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Chapter 4: Algorithm Overview 43implementation of Guillaume’s algorithm [Guillaume, 2004]. He presents a moreefficient algorithm for the decomposition of 3D arbitrary triangle meshes into surfacepatches. The algorithm is based on the curvature tensor field analysis and presents twodistinct complementary steps: a region-based segmentation, which decomposes theobject into known and near constant curvature patches and a boundary rectificationbased on curvature tensor directions, which correct boundaries by suppressing theirartifacts and discontinuities.We then analyze each surface patch individually to compute the CVM, which is theentropy of curvature. We compute the Gaussian curvature on each of those surfacepatches. The kernel density of the Gaussian curvature is estimated. We optimize thebandwidth of the kernel density using the plug-in method to ensure stability in theresolution normalized entropy. This log scale measure from the curvature density isthe curvature variation measure (CVM). We then combine the surface connectivityinformation and the curvature variation measure into a single graph representation.We call our CVM algorithm a curvature-based approach because the segmentation andthe description require computation of curvature. (Curvedness is a function ofprincipal curvatures.) However, the surface variation measure that we describe is notinvariant to scale. We would like to emphasize that our algorithm can be used fordescribing occluded scenes as well but at the cost of partial graph matching if we haveto attempt object recognition.4.2 Building Blocks of the CVM algorithmAs background, we first present a brief overview of surface curvature in the importantcontext of differential geometry. We in particular deal with curvature of a surface inSection 4.2.1 because we have assumed that curvature intrinsically describes the localshape of that surface. The differential geometry section helps us understand curvatureestimation on triangle meshes. We present a brief survey on curvature estimationtechniques in Section 4.2.2 and then discuss the theory behind the other buildingblocks of the algorithm in Section 4.2.3 and Section 4.2.4.4.2.1 Differential Geometry of Curves and SurfacesFirst, let us consider the continuous case for 2D curves. Using [Carmo, 1976], wearbitrarily define a planar curve α: I R 2 parameterized by arc length s such that wehave α(s). We carefully choose, without loss of generality, this parameterization suchthat the vector field T = α’ has unit length. With this construction, the derivative T’ =α" measures the way the curve is turning in R 2 and we term T’ the curvature vectorfield. Since T’ is always orthogonal to T, that is normal to, we can write that T’=κN