoon and Fu 1977; Zahn and Roskies 1972; Pavlidis 1978) and extensions of moments (Khotanzad and Hong 1990) were introduced. Stochastic descriptions of shapes were introduced by Kashyap and Chellappa (1981) using the notion of circular autoregressive (CAR) models, a special case of cyclostationary process used in the article under discussion. The CAR models were used for OCR applications by Dubois and Glanz (1986). Some of the shape descriptors (Zahn and Roskies 1972) were naturally invariant to transformations such as rotation, scaling, and translation, whereas for many others, simple (Kashyap and Chellappa 1981) to complicated transformations (Khotanzad and Hong 1990) of the shape parameters were used. Recently, the procrustes matching algorithm has been extended to tackle small errors in feature correspondence using the softassign Procrustes matching algorithm (Rangarajan, Chui, and Bookstein 1997) which simultaneously establishes correspondences and determines the Procrustes fit. Another approach for shape matching produces a dense shape descriptor called the shape-context (Belongie, Malik, and Puzicha 2002) which captures the distribution of all pairs of points on the boundary of a shape. This method has recently been extended in order to make the shape-context feature invariant to articulations by using the inner-distance shape context (Ling and Jacobs 2007). In passing, we would like to note that the representation used by the authors is similar to the tangent angle versus arc length used by Zahn and Roskies (1972). Some of these descriptors are sensitive to where the origin of tracing is placed, as in the representation used by Kaziska and Srivastava. Because most image analysis and computer vision researchers have not been exposed to differential geometry, shape-matching techniques have been derived using linear techniques and Euclidean metrics. The notion of using statistical shape theory to analyze gait sequences has been of recent interest. Landmark-based descriptions were considered by Vaswani, Roy-Chowdhury, and