Progressive Refinement With Topological Simplification

Progressive Refinement with Topological Simplification: A Theoretical Framework ERGUN AKLEMAN Visualization Laboratory Colleg e of Architectu re J IANER CHEN Departmen t of Compute r Science Colleg e of Engineer ing Abstract This paper presents a theoretical framework for progressive refinement of manifold meshes with topological simplification. We demonstrate that topology changes are not int uit iv e and the ref ore a great dea l of car e is necessary for handling topological simp lification. We illustrate non- intutive nature of topology changes with several examples. We also show how to use the non-intutive nature of the topology changes as an advantage and develop a theretical framewo rk for progr essi ve refine ment with t opolo gical simplification. 1 Intr odu ct ion Our goal in this paper is to develop a theoretical framework for progressive refinement schemes with topological simplifications. Level-of-deta il (LOD) representations ha ve recently become popular in computer graphics [9, 14, 10, 19, 18, 17]. Among the LOD repr esent atio ns progres siv e refinement is particularly useful since it allows continuous change [18, 17]. Although topolog ical simpl ificat ion can prov ide bette r simp lifica tion and effic ient repr esent atio n, thereexists no fra mework that pr ovid es both progr essive refinement and topological simplification. Ignoring topological simplification is not special to progressi ve meshes. In fact, except one recent development (Hybrid meshes) [12] topological simplification is not even considered in other LOD representations. Moreover , Hybrid meshes is a user- assisted LOD approach and cannot automatically guarantee topological simplification. As an example of application of progressive refinement with topology simplification, let us consider a city that consists of huge number of buildings or a forest that is made up of lots of trees. It is not reall y helpful to simpl ify each Address: 216 Langford Center, College Station, Texas 77843- 3137. email: [email protected]. phone: +(409) 845-6599. Supported in part by the Texas A&M, Scholarly & Creative Activities Program. bui lding or each tree to a simp le polyhed ron. At the end, there will still be a huge number of polyhedra. The most vi- able alte rnati ve is t o comb ine sev eral bui lding s to create one composite building and to combine a group of trees to create one composite tree. In this way, from the distance a city block that consists of a set of buildings will be represented by one composite bui lding and a cluster of trees will be represented by one composite tree. Moreover , this refinement should be progressive, i.e. the change must be gradual. For instance, let us say that the forest has 100 trees. The number of trees should gradually reduce; and so on. At the simplest lev el the whol e cit y or fore st should be represented by a singl e buildi ng or tre e. Then this building or tree should further be progressively refined into simplest polyhedra that resemble the shape of the building or tree. This example demonstrate progressive refinement with topology simplification will definitely be beneficial for a wide variety of applications that includes games, mesh data transmission, rendering. In this paper , we dev elop a theor etic al framework for progressive refinement with topological simplification. We will demonstra te that topology changes are no t intuitive and therefore a great deal of care is necessar y for handling topological simplificat ion. We show that the same operation can both clo se a hole and cut a handle. Moreo ver , all topology change operators can both complicate and simplify the topol ogy. Our framework uses the non-i ntuti ve nature of topol ogy chang es as its adva ntageand guara ntees high com- pression rates with topological simplication. 1.1 Pr evi ous Work Render ing comp lex geometric mode ls at in terac tiv e rates has always been a challenging problem in computer graphics. Improving t he rendering performance alone could not solve this problem since the complexity of geometric models increases with rendering per formance. In other words, as the rendering perfo rmance impr oves , more and more complicated geometric models will be used. It is observed 1

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Progressive Refinement With Topological Simplification