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A Theory of Specular Surface Geometry
, 1996
"... A theoretical framework is introduced for the perception of specular surface geometry. When an observer moves in threedimensional space, real scene features such as surface markings remain stationary with respect to the surfaces they belong to. In contrast, a virtual feature which is the specular r ..."
Abstract

Cited by 74 (2 self)
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A theoretical framework is introduced for the perception of specular surface geometry. When an observer moves in threedimensional space, real scene features such as surface markings remain stationary with respect to the surfaces they belong to. In contrast, a virtual feature which is the specular reflection of a real feature, travels on the surface. Based on the notion of caustics, a feature classification algorithm is developed that distinguishes real and virtual features from their image trajectories that result from observer motion. Next, using support functions of curves, a closedform relation is derived between the image trajectory of a virtual feature and the geometry of the specular surface it travels on. It is shown that, in the 2D case, where camera motion and the surface profile are coplanar, the profile is uniquely recovered by tracking just two unknown virtual features. Finally, these results are generalized to the case of arbitrary 3D surface profiles that are traveled by virtual features when camera motion is not confined to a plane. This generalization includes a number of mathematical results that substantially enhance the present understanding of specular surface geometry. An algorithm is developed that uniquely recovers 3D surface profiles using a single virtual feature tracked from the occluding boundary of the object. All theoretical derivations and proposed algorithms are substantiated by experiments.
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 55 (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...
Interactive Simulation of the Human Eye Depth of Field and Its Correction by Spectacle Lenses
"... This paper describes a fast rendering algorithm for verification of spectacle lens design. Our method simulates refraction corrections of astigmatism as well as myopia or presbyopia. Refraction and defocus are the main issues in the simulation. For refraction, our proposed method uses pervertex bas ..."
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This paper describes a fast rendering algorithm for verification of spectacle lens design. Our method simulates refraction corrections of astigmatism as well as myopia or presbyopia. Refraction and defocus are the main issues in the simulation. For refraction, our proposed method uses pervertex basis ray tracing which warps the environment map and produces a realtime refracted image which is subjectively as good as ray tracing. Conventional defocus simulation was previously done by distribution ray tracing and a realtime solution was impossible. We introduce the concept of a blur field, which we use to displace every vertex according to its position. The blurring information is precomputed as a set of field values distributed to voxels which are formed by evenly subdividing the perspective projected space. The field values can be determined by tracing a wavefront from each voxel through the lens and the eye, and by evaluating the spread of light at the retina considering the best human accommodation effort. The blur field is stored as texture data and referred to by the vertex shader that displaces each vertex. With an interactive frame rate, blending the multiple rendering results produces a blurred image comparable to distribution ray tracing output. Categories and Subject Descriptors (according to ACM CCS): I.3.3 [Computer Graphics]: Picture/Image Generation 1.
Minkowski Combinations of Complex Sets Geometry, Algorithms, and Applications
"... x1. Preamble The evolution of mathematics has been shaped by a constantlychanging relationship between algebra and geometry. These subjects have, at times, vied for supremacy in mathematical discourse at other times, they have served to reveal profound new insights and perspectives to each other. ..."
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x1. Preamble The evolution of mathematics has been shaped by a constantlychanging relationship between algebra and geometry. These subjects have, at times, vied for supremacy in mathematical discourse at other times, they have served to reveal profound new insights and perspectives to each other. In the first mathematicallyadept civilization, that of ancient Mesopotamia, skilled algebraists were fully conversant with the manipulation of equations (and in certain cases the extraction of their roots), but were apparently less interested in problems of geometry. Conversely, the ancient Greeks sought refuge from the mysteries of irrational numbers (an inevitable consequence of basic algebraic operations) in purely geometrical constructions. Rene Descartes (15961650) liberated geometry from the confines of mere rulerandcompass constructions. The arithmetization of geometry, through the introduction of coordinates, opened new worlds of unimagined subtlety and intricacy to systematic exploration. With the development of
(Guest Editors) Using Wavefront Tracing for the Visualization and Optimization of Progressive Lenses
"... Progressive addition lenses are a relatively new approach to compensate for defects of the human visual system. While traditional spectacles use rotationally symmetric lenses, progressive lenses require the specification of freeform surfaces. This poses difficult problems for the optimal design and ..."
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Progressive addition lenses are a relatively new approach to compensate for defects of the human visual system. While traditional spectacles use rotationally symmetric lenses, progressive lenses require the specification of freeform surfaces. This poses difficult problems for the optimal design and its visual evaluation. This paper presents two new techniques for the visualization of optical systems and the optimization of progressive lenses. Both are based on the same wavefront tracing approach to accurately evaluate the refraction properties of complex optical systems. We use the results of wavefront tracing for continuously refocusing the eye during rendering. Together with distribution ray tracing, this yields highquality images that accurately simulate the visual quality of an optical system. The design of progressive lenses is difficult due to the tradeoff between the desired properties of the lens and unavoidable optical errors, such as astigmatism and distortions. We use wavefront tracing to derive an accurate error functional describing the desired properties and the optical error across a lens. Minimizing this error yields optimal freeform lens surfaces. While the basic approach is much more general, in this paper, we describe its application to the particular problem of designing and evaluating progressive lenses and demonstrate the benefits of the new approach with several example images. 1.
Optics, Mechanics and HamiltonJacobi Skeletons
"... In an effort to articulate models for the intuitive representation and manipulation of 2D and 3D forms, Blum (1967, 1973) invented the notion of a skeleton. His insight was to consider a disc as a basic geometric primitive and to use it to describe the reflective symmetries of an object. This repres ..."
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In an effort to articulate models for the intuitive representation and manipulation of 2D and 3D forms, Blum (1967, 1973) invented the notion of a skeleton. His insight was to consider a disc as a basic geometric primitive and to use it to describe the reflective symmetries of an object. This representation became very popular in a variety of fields including computer vision, computer aided design, graphics and medical image analysis. In this article we survey the basic properties of skeletons and show how considerations from classical mechanics and geometric optics lead to new insights into how to compute them. We detail the algorithms we have developed in this regard and present several numerical examples to illustrate their use.
Path Jacobians: Theory and Applications
, 1998
"... In accordance with Fermat's Variation Principle, a ray path connecting two arbitrary points in a scene via multiple reflectors is given by a nonlinear system. If we fix one of the two points and let the other change, the system can be considered as a function relating the reflection points alo ..."
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In accordance with Fermat's Variation Principle, a ray path connecting two arbitrary points in a scene via multiple reflectors is given by a nonlinear system. If we fix one of the two points and let the other change, the system can be considered as a function relating the reflection points along the path to the varying point. In this paper, we present the concept of path Jacobians for such a ray path, which are the derivatives of the reflection points along the path with respect to its varying endpoint. Path Jacobians provide a firstorder approximation to the path perturbation as a result of the changes of the endpoint and set a mathematical foundation for exploiting path coherence in incremental rendering. We also derive the analytic expression for path Jacobians from the Implicit Function Theorem. To illustrate its use, three related applications are described: image warping, stereoscopic ray tracing and caustic contour generation. 1 Introduction Ray tracing is an attractive meth...