- Author:
- Amitabh Varshney
This dissertation explores some techniques for automatic approximation of geometric objects. My thesis is that using and extending concepts from computational geometry can help us in devising efficient and parallelizable algorithms for automatically constructing useful detail hierarchies for geometric objects. We have demonstrated this by developing new algorithms for two kinds of geometric approximation problems that have been motivated by a single driving problem--the efficient computation and display of smooth solvent-accessible molecular surfaces. The applications of these detail hierarchies are in biochemistry and computer graphics.
The smooth solvent-accessible surface of a molecule is useful in studying the structure and interactions of proteins, in particular for attacking the protein-substrate docking problem. We have developed a parallel linear-time algorithm for computing molecular surfaces. Molecular surfaces are equivalent to the weighted $\alpha$-hulls. Thus our work is potentially useful in the application areas of $\alpha$-hulls which include astronomy and surface modeling, besides biochemistry.
We have defined the concept of interface surfaces and developed efficient algorithms for computation of surfaces at the interface of two or more molecular units. Interface surfaces are useful for visualizing the inter and intra-molecular interfaces and for characterizing the fit, or complementarity, of molecular interfaces.
We have developed an algorithm for simplification of polygonal meshes. The simplified polygonal mesh has the following properties: (a) every point on it is within a user-specifiable distance $\epsilon$ from the input mesh, (b) it is topologically consistent with the input mesh (i.e. both have the same genus), (c) its vertices are a subset of the vertices of the input mesh, and (d) it is within a computable factor in complexity (in terms of number of faces) of the optimal mesh that satisfies (a), (b), and (c) (computing the optimal mesh is known to be NP-hard). We have accomplished this by transforming our problem to the set-partitioning problem.
Cited By
Cardoze D An optimal algorithm for the minimum edge cardinality cut surface problem Proceedings of the nineteenth annual symposium on Computational geometry, (338-343)- Guéziec A, Taubin G, Lazarus F and Horn B (2001). Cutting and Stitching, IEEE Transactions on Visualization and Computer Graphics, 7:2, (136-151), Online publication date: 1-Apr-2001.
- Guéziec A (1999). Locally Toleranced Surface Simplification, IEEE Transactions on Visualization and Computer Graphics, 5:2, (168-189), Online publication date: 1-Apr-1999.
- Guéziec A, Taubin G, Lazarus F and Horn W Converting sets of polygons to manifold surfaces by cutting and stitching Proceedings of the conference on Visualization '98, (383-390)
- Kumar S, Manocha D, Zhang H and Hoff K Accelerated walkthrough of large spline models Proceedings of the 1997 symposium on Interactive 3D graphics, (91-ff.)
- Krus M, Bourdot P, Guisnel F and Thibault G (1997). Levels of detail & polygonal simplification, XRDS: Crossroads, The ACM Magazine for Students, 3:4, (13-19), Online publication date: 1-May-1997.
- Cohen J, Manocha D and Olano M Simplifying polygonal models using successive mappings Proceedings of the 8th conference on Visualization '97, (395-ff.)
- Xia J and Varshney A Dynamic view-dependent simplification for polygonal models Proceedings of the 7th conference on Visualization '96, (327-ff.)
- Cohen J, Varshney A, Manocha D, Turk G, Weber H, Agarwal P, Brooks F and Wright W Simplification envelopes Proceedings of the 23rd annual conference on Computer graphics and interactive techniques, (119-128)
- He T, Hong L, Kaufman A, Varshney A and Wang S Voxel based object simplification Proceedings of the 6th conference on Visualization '95
- Silva C, Mitchell J and Kaufman A Automatic generation of triangular irregular networks using greedy cuts Proceedings of the 6th conference on Visualization '95
- Eck M, DeRose T, Duchamp T, Hoppe H, Lounsbery M and Stuetzle W Multiresolution analysis of arbitrary meshes Proceedings of the 22nd annual conference on Computer graphics and interactive techniques, (173-182)
Index Terms
- Hierarchical geometric approximations
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