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SPM 2007, Beijing, China, June 04 – 06, 2007. © 2007 ACM 978-1-59593-666-0/07/0006 $5.00 Salient Critical Points for Meshes Yu-Shen Liu * Min Liu † Daisuke Kihara † Karthik Ramani † Purdue University, West Lafayette, Indiana, USA (a) (b) (c) (d) Figure 1: Salient critical points (the blue, red, and green points are minimum, maximum, and saddles, respectively). (a) The back of the lion head model with large noise and small hair textures, and the corresponding mean curvature visualization. (b) The mean curvature function yields 7,629 critical points due to the curvature function’s sensitivity to noise. (c) The corresponding mesh saliency. (d) Salient critical points with lower number. Since mesh saliency in (c) captures the hair texture and negates the noisy curvature in (a), our method based on saliency selects the more interesting critical points in the important region. In color images shown in this paper, warmer colors (reds and yellows) show high curvature or saliency and cooler colors (blues) show low curvature or saliency. Abstract A novel method for extracting the salient critical points of meshes, possibly with noise, is presented by combining mesh saliency with Morse theory. In this paper, we use the idea of mesh saliency as a measure of regional importance for meshes. The proposed method defines the salient critical points in a scalar function space using a center-surround filter operator on Gaussian-weighted average of the scalar of vertices. Compared to using a purely geometric measure of shape, such as curvature, our method yields more satisfactory results with the lower number of critical points. We demonstrate the effectiveness of this approach by comparing our results with the results of the conventional approaches in a number of examples. Furthermore, this work has a variety of potential applications. We give a direct application to the hierarchical topological represen- tation for meshes by combining the salient critical points with the Morse-Smale complex. Keywords: critical points, saliency, Morse theory 1 Introduction Morse theory is a powerful mathematical tool for determining the topology of a manifold from the critical points of one suitable scalar function on the manifold. Recently, discrete Morse theory * e-mail: liuyushen00@gmail.com † e-mail: {liu66, dkihara, ramani}@purdue.edu [Banchoff 1970; Edelsbrunner et al. 2003] on a triangulated mani- fold has also became an active research area in computer graphics and computational geometry using different real scalar functions [Edelsbrunner et al. 2003; Bremer et al. 2004; Dong et al. 2006; Natarajan et al. 2006; Ni et al. 2004]. In discrete Morse theory, ex- tracting the critical points of 3D meshes is an important problem. However, a poor choice of this real function can lead to a com- plex configuration of a high number of critical points due to noise. In addition, some salient critical points on important regions might also be missed when some methods of smoothing (or fairing) Morse function are used. Recently, mesh saliency [Lee et al. 2005], as a measure of regional importance for meshes, has been derived from 2D image techniques. In this paper, we focus on the problem of extracting the salient critical points of meshes by combining mesh saliency with Morse theory. Morse theory, which was originally devised for smooth functions on manifolds, connects the differential geometry of a surface with its algebraic topology. Morse theory has been extended to piecewise linear functions on triangulated meshes [Banchoff 1970]. Given a real-valued function over some shape, discrete Morse theory de- scribes the connectedness of the shape from the configuration of the points where the function’s gradient vanishes, its so-called crit- ical points (e.g. minima, maxima, saddles). Extracting the critical points of 3D meshes is an important problem in discrete Morse the- ory. Some applications are strongly dependent on the quality of the critical points, such as quadrilateral remeshing [Dong et al. 2006] and surface segmentation [Natarajan et al. 2006]. However, there are still two problems in extracting critical points from meshes due to noise. One problem is that a poor choice of Morse function can yield many more critical points due to the scalar function’s sensitiv- ity to surface noise [Ni et al. 2004]. Figure 1(b) shows an example for the lion head model with large noise, in which a mean curvature function yields 7,629 critical points. These extra critical points are caused by the poor curvature function’s sensitivity to noise and hair textures on the back of this model. The other problem is that some 277