Luiz Velho’s Legacy on Geometry Processing Prof. Afonso Paiva ICMC-USP, S˜ ao Carlos, SP, Brazil apneto@icmc.usp.br Prof. Marcelo Ferreira Siqueira DMAT-UFRN, Natal, RN, Brazil mfsiqueira@mat.ufrn.br
Outline The 80's The 90's 2000-2004 2005-2010 2011-2017 Beyond geometry processing 2
The 80's Geometry processing ≈ geometric modeling + computational geometry • Engineering (CAD/CAM/CAE) and graphics applications. 3
The 80's Geometry processing ≈ geometric modeling + computational geometry • Parametric models (B´ ezier and B-spline surfaces, NURBS). 4
The 80's Geometry processing ≈ geometric modeling + computational geometry • How to optimally tessellate NURBS? Needed for nice and fast rendering... 5
The 80's Geometry processing ≈ geometric modeling + computational geometry • Still an important problem to this date! 6
The 80's Luiz Velho was involved with the animation industry and started working with geometric modeling in the late 80’s and early 90’s. More precisely, he got inter- ested in a yet unexplored mathematical model for curves, surfaces, and solids: [Velho, Gomes and de Figueiredo, 2002] 7
The 80's How to tessellate implicit surfaces? • Polygonization methods (e.g., Marching Cubes ). • Needed for rendering and for representation conversion. [Lorensen and Cline, SIGGRAPH, 1987] 8
The 80's How to tessellate implicit surfaces? • Early algorithms were based on regular, uniform grids. • Inappropriate for polygonizing surfaces with high curvature variation. [Lorensen and Cline, SIGGRAPH, 1987] 9
The Early 90's How to tessellate implicit surfaces? • Mesh adaptivity was required for optimizing memory usage. • But mesh adaptivity was a lot more complex... 10
The Early 90's How to tessellate implicit surfaces? • Luiz Velho provided one of the first solutions for the polygonization problem: Luiz Velho. Adaptive polygonization of implicit surfaces using simplicial decomposition and boundary constraints. Proceedings of the Eurographics’90 , pages 125-136, September 1990 . 11
The Early 90's First Brazilian to publish a paper in the prestigious Eurographics conference! 12
The Mid 90's How to tessellate implicit surfaces? • Adaptivity required more complex data structures and made the problem of generat- ing conformal triangle meshes more difficult. But, what if you sole purpose is render- ing ? [Velho, JGT, 1996] 13
The Mid 90's How to tessellate implicit surfaces? • Luiz Velho had a clever insight, which led him to developing a very efficient and sim- ple algorithm: Luiz Velho. Simple and efficient polygonization of implicit surfaces. Journal of Graphics Tools , 1(2): 5-25, 1996 . T-splines were inspired by a rather similar insight! [Velho, JGT, 1996] 14
The Mid 90's Revisiting the tessellation problem for parametric surfaces... • Luiz Velho and Luiz Henrique de Figueiredo offered a solution for the problem of tessellating parametric surfaces: L. Velho and L. H. de Figueiredo. Optimal adaptive ap- proximation of parametric surfaces. Proceedings of SIBGRAPI’96 , pages 127-133, October 1996 . 15
The Mid 90's What if the purpose is representation conversion? • Luiz Velho and Jonas Gomes provided an approach to convert parametric to implicit representation: L. Velho and J. Gomes. Approximate conversion of parametric to implicit surfaces. Proceedings of Implicit Surfaces ’95 , pages 77-96, Grenoble, France, April 1995 . Misha Kazhdan (Jonhs Hopkins University) adopted a similar idea for Poisson surface reconstruction from point cloud more than 10 years later (SGP 2006). 16
The Mid 90's Luiz Velho received his Ph.D. degree from the University of Toronto in 1994, but he was already a established researcher and world-renowned expert in computer animation before starting his Ph.D. studies. His Ph.D. advisor, Dimetri Terzopoulos, was a pioneer in com- bining techniques from computer graphics, image processing, and computer vision to solve problems in each of these areas. Luiz Velho is likely to have inherited the same holistic view from him! 17
The Late 90's Tessellating both implicit and parametric surfaces with a single algorithm: • Luiz Velho, Luiz Henrique and Jonas Gomes provided a unified approach to tessellate both implicit and parametric surfaces: L. Velho, L. H. de Figueiredo and J. Gomes. A Unified Approach for Hierarchical Adaptive Tessellation of Surfaces. ACM ToG , 18(4): 329- 360, 1999 . 18
The Late 90's At this point in time the graphics community started focusing on polygonal meshes , and what we mean today by geometry processing did not exist back then. The Stanford’s Digital Michelangelo Project opened up a huge window of research opportunities. It was time to solve problems such as mesh simplification, mesh compression, mesh transmission, and mesh smoothing, mesh reconstruction from point cloud, and noise removal in the graphics context . Mesh processing demanded efficient data structures and mesh operators . Luiz Velho was one of the first researchers to offer a powerful data structure to represent multi-resolution and adaptive meshes, which combined elements from combinatorial topology and graph theory. 19
2000 to 2004 The mesh processing work developed by Luiz Velho from the late 90’s until 2004 was influ- enced by some ideas from another hot research topic in the graphics community: subdivision surfaces . Catmull-Clark, 1978 Loop, 1987 Pixar, 1997 20
2000 to 2004 The mesh processing work developed by Luiz Velho from the late 90’s until 2004 was influ- enced by some ideas from another hot research topic in the graphics community: subdivision surfaces . 21
2000 to 2004 The paper co-authored with Jonas Gomes in 1999, ( L. Velho and J. Gomes. Quasi 4-8 Subdivi- sion Surfaces, Proceedings of SIBGRAPI ’99 , pages 7-19, Campinas, SP, Brazil ), introduced the 4-8 meshes: 22
2000 to 2004 and a refinement scheme: 23
2000 to 2004 The previous regular refinement scheme was modified to give rise to a semi-regular and hier- archical scheme, which allowed for the extraction of conformal variable-resolution meshes based on spatially varying adaptation functions (see paper L. Velho. Semi-regular 4-8 refine- ment and box spline surfaces. Proceedings of the SIBGRAPI 2000 , pages 131-138, Gramado, RS, Brazil ). 24
2000 to 2004 The implementation of the algorithms in the papers showed before was made possible by the use of a powerful data structure for representing multi-triangulations (the so-called 4-k meshes ). L. Velho and J. Gomes. Variable resolution 4-k meshes: Concepts and Applications. CGF, 19(4): 195-214, 2000. 25
2000 to 2004 In 2001, Luiz Velho described an algorithm for mesh simplification which is significantly influenced by the 4-8 meshes. Simplification can be seen as the inverse of the refinement operation. L. Velho. Mesh Simplification using Four-Face Clusters, Proceedings of the SMI’2001. First Brazilian to publish a paper in the SMI conference! 26
2000 to 2004 The original data structure was later modified to give rise to the so-called A48 data structure . The latter need not explicitly store the local operations that dynamically modify an initial mesh. The A48 data structure constitutes an improvement over all data structures available back then. More importantly, it could be used to more efficiently implement the most popular algorithms for mesh simplification and mesh refinement (such as QEM and progressive meshes). Most research work developed at IMPA from 2000 on were based on the A48 data structure. L. Velho. A Dynamic Adaptive Mesh Library Based on Stellar Operators. JGT, 9(2): 1-2, 2004. 27
2000 to 2004 Stellar operations : simplicity and completeness L. Velho. Stellar Subdivision Grammars. Proceedings of the SGP , 2003. 28
2000 to 2004 Stellar operations : simplicity and completeness L. Velho. Stellar Subdivision Grammars. Proceedings of the SGP , 2003. 29
2000 to 2004 First Brazilian to publish a paper in the prestigious Symposium on Geometry Processing! 30
2000 to 2004 Revisiting the mesh simplification problem... • A. Vieira, T. Lewiner, L. Velho , H. Lopes, and G. Tavares. Stellar Mesh Simplification Using Probabilistic Optimization. Computer Graphics Forum , 23(4): 825-838, 2004 . 31
2000 to 2004 A paper from 2001 gave a proof that 4-8 subdivision surfaces are C 4 - continuous everywhere, except at extraordinary vertices where they are C 1 - continuous: L. Velho and D. Zorin. 4-8 Subdivision. CAGD , 18(5):397–427, 2001 . 32
2005 to 2010 In this moment, discrete differential geometry and spectral mesh processing become popular in computer graphics, and a modern book about geometry processing is released . [Grinspun et al. , SIGGRAPH, 2005] [L´ evy and Zhang, SIGGRAPH, 2010] 33
2005 to 2010 From 2005 to 2010, Luiz Velho worked on a variety of important problems in geometry processing: surface reconstruction from point cloud , discrete geodesics , mesh compression , adapted dynamic meshes , mesh segmentation , and geometry sam- pling . A remarkable fact is that most papers published in the above period resulted from the work of his PhD students and from collaborations with other re- searchers from Brazilian institutions, such UFRJ, PUC-Rio, Unicamp, and USP. Luiz Velho also kept alive his fruitful collaboration with his IMPA peers (Jonas Gomes, Luiz Henrique de Figueiredo, and Paulo Cezar Carvalho), and their students. 34
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