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Mesh parameterization is a key problem in digital geometry processing. By cutting a surface along a set of edges (a seam), one can map an arbitrary topology surface mesh to a single chart. Unfortunately, high distortion occurs when protrusions of the surface (such as fingers of a hand and horses’ legs) are flattened into a plane. This paper presents a novel skeleton-based algorithm for computing a...
We define a parameterization for an n-holed tori based on the hyperbolic polygon. We model the domain using a manifold with 2n+2 charts, and linear fractional transformations for transition functions. We embed the manifold using standard spline techniques to produce a surface.
This paper shows how difficulties with singularities encountered in the geometry of bifurcating surfaces can be overcome and the shape parameterized efficiently enabling a designer to create and manipulate such geometries within an interactive environment.
In this paper, we present general closed form equations for directly computing the position of a vertex at different subdivision levels for both triangular and quadrilateral meshes. These results are obtained using simple computations and they lead to very useful applications, especially for adaptive subdivision. We illustrate our method on Loop’s and Catmull-Clark’s subdivision schemes.
We discuss an optimisation procedure for triangulated polyhedral surfaces (referred to as (2-3)D triangulations) which allows us to process self–intersecting surfaces. As an optimality criterion we use minimisation of total absolute extrinsic curvature (MTAEC) and as a local transformation – a diagonal flip, defined in a proper way for (2-3)D triangulations. This diagonal flip is a natural generalisation...
In this paper we study subdivision from a graph-theoretic point of view. In particular, we study the chromatic numbers of subdivision meshes, that is the number of distinct colors we need for a vertex, face or edge coloring of a subdivision mesh. We show that, unlike the size, the chromatic numbers of subdivision meshes are not larger than the corresponding chromatic numbers of the initial mesh and...
Watermarking is to embed a structure called a watermark into the target data such as images. The watermark can be used, for example, in order to secure the copyright and detect tampering. This paper presents a new robust watermarking method that adds a watermark into a 3D polygonal mesh in the spectral domain. In this algorithm, a shape of a 3D polygonal model is regarded as a sequence of vertices...
This paper proposes a new method for mesh data compression. In this method, the original mesh is fitted by a subdivision surface. Thus, our method approximates irregular meshes to semi-regular meshes. The volume bounded by the mesh data is first partitioned into star-shaped volumes, and then each star-shaped volume is approximated. For the approximation we establish a nearly one-to-one correspondence...
Current scan technologies provide huge data sets which have to be processed considering several application constraints. The different steps required to achieve this purpose use a structured approach where fundamental tasks, e.g. surface reconstruction, multi-resolution simplification, smoothing and editing, interact using both the input mesh geometry and topology. This paper is twofold; firstly,...
Within this paper a technique for model-based 3D hand tracking is presented. A hand model is built from a set of truncated quadrics, approximating the anatomy of a real hand with few parameters. Given that the projection of a quadric onto the image plane is a conic, the contours can be generated efficiently. These model contours are used as shape templates to evaluate possible matches in the current...
In this paper we describe a new shape-from-shading method. We show how the parallel transport of surface normals can be used to impose curvature consistency and also to iteratively update surface normal directions so as to improve the brightness error. We commence by showing how to make local estimates of the Hessian matrix from surface normal information. With the local Hessian matrix to hand, we...
This paper describes a graph-spectral method for 3D surface integration. The algorithm takes as its input a 2D field of surface normal estimates, delivered, for instance, by a shape-from-shading or shape-from-texture procedure. We commence by using the Mumford-Shah energy function to obtain transition weights for pairs of sites in the field of surface normals. The weights depend on the sectional curvature...
This paper presents a more robust reconstruction algorithm to solve the genus restriction of displaced subdivision surface (DSS) from unorganized points. DSS is a useful mesh representation to guarantee the memory efficiency by storing a vertex position as one scalar displacement value, which is measured from the original mesh to its parametric domain. However, reconstructing DSS from unorganized...
Laser scans of real objects produce data sets (point clouds) which may have holes, due to problems with visibility or with the optical properties of the surface. We describe a method for detecting and filling these holes. After detecting the boundary of the hole, we fit an algebraic surface patch to the neighbourhood and sample auxiliary points. The method is able to reproduce technically important...
The problem of detecting and eliminating self-intersections in offset curves is a fundamental question that has attracted numerous researchers over the years. The interest has resulted in copious publications on the subject. Unfortunately, the detection of self-intersections in offset curves, and more so, the elimination of these self-intersections are difficult problems with less than satisfactory...
Surfaces in line space are called line congruences. We consider several special line congruences forming a fibration of the three–dimensional space. These line congruences correspond to certain special algebraic surfaces. Using rational mappings associated with the line congruences, it is possible to generate rational curves and surfaces on them. This approach is demonstrated for quadric surfaces,...
In their seminal paper Boehm et al [3] show how box splines over regular bivariate grids are defined by coefficients (control points) associated with centres outside the region being defined (“phantom points”), as well as with those inside. If we apply the pure subdivision rules derived from the box splines, this means that the configuration shrinks at every step from the original coarse lattice,...
We study the Plateau problem restricted to polynomial surfaces using techniques coming from the theory of Computer Aided Geometric Design. The results can be used to obtain polynomial approximations to minimal surfaces. The relationship between harmonic Bézier surfaces and minimal surfaces with free boundaries is shown.
The paper considers a particular general class of parametrised path functions used in computer graphics, geometric modeling and approximation theory for the construction of curves and surfaces. General methods are developed for the identification of the conditions under which parameter transformations preserve the path geometry. The determination of these ‘parameter symmetries’ is shown to be equivalent...
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