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118
Tracing Ray Differentials
, 1999
"... Antialiasing of ray traced images is typically performed by supersampling the image plane. While this type of filtering works well for many algorithms, it is much more efficient to perform filtering locally on a surface for algorithms such as texture mapping. In order to perform this type of filteri ..."
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Cited by 62 (0 self)
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Antialiasing of ray traced images is typically performed by supersampling the image plane. While this type of filtering works well for many algorithms, it is much more efficient to perform filtering locally on a surface for algorithms such as texture mapping. In order to perform this type of filtering, one must not only trace the ray passing through the pixel, but also have some approximation of the distance to neighboring rays hitting the surface (i.e., a ray's footprint). In this paper, we present a fast, simple, robust scheme for tracking such a quantity based on ray differentials, derivatives of the ray with respect to the image plane. CR Categories and Subject Descriptors: I.3.7 [Computer Graphics]: ThreeDimensional Graphics and Realism  color, shading, shadowing, and texture; raytracing. 1 INTRODUCTION Ray tracing [18] is an image generation technique that is able to accurately model many phenomena which are difficult or impossible to produce with a traditional graphics pip...
Explicit Surface Remeshing
, 2003
"... We present a new remeshing scheme based on the idea of improving mesh quality by a series of local modifications of the mesh geometry and connectivity. Our contribution to the family of local modification techniques is an areabased smoothing technique. Areabased smoothing allows the control of bo ..."
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Cited by 59 (6 self)
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We present a new remeshing scheme based on the idea of improving mesh quality by a series of local modifications of the mesh geometry and connectivity. Our contribution to the family of local modification techniques is an areabased smoothing technique. Areabased smoothing allows the control of both triangle quality and vertex sampling over the mesh, as a function of some criteria, e.g. the mesh curvature. To perform local modifications of arbitrary genus meshes we use dynamic patchwise parameterization. The parameterization is constructed and updated onthefly as the algorithm progresses with local updates. As a postprocessing stage, we introduce a new algorithm to improve the regularity of the mesh connectivity. The algorithm is able to create an unstructured mesh with a very small number of irregular vertices. Our remeshing scheme is robust, runs at interactive speeds and can be applied to arbitrary complex meshes.
Partial differential equations and MongeKantorovich mass transfer (surveypaper
 Current Developments in Mathematics, 1997, International Press
, 1999
"... 1.1 Optimal mass transfer 1.2 Relaxation, duality ..."
Illumination from Curved Reflectors
, 1992
"... A technique is presented to compute the reflected illumination from curved mirror surfaces onto other surfaces. In accordance with Fermat's principle, this is equivalent to finding extremal paths from the light source to the visible surface via the mirrors. Once pathways of illumination are fou ..."
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Cited by 56 (0 self)
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A technique is presented to compute the reflected illumination from curved mirror surfaces onto other surfaces. In accordance with Fermat's principle, this is equivalent to finding extremal paths from the light source to the visible surface via the mirrors. Once pathways of illumination are found, irradiance is computed from the Gaussian curvature of the geometrical wavefront. Techniques from optics, differential geometry and interval analysis are applied to solve these problems. CR Categories and Subject Descriptions: I.3.3 [ Computer Graphics ]: Picture/Image Generation; I.3.7 [ Computer Graphics ]: ThreeDimensional Graphics and Realism General Terms: Algorithms Additional Keywords and Phrases: Caustics, Differential Geometry, Geometrical Optics, Global Illumination, Interval Arithmetic, Ray Tracing, Wavefronts 1. Introduction Ray tracing provides a straightforward means for synthesizing realistic images on the computer. A scene is first modeled, usually by a collection of implici...
Smooth and Efficient Zooming and Panning
"... Large 2D information spaces, such as maps, images, or abstract visualizations, require views at various level of detail: Close ups to inspect details, overviews to maintain (literally) an overview. Users often switch between these views. We discuss how smooth animations from one view to another can ..."
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Cited by 41 (2 self)
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Large 2D information spaces, such as maps, images, or abstract visualizations, require views at various level of detail: Close ups to inspect details, overviews to maintain (literally) an overview. Users often switch between these views. We discuss how smooth animations from one view to another can be defined. To this end, a metric on the effect of simultaneous zooming and panning is de fined, based on an estimate of the perceived velocity. Optimal is defined as smooth and efficient. Given the metric, these terms can be translated into a computational model, which is used to calculate an analytic solution for optimal animations. The model has two free parameters: animation speed and zoom/pan tradeoff. A user experiment to find good values for these is described.
Stereo Matching as a NearestNeighbor Problem
 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE
, 1998
"... We propose a representation of images, called intrinsic curves, that transforms stereo matching from a search problem into a nearestneighbor problem. Intrinsic curves are the paths that a set of local image descriptors trace as an image scanline is traversed from left to right. Intrinsic curves ar ..."
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Cited by 20 (2 self)
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We propose a representation of images, called intrinsic curves, that transforms stereo matching from a search problem into a nearestneighbor problem. Intrinsic curves are the paths that a set of local image descriptors trace as an image scanline is traversed from left to right. Intrinsic curves are ideally invariant with respect to disparity. Stereo correspondence then becomes a trivial lookup problem in the ideal case. We also show how to use intrinsic curves to match real images in the presence of noise, brightness bias, contrast fluctuations, moderate geometric distortion, image ambiguity, and occlusions. In this case, matching becomes a nearestneighbor problem, even for very large disparity values.
Nonlinear Partial Least Squares
, 1995
"... We propose a new nonparametric regression method for highdimensional data, nonlinear partial least squares (NLPLS). NLPLS is motivated by projectionbased regression methods, e.g., partial least squares (PLS), projection pursuit (PPR), and feedforward neural networks. The model takes the form of a ..."
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Cited by 16 (0 self)
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We propose a new nonparametric regression method for highdimensional data, nonlinear partial least squares (NLPLS). NLPLS is motivated by projectionbased regression methods, e.g., partial least squares (PLS), projection pursuit (PPR), and feedforward neural networks. The model takes the form of a composition of two functions. The first function in the composition projects the predictor variables onto a lowerdimensional curve or surface yielding scores, and the second predicts the response variable from the scores. We implement NLPLS with feedforward neural networks. NLPLS will often produce a more parsimonious model (fewer score vectors) than projectionbased methods, and the model is well suited for detecting outliers and future covariates requiring extrapolation. The scores are also shown to have useful interpretations. We also extend the model for multiple response variables and discuss situations when multiple response variab...
On skew loops, skew branes and quadratic hypersurfaces
 Moscow Math. J
, 2003
"... A skew brane is an immersed codimension 2 submanifold in affine space, free from pairs of parallel tangent spaces. Using Morse theory, we prove that a skew brane cannot lie on a quadratic hypersurface. We also prove that there are no skew loops on embedded ruled developable discs in 3space. MSC: 53 ..."
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Cited by 13 (5 self)
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A skew brane is an immersed codimension 2 submanifold in affine space, free from pairs of parallel tangent spaces. Using Morse theory, we prove that a skew brane cannot lie on a quadratic hypersurface. We also prove that there are no skew loops on embedded ruled developable discs in 3space. MSC: 53A05, 53C50, 58E05 Key words: skew loops and skew branes, quadratic hypersurfaces, double normals, Morse theory, developable surfaces To V. I. Arnold on the occasion of the 65th anniversary 1 Introduction and formulation of results In a lecture, given in 1966, H. Steinhaus asked the question: Do there exist smooth closed space curves without a pair of parallel tangents? Call such a curve a skew loop. Steinhaus conjectured that skew loops did not exist. The problem was solved by B. Segre in [5, 6]. Segre constructed examples of
General helices and a theorem of Lancret
 Proc. A. M. S
, 1997
"... Abstract. We present a theorem of Lancret for general helices in a 3dimensional realspaceform which gives a relevant difference between hyperbolic and spherical geometries. Then we study two classical problems for general helices in the 3sphere: the problem of solving natural equations and the c ..."
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Cited by 12 (1 self)
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Abstract. We present a theorem of Lancret for general helices in a 3dimensional realspaceform which gives a relevant difference between hyperbolic and spherical geometries. Then we study two classical problems for general helices in the 3sphere: the problem of solving natural equations and the closed curve problem. 1.
Rational approximation of rotation minimizing frames using Pythagoreanhodograph cubics
 J. GEOM. GRAPHICS
, 1999
"... This article is devoted to the rotation minimizing frames that are associated with spatial curves. Firstly we summarize some results concerning the differential geometry of the sweeping surfaces which are generated by these frames (the socalled profile or moulding surfaces). In the second part of t ..."
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Cited by 11 (3 self)
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This article is devoted to the rotation minimizing frames that are associated with spatial curves. Firstly we summarize some results concerning the differential geometry of the sweeping surfaces which are generated by these frames (the socalled profile or moulding surfaces). In the second part of the article we describe a rational approximation scheme. This scheme is based on the use of spatial Pythagorean hodograph (PH) cubics (also called cubic helices) as spine curves. We discuss the existence of solutions and the approximation order of G 1 Hermite interpolation with PH cubics. It is shown that any spatial curve can approximately be converted into cubic PH spline form. By composing the rational FrenetSerret frame of these curves with suitable rotations around the tangent we develop a highly accurate rational approximation of the rotation minimizing frame. This leads to an approximate rational representation of profile surfaces.