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"abstract": "Accurate curvature estimation in discrete surfaces is an important problem with numerous applications. Curvature is an indicator of ridges and can be used in applications such as shape analysis and recognition, object segmentation, adaptive smoothing, anisotropic fairing of irregular meshes, and anisotropic texture mapping. In this paper, a new framework is proposed for accurate curvature estimation in discrete surfaces. The proposed framework is based on a local directional curve sampling of the surface where the sampling frequency can be controlled. This local model has a large number of degrees of freedoms compared with known techniques and, so, can better represent the local geometry. The proposed framework is quantitatively evaluated and compared with common techniques for surface curvature estimation. In order to perform an unbiased evaluation in which smoothing effects are factored out, we use a set of randomly generated Bezier surface patches for which the curvature values can be analytically computed. It is demonstrated that, through the establishment of sampling conditions, the error in estimations obtained by the proposed framework is smaller and that the proposed framework is less sensitive to low sampling density, sampling irregularities, and sampling noise.",
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"normalizedAbstract": "Mesh parameterization is a fundamental technique in computer graphics. Our paper focuses on solving the problem of finding the best discrete conformal mapping that also minimizes area distortion. Firstly, we deduce an exact analytical differential formula to represent area distortion by curvature change in the discrete conformal mapping, giving a dynamic Poisson equation. Our result shows the curvature map is invertible. Furthermore, we give the explicit Jacobi matrix of the inverse curvature map. Secondly, we formulate the task of computing conformal parameterizations with least area distortions as a constrained nonlinear optimization problem in curvature space. We deduce explicit conditions for the optima. Thirdly, we give an energy form to measure the area distortions, and show it has a unique global minimum. We use this to design an efficient algorithm, called free boundary curvature diffusion, which is guaranteed to converge to the global minimum. This result proves the common belief that optimal parameterization with least area distortion has a unique solution and can be achieved by free boundary conformal mapping. Major theoretical results and practical algorithms are presented for optimal parameterization based on the inverse curvature map. Comparisons are conducted with existing methods and using different energies. Novel parameterization applications are also introduced.",
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"content": "Abstract—This paper proposes a novel hypergraph skeletal representation for 3D shape based on a formal derivation of the generic structure of its medial axis. By classifying each skeletal point by its order of contact, we show that, generically, the medial axis consists of five types of points, which are then organized into sheets, curves, and points: 1) sheets (manifolds with boundary) which are the locus of bitangent spheres with regular tangency A_1^2 (A_k^n notation means n distinct k{\\hbox{-}}{\\rm{fold}} tangencies of the sphere of contact, as explained in the text); two types of curves, 2) the intersection curve of three sheets and the locus of centers of tritangent spheres, A_1^3, and 3) the boundary of sheets, which are the locus of centers of spheres whose radius equals the larger principal curvature, i.e., higher order contact A_3 points; and two types of points, 4) centers of quad-tangent spheres, A_1^4, and 5) centers of spheres with one regular tangency and one higher order tangency, A_1A_3. The geometry of the 3D medial axis thus consists of sheets (A_1^2) bounded by one type of curve (A_3) on their free end, which corresponds to ridges on the surface, and attached to two other sheets at another type of curve (A_1^3), which support a generalized cylinder description. The A_3 curves can only end in A_1A_3 points where they must meet an A_1^3 curve. The A_1^3 curves meet together in fours at an A_1^4 point. This formal result leads to a compact representation for 3D shape, referred to as the medial axis hypergraph representation consisting of nodes (A_1^4 and A_1 A_3 points), links between pairs of nodes (A_1^3 and A_3 curves) and hyperlinks between groups of links (A_1^2 sheets). The description of the local geometry at nodes by itself is sufficient to capture qualitative aspects of shapes, in analogy to 2D. We derive a pointwise reconstruction formula to reconstruct a surface from this medial axis hypergraph together with the radius function. Thus, this information completely characterizes 3D shape and lays the theoretical foundation for its use in recognition, morphing, design, and manipulation of shapes.",
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"abstract": "A new method for finding the locus of parallel vectors is presented, called PVsolve. A parallel-vector operator has been proposed as a visualization primitive, as several features can be expressed as the locus of points where two vector fields are parallel. Several applications of the idea have been reported, so accurate and efficient location of such points is an important problem. Previously published methods derive a tangent direction under the assumption that the two vector fields are parallel at the current point in space, then extend in that direction to a new point. PVsolve includes additional terms to allow for the fact that the two vector fields may not be parallel at the current point, and uses a root-finding approach. Mathematical analysis sheds new light on the feature flow field technique (FFF) as well. The root-finding property allows PVsolve to use larger step sizes for tracing parallel-vector curves, compared to previous methods, and does not rely on sophisticated differential equation techniques for accuracy. Experiments are reported on fluid flow simulations, comparing FFF and PVsolve.",
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"fullName": "Allen Van Gelder",
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"abstract": "This article proposes HaptoMapping, a projection-based visuo-haptic augmented reality (VHAR) system, that can render visual and haptic content independently and present consistent visuo-haptic sensations on physical surfaces. HaptoMapping controls wearable haptic displays by embedded control signals that are imperceptible to the user in projected images using a pixel-level visible light communication technique. The prototype system is comprised of a high-speed projector and three types of haptic devices—finger worn, stylus, and arm mounted. The finger-worn and stylus devices present vibrotactile sensations to a user's fingertips. The arm-mounted device presents stroking sensations on a user's forearm using arrayed actuators with a synchronized hand projection mapping. We identified that the developed system's maximum latency of haptic from visual sensations was 93.4 ms. We conducted user studies on the latency perception of our VHAR system. The results revealed that the developed haptic devices can present haptic sensations without user-perceivable latencies, and the visual-haptic latency tolerance of our VHAR system was 100, 159, 500 ms for the finger-worn, stylus, and arm-mounted devices, respectively. Another user study with the arm-mounted device discovered that the visuo-haptic stroking system maintained both continuity and pleasantness when the spacing between each substrate was relatively sparse, such as 20 mm, and significantly improved both the continuity and pleasantness at 80 and 150 mm/s when compared to the haptic only stroking system. Lastly, we introduced four potential applications in daily scenes. Our system methodology allows for a wide range of VHAR application design without concern for latency and misalignment effects.",
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