Bounded Radiosity with Affine Arithmetic

implemented in Vision

Affine Arithmetic

Affine arithmetic (AA) has been introduced by J. L. D. Comba and J. Stolfi. It enhances the idea of standard interval arithmetic (IA) to compute conservative bounds for formulas applied to uncertain quantities. In contrast to IA, AA automatically keeps track of the result's dependency on the sources of error of the initial quantities. In this way the use of affine arithmetic significantly improves the quality of calculated bounds, unfortunately at higher computational cost. Uncertain quantities are represented by affine forms for which all standard arithmetic operations are provided.

A collection of three affine forms builds a 3D affine form. It is possible to perform any kind of operation to 3D affine forms that is allowed on simple vectors. These operations will yield other affine forms.Three-dimensional affine forms can be used to bound solid objects. The range of a 3D affine form equals to the projection of a n-dimensional hypercube and thus has a point-symmetrical structure. Due to this, not all objects can be represented exactly, but for some of them an optimal representation may be found. These are rectangular parameterized surfaces, triangles and convex polygons. Other objects like clusters are simply approximated by their bounding boxes.


A random 3D affine form	Bounding a circle segment

Bounding a polygon

Bounded Radiosity

Based on Stamminger's bounded radiosity algorithm which uses IA to calculates conservative bounds on the patch-to-patch form factor in order to allow error-driven refinement, this approach uses affine arithmetic to tighten the computed bounds. The algorithm works on all kinds of objects that allow subdivision, even on clusters and curved surfaces. The spatial extent and normals of each object are represented by 3D affine forms. Thus, the use of affine arithmetic during evaluation of the form factor kernel, treating solid objects like single points, results in bounds on the patch-to-patch form factor between two objects.

It was not possible to apply affine arithmetic to all tasks of the hierarchical radiosity algorithm because sometimes upper and lower bound of an interval are computed in separate ways neglecting the additional information available from the affine form. Although the approach succeeds in gaining tighter bounds, the additional effort performing the affine arithmetic operations slows down the algorithm compared to Stamminger's implementation.

Here some pictures

A simple scene

A scene with complex clusters

A scene with curved surfaces

Literature

J. L. D. Comba and J. Stolfi. Affine arithmetic and its application to computer graphics. In Anais do VII Sibgrapi, pages 9-18, 1993.

M. Stamminger et al. Bounded radiosity - illumination on general surfaces and clusters. In Computer Graphic Forum (EUROGRAPHICS '97 Proceedings), 16(3), September 1997.

M. Stamminger et al. Bounded clustering - finding good bounds on clustered light transport. In Proc. Pacific Graphics '98. IEEE Computer Society Press, 1998

Last modified: Tue Sep 7 16:03:26 MDT 1999 by Hendrik Lensch