Summary. This paper provides a tutorial and survey of methods for parameterizing surfaces with a view to applications in geometric modelling and computer graphics. We gather various concepts from differential geometry which are relevant to surface mapping and use them to understand the strengths and weaknesses of the many methods for parameterizing piecewise linear surfaces and their relationship to one another. 1

Abstract—We developed a general method for global conformal parameterizations based on the structure of the cohomology group of holomorphic one-forms for surfaces with or without boundaries (Gu and Yau, 2002), (Gu and Yau, 2003). For genus zero surfaces, our algorithm can find a unique mapping between any two genus zero manifolds by minimizing the harmonic energy of the map. In this paper, we apply the algorithm to the cortical surface matching problem. We use a mesh structure to represent the brain surface. Further constraints are added to ensure that the conformal map is unique. Empirical tests on magnetic resonance imaging (MRI) data show that the mappings preserve angular relationships, are stable in MRIs acquired at different times, and are robust to differences in data triangulation, and resolution. Compared with other brain surface conformal mapping algorithms, our algorithm is more stable and has good extensibility. Index Terms—Brain mapping, conformal map, landmark matching, spherical harmonic transformation. I.

Good parameterizations are of central importance in many digital geometry processing tasks. Typically the behavior of such processing algorithms is related to the smoothness of the parameterization and how much distortion it contains, i.e., how rapidly the derivatives of the parameterization change. Since a parameterization maps a bounded region of the plane to the surface, a parameterization for a surface which is not homeomorphic to a disc must be made up of multiple pieces. We present a novel parameterization algorithm for arbitrary topology surface meshes which computes a globally smooth parameterization with low distortion. We optimize the patch layout subject to criteria such as shape quality and parametric distortion, which are used to steer a mesh simplification approach for base complex construction. Global smoothness is achieved through simultaneous relaxation over all patches, with suitable transition functions between patches incorporated into the relaxation procedure. We demonstrate the quality of our parameterizations through numerical evaluation of distortion measures; the rate distortion behavior of semi-regular remeshes produced with these parameterizations; and a comparison with globally smooth subdivision methods. The numerical algorithms required to compute the parameterizations are robust and run on the order of minutes even for large meshes.

We consider the problem of creating a map between two arbitrary triangle meshes. Whereas previous approaches compose parametrizations over a simpler intermediate domain, we directly create and optimize a continuous map between the meshes. Map distortion is measured with a new symmetric metric, and is minimized during interleaved coarse-to-fine refinement of both meshes. By explicitly favoring low inter-surface distortion, we obtain maps that naturally align corresponding shape elements. Typically, the user need only specify a handful of feature correspondences for initial registration, and even these constraints can be removed during optimization. Our method robustly satisfies hard constraints if desired. Inter-surface mapping is shown using geometric and attribute morphs. Our general framework can also be applied to parametrize surfaces onto simplicial domains, such as coarse meshes (for semi-regular remeshing), and octahedron and toroidal domains (for geometry image remeshing). In these settings, we obtain better parametrizations than with previous specialized techniques, thanks to our fine-grain optimization.

In this paper, we propose a new quadrilateral remeshing method for manifolds of arbitrary genus that is at once general, flexible, and efficient. Our technique is based on the use of smooth harmonic scalar fields defined over the mesh. Given such a field, we compute its gradient field and a second vector field that is everywhere orthogonal to the gradient. We then trace integral lines through these vector fields to sample the mesh. The two nets of integral lines together are used to form the polygons of the output mesh. Curvature-sensitive spacing of the lines provides for anisotropic meshes that adapt to the local shape. Our scalar field construction allows users to exercise extensive control over the structure of the final mesh. The entire process is performed without computing an explicit parameterization of the surface, and is thus applicable to manifolds of any genus without the need for cutting the surface into patches. Key words: quad-dominant remeshing, gradient flow tracing, harmonic fields, harmonic 1-forms 1

Figure 1: Deformation using Moving Least Squares. Original image with control points shown in blue (a). Moving Least Squares deformations using affine transformations (b), similarity transformations (c) and rigid transformations (d). We provide an image deformation method based on Moving Least Squares using various classes of linear functions including affine, similarity and rigid transformations. These deformations are realistic and give the user the impression of manipulating real-world objects. We also allow the user to specify the deformations using either sets of points or line segments, the later useful for controlling curves and profiles present in the image. For each of these techniques, we provide simple closed-form solutions that yield fast deformations, which can be performed in real-time.

We introduce a novel method for the construction of discrete conformal mappings from (regions of) embedded meshes to the plane. Our approach is based on circle patterns, i.e., arrangements of circles---one for each face---with prescribed intersection angles. Given these angles the circle radii follow as the unique minimizer of a convex energy. The method has two principal advantages over earlier approaches based on discrete harmonic mappings: (1) it supports very flexible boundary conditions ranging from natural boundaries to control of the boundary shape via prescribed curvatures; (2) the solution is based on a convex energy as a function of logarithmic radius variables with simple explicit expressions for gradients and Hessians, greatly facilitating robust and efficient numerical treatment. We demonstrate the versatility and performance of our algorithm with a variety of examples.

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We introduce an algorithm for automatic computation of global parameterizations on arbitrary simplicial 2-manifolds whose parameter lines are guided by a given frame field, for example by principal curvature frames. The parameter lines are globally continuous, and allow a remeshing of the surface into quadrilaterals. The algorithm converts a given frame field into a single vector field on a branched covering of the 2-manifold, and generates an integrable vector field by a Hodge decomposition on the covering space. Except for an optional smoothing and alignment of the initial frame field, the algorithm is fully automatic and generates high quality quadrilateral meshes.

The Willmore energy of a surface, � (H 2 − K)dA, as a function of mean and Gaussian curvature, captures the deviation of a surface from (local) sphericity. As such this energy and its associated gradient flow play an important role in digital geometry processing, geometric modeling, and physical simulation. In this paper we consider a discrete Willmore energy and its flow. In contrast to traditional approaches it is not based on a finite element discretization, but rather on an ab initio discrete formulation which preserves the Möbius symmetries of the underlying continuous theory in the discrete setting. We derive the relevant gradient expressions including a linearization (approximation of the Hessian), which are required for non-linear numerical solvers. As examples we demonstrate the utility of our approach for surface restoration, n-sided hole filling, and non-shrinking surface smoothing.