CSC 274b-aa
Spring 2000
Courtney Christman
Sophomore, Class of 2002
CSC 274: Computational Geometry
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Chapter 8, Arrangements and Duality Supersampling in Ray Tracing
8.1: Computing the Discrepancy

A Brief Description of the Problem:

A picture on a screen is made of pixels. A third-dimension scene consists of objects, a light source, and a view point. The goal is to render the scene by determining for every pixel on the screen which object is visible at that pixel, and determining the intensity of light that the object emits in the direction of the view point at that particular point.
This is done through ray tracing. Ray tracing is a way of determining the visible object at each pixel. Once the visible object is determined, one has to compute the intensity of light emitted by the object at the visible point. You must do this without forgetting to take into account how much light the point recieves from the light surce, either directly or indirectly via reflections on the other objects. However, this method has many flaws. If the light source covers 49% of the square pixel, no changes will be made to the pixel. But if the light source covers 51% of the pixel, then the entire pixel will show the effect of the light source. Also, if ther are two objects in the pixel, then the pixel will show a mixture of the object intensities.
A remedy to this problem is supersampling. Supersampling involves taking random generated sampling points. The discrepancy of the sample set with respect to the object is the difference between the percentage of hits for an object and teh percentage of pixel area where the object is visible but small. The new goal is to calculate the discrepancy of the sample set. The maximum discrepancy over all possible ways that an object can be visible inside the pixel to be small.
Discrepancy of a point is defined with respect to a class of objects. Third-dimension curved objects are make of polygonal meshes. Two-dimensional projections are of the facts of polyhedra. We are interested in the discrepancy with respect to the class of polygons. If a pixel is intersected by one polygon edge, the polygon behaves inside the pixel like a half plane. A theorum proves that h (infinite set of all possible closed half-planes) contains EITHER one point in in the set of sample points of models in a pixel OR h contains tow or more points of S on its boundary. The results are that the half-plain discrepancy of a set of n sample points in a models of pixels in the unit square can be computed in O(n^2) time.

Glossary of Geometric Terms

1. ray tracing, Unfamiliar
2. rendering the scene, Unfamiliar
3. supersamplling, Unfamiliar
4. discrepancy of the sample set with respect to the object, Unfamiliar
5. discrepancy of the sample set, Unfamiliar
6. discrete measure, Unfamiliar
7. continuous measure, Unfamiliar
8. pixel, Familiar
The meanings of these words is clearly explained in my summary of the section.

Key Words for Describing the Problem

1. ray tracing
2. supersampling

1. A Web Search based on Keywords
a. The Internet Raytracing Competition
b. Relativistic Ray-Tracing
c. Basic Ray Tracer
d. Supersampling

2. Bibliographic Search in the Computational Geometry database
'ray tracing' yielded 40 results, here are 2

author = "J. Amanatides"
title = "Ray Tracing with Cones"
journal = "Comput. Graph."
volume = 18
number = 3
year = 1984
pages = "129--135"
note = "Proc. SIGGRAPH '84"
update = "97.11 bibrelex"

author = "J. Arvo and D. Kirk"
title = "Fast ray tracing by ray classification"
editor = "M. C. Stone" year = 1987
pages = "55--63"
update = "97.11 bibrelex"

'rendering the scene' yielded 3 results

author = "M. Pharr and C. Kolb and R. Gershbein and P. Hanrahan"
title = "Rendering complex scenes with memory-coherent ray tracing"
booktitle = "Proc. SIGGRAPH '97"
series = "Computer Graphics Proceedings, Annual Conference Series"
organization = "ACM SIGGRAPH"
publisher = "Addison-Wesley"
year = 1997
pages = "101--108"
update = "98.07 agarwal, 98.03 agarwal"

author = "J. R. Rossignac and P. Borrel"
title = "Multi-Resolution {3D} Approximations for Rendering Complex Scenes"
type = "Technical {Report}"
number = "RC 17697 (\#77951)"
institution = "IBM Research Division, T. J. Watson Research Center"
address = "Yorktown Heights, NY 10958"
year = 1992
update = "95.01 mitchell"
author = "S. M. Rubin and T. Whitted"
title = "A $3$-dimensional representation for fast rendering of complex scenes"
journal = "Comput. Graph."
volume = 14
number = 3
year = 1980
pages = "110--116"
note = "Proc. SIGGRAPH '80"

'supersampling" yielded one result:

author = "D. Dobkin and D. Mitchell"
title = "Random-edge discrepancy of supersampling patterns"
booktitle = "Graphics Interface '93"
year = 1993
keywords = "discrepancy"
update = "97.03 schwarzkopf"

Time Assessment

This assignment took me 7 hours to complete.