Vis Comput (2011) 27: 429–439 DOI 10.1007/s00371-011-0582-y ORIGINAL ARTICLE Multi-scale anisotropic heat diffusion based on normal-driven shape representation Shengfa Wang · Tingbo Hou · Zhixun Su · Hong Qin Published online: 19 April 2011 © Springer-Verlag 2011 Abstract Multi-scale geometric processing has been a pop- ular and powerful tool in graphics, which typically em- ploys isotropic diffusion across scales. This paper proposes a novel method of multi-scale anisotropic heat diffusion on manifold, based on the new normal-driven shape represen- tation and Edge-weighted Heat Kernels (EHK). The new shape representation, named as Normal-Controlled Coordi- nates (NCC), can encode local geometric details of a vertex along its normal direction and rapidly reconstruct surface geometry. Moreover, the inner product of NCC and its corre- sponding vertex normal, called Normal Signature (NS), de- fines a scalar/heat field over curved surface. The anisotropic heat diffusion is conducted using the weighted heat kernel convolution governed by local geometry. The convolution is computed iteratively based on the semigroup property of heat kernels toward accelerated performance. This dif- fusion is an efficient multi-scale procedure that rigorously conserves the total heat. We apply our new method to multi- scale feature detection, scalar field smoothing and mesh de- noising, and hierarchical shape decomposition. We conduct various experiments to demonstrate the effectiveness of our method. Our method can be generalized to handle any scalar field defined over manifold. Keywords Normal-Controlled Coordinates · Normal signature · Anisotropic diffusion · Edge-weighted heat kernel · Multi-scale feature extraction S. Wang ( ) · Z. Su Dalian University of Technology, Dalian 116024, China e-mail: shengfawang@gmail.com S. Wang · T. Hou · H. Qin Stony Brook University (SUNY), Stony Brook, NY 11794-4400, USA 1 Introduction Recently, heat kernels and their utilities in diffusion start to gain momentum for geometric information processing in graphics, with applications in various research prob- lems, including multi-scale feature detection [31], scalar field smoothing [26], automatic diffusion [8], shape match- ing [25], shape retrieval [2], etc. A typical diffusion process is conducted by convoluting an initial scalar/heat field with heat kernels. The advantage of such process is that both dif- fusion and its kernel function afford robust multi-scale pro- cessing on manifold, with intrinsic property of isometric- invariance. While these techniques have shown promise in multi-scale shape analysis, there still remain certain limi- tations in the current state-of-the-art, including initial heat field design, anisotropic diffusion, short-time scale behav- ior, improved performance, etc. First, existing work oftentimes emphasizes the compre- hensive studies of kernel functions and their properties, while paying far less attention to the initial heat design. The initial field is a scalar function defined on the surface at time t = 0. It will gradually diffuse on the surface along t , result- ing in different scales. The initial field is frequently assigned by using some simple characteristics such as curvature [20], texture [13], or other surface measurements [28, 32]. These characteristics lack sufficient information to describe and re- construct the shape. Moreover, they are sensitive to scale changes, which goes against the scale-invariant nature of diffusion. To better depict the characteristics of a surface, an informative and stable initial field is strongly desirable. Sec- ond, all the previous heat diffusions are isotropic in nature, which are based on isotropic heat kernels on manifold. Yet, anisotropic diffusion is much more desirable in many cases, such as smoothing and feature finding on surfaces with sharp edges. Anisotropic diffusion, which is much more power-