## Computing Exact Shadow Irradiance Using Splines (1999)

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Venue: | PROCEEDINGS OF SIGGRAPH 99 |

Citations: | 11 - 6 self |

### BibTeX

@INPROCEEDINGS{Stark99computingexact,

author = {Michael M. Stark and Elaine Cohen and Tom Lyche and Richard F. Riesenfeld},

title = {Computing Exact Shadow Irradiance Using Splines},

booktitle = {PROCEEDINGS OF SIGGRAPH 99},

year = {1999},

pages = {155--164},

publisher = {}

}

### OpenURL

### Abstract

We present a solution to the general problem of characterizing shadows in scenes involving a uniform polygonal area emitter and a polygonal occluder in arbitrary position by manifesting shadow irradiance as a spline function. Studying generalized prism-like constructions generated by the emitter and the occluder in a fourdimensional (shadow) space reveals a simpler intrinsic structure of the shadow as compared to the more complicated 2D projection onto a receiver. A closed form expression for the spline shadow irradiance function is derived by twice applying Stokes' theorem to reduce an evaluation over a 4D domain to an explicit formula involving only 2D faces on the receiver, derived from the scene geometry.

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Citation Context ...ion, that is, they assume that an unoccluded radiosity kernel can be multiplied against a visibility factor, which gives the fraction of the area of the light source element visible from the receiver =-=[9, 24, 40]-=-. The point to polygon form factor is assumed to have constant exitance on the source. Then it is separated into the unoccluded point to polygon form factor integral and a visibility factor integral. ... |

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Citation Context ...ce. The shadow buffer creates a discrete depth image from the point light source view [43], using only a single point. Shadow volumes are constructed from a point light source and polygonal occluders =-=[13]-=- or curved boundary occluders [27], from which a point can be tested for its visibility in object space. Soft shadows can be approximated in a similar manner to the ray tracing method, by breaking the... |

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Citation Context ...m solution for the irradiance function. 1.1 Related Work Accurate computation of shadows has been a rich and active area of graphics research since near the beginning of the field. In 1990 Woo et al. =-=[44]-=- made an extensive survey of relevant techniques. Since then many techniques have been aimed at faster or more accurate computation. Methods that are based on a single point light source create hard s... |

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Citation Context ... selecting a set of sample points on the source and casting several rays towards them from the surface [11, 36]. Exact evaluation can be obtained by clipping all the occluders against the area source =-=[18, 33]-=-. Several methods consider the view from a point light source. The shadow buffer creates a discrete depth image from the point light source view [43], using only a single point. Shadow volumes are con... |

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Citation Context ...the values are interpolated across each mesh element. Meshing surfaces without considering the shadow structure can lead to inaccurate solutions and "shadow leaks". The discontinuity meshing=-= approach [17, 25, 30]-=- was developed to avoid these problems. For each source, a discontinuity mesh is created on the scenes surfaces (receivers). The mesh includes all lines where the illumination function has discontinui... |

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Citation Context ...the values are interpolated across each mesh element. Meshing surfaces without considering the shadow structure can lead to inaccurate solutions and "shadow leaks". The discontinuity meshing=-= approach [17, 25, 30]-=- was developed to avoid these problems. For each source, a discontinuity mesh is created on the scenes surfaces (receivers). The mesh includes all lines where the illumination function has discontinui... |

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Citation Context ... the light source [42]. Soft shadows caused by area light sources are simulated by stochastically selecting a set of sample points on the source and casting several rays towards them from the surface =-=[11, 36]-=-. Exact evaluation can be obtained by clipping all the occluders against the area source [18, 33]. Several methods consider the view from a point light source. The shadow buffer creates a discrete dep... |

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Citation Context ... our result. The polyhedral spline is defined as a distribution. A geometric interpretation of this definition involves regarding the splines as crosssectional volumes of higher dimensional polyhedra =-=[14]-=-. The more familiar univariate and tensor product B-splines are special cases of the polyhedral spline. To introduce the fundamental ideas of our approach, we begin with an analysis in Flatland [26], ... |

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Citation Context ... point to polygon form factor is assumed to have constant exitance on the source. Then it is separated into the unoccluded point to polygon form factor integral and a visibility factor integral. Arvo =-=[1]-=- provided hope that a closed-form formula for shadow irradiance might exist by developing a clean, elegant closed-form formula for the irradiance Jacobian due to a partially occluded polygonal source.... |

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Citation Context ...which a point can be tested for its visibility in object space. Soft shadows can be approximated in a similar manner to the ray tracing method, by breaking the light source into several point samples =-=[7]-=-. The radiosity method theoretically includes the computation of soft shadows. However, the method requires the reconstruction of irradiance functions, for which no general closed form is known. Metho... |

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Citation Context ...a [14]. The more familiar univariate and tensor product B-splines are special cases of the polyhedral spline. To introduce the fundamental ideas of our approach, we begin with an analysis in Flatland =-=[26]-=-, where the higher dimension construction can be directly visualized and more easily understood. Then we work through the construction of a generalized perspective prism in 4D associated with an eleme... |

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Citation Context ...d shadow research. It was recognized quite early that solutions could be improved if the form factors could be computed analytically. Analytical solutions have been determined for specific geometries =-=[28, 35]-=-. Generally such solutions assume that each surface is flat and fully visible from the other. Baum et al [3] proposed computing the unoccluded point to polygon form factor analytically. To avoid viola... |

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A Catalog of Radiation Configuration Factors
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Citation Context ...d shadow research. It was recognized quite early that solutions could be improved if the form factors could be computed analytically. Analytical solutions have been determined for specific geometries =-=[28, 35]-=-. Generally such solutions assume that each surface is flat and fully visible from the other. Baum et al [3] proposed computing the unoccluded point to polygon form factor analytically. To avoid viola... |

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Citation Context ...in geometric modeling [19]. McCool has used polyhedral splines for signal processing and rendering analytically antialiased polygons, and also suggested they might be used for penumbral approximation =-=[31, 32]. 2 FLATLA-=-ND We begin by carrying through our formulation in Edwin Abbott's "Flatland." In Flatland, surfaces become plane curves, planes become lines, and polygons become line segments [26]. Irradian... |

17 |
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(Show Context)
Citation Context ...d shadow research. It was recognized quite early that solutions could be improved if the form factors could be computed analytically. Analytical solutions have been determined for specific geometries =-=[28, 35]-=-. Generally such solutions assume that each surface is flat and fully visible from the other. Baum et al [3] proposed computing the unoccluded point to polygon form factor analytically. To avoid viola... |

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(Show Context)
Citation Context ...ution to the general differential area to polygon form factor in the presence of a polygonal occluder, by representing the induced irradiance as a spline function. We generalize the polyhedral spline =-=[4, 16, 32]-=- to obtain our result. The polyhedral spline is defined as a distribution. A geometric interpretation of this definition involves regarding the splines as crosssectional volumes of higher dimensional ... |

16 | Structured Sampling and Reconstruction of Illumination for Image Synthisis
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(Show Context)
Citation Context ...the values are interpolated across each mesh element. Meshing surfaces without considering the shadow structure can lead to inaccurate solutions and "shadow leaks". The discontinuity meshing=-= approach [17, 25, 30]-=- was developed to avoid these problems. For each source, a discontinuity mesh is created on the scenes surfaces (receivers). The mesh includes all lines where the illumination function has discontinui... |

12 |
Recurrence Relations for Multivariate B-Splines
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(Show Context)
Citation Context ...ution to the general differential area to polygon form factor in the presence of a polygonal occluder, by representing the induced irradiance as a spline function. We generalize the polyhedral spline =-=[4, 16, 32]-=- to obtain our result. The polyhedral spline is defined as a distribution. A geometric interpretation of this definition involves regarding the splines as crosssectional volumes of higher dimensional ... |

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7 | Shadow Volume Generation from Free Form Surfaces
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(Show Context)
Citation Context ...screte depth image from the point light source view [43], using only a single point. Shadow volumes are constructed from a point light source and polygonal occluders [13] or curved boundary occluders =-=[27]-=-, from which a point can be tested for its visibility in object space. Soft shadows can be approximated in a similar manner to the ray tracing method, by breaking the light source into several point s... |

6 |
Radiosity in flatland. Computer Graphics Forum
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(Show Context)
Citation Context ...a [14]. The more familiar univariate and tensor product B-splines are special cases of the polyhedral spline. To introduce the fundamental ideas of our approach, we begin with an analysis in Flatland =-=[26]-=-, where the higher dimension construction can be directly visualized and more easily understood. Then we work through the construction of a generalized perspective prism in 4D associated with an eleme... |

6 | Analytic Signal Processing for Computer Graphics using Multivariate Polyhedral Splines
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(Show Context)
Citation Context ...ution to the general differential area to polygon form factor in the presence of a polygonal occluder, by representing the induced irradiance as a spline function. We generalize the polyhedral spline =-=[4, 16, 32]-=- to obtain our result. The polyhedral spline is defined as a distribution. A geometric interpretation of this definition involves regarding the splines as crosssectional volumes of higher dimensional ... |

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1 |
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Citation Context ...llig's original paper [4] contains references and a recurrence relation valid for more general polyhedra than simplices, boxes, and cones. Simplex and box splines have been used in geometric modeling =-=[19]-=-. McCool has used polyhedral splines for signal processing and rendering analytically antialiased polygons, and also suggested they might be used for penumbral approximation [31, 32]. 2 FLATLAND We be... |