Boolean Operations for Free-form Models Represented in Geometry Images Yan Fu and Bingfeng Zhou (Presented by Jie Feng) Institute of Computer Science & Technology Peking University April 17, 2008

ICST, Peking University Outline Motivation Geometry Images Boolean Operation on Geometry Images Mesh Intersection Pixel Classification Triangulation Experimental Results Summary 2008-04-17 ICST, Peking University

ICST, Peking University Motivation Boolean operation of solid models is a key algorithm for geometry modeling Boolean operation algorithms based on different representations Level-set Multi-resolution meshes Hybrid of implicit and explicit models Volumetric representation Geometry images –– higher efficiency 2008-04-17 ICST, Peking University

ICST, Peking University Geometry Images Geometry Images [Gu et al. 2002] Store the geometry information as (r,g,b) in a 2D image Avoid the storage of connectivity information 2008-04-17 ICST, Peking University

Boolean Operation on Geometry Images Reconstructed meshes gA-1 gB-1 Boolean operation GA GB Geometry images Resulting model gA gB Definition: p: M->D (M: 3d model, D: 2d domain) M~G (geometry image) D~g (2d grid) g projected back to 3d surface -> g-1 2008-04-17 ICST, Peking University

Step 1. Mesh Intersection Calculate intersection lines between gA-1 and gB-1 → Boundary grid lines (3D) Project boundary grid lines to gA and gB → Closed paths in the parameter domain (2D) Quadtree scheme to accelerate the triangle intersection 2008-04-17 ICST, Peking University

Step 2. Pixel Classification Closed paths in parameter domain divide the pixels of a geometry image into subsets Decide the inside/outside property of each subset by a boundary-fill scheme 2008-04-17 ICST, Peking University

ICST, Peking University Step 3. Triangulation Intersection lines are used as the constrained edges in constrained Delaunay triangulation Resulting triangle meshes are transformed into 3D space and merged together The final triangle mesh can be further converted into a geometry image 2008-04-17 ICST, Peking University

ICST, Peking University Experimental Results 2008-04-17 ICST, Peking University

ICST, Peking University Experimental Results Timing for Boolean operations Results Size of G.I. G.I. Without quadtree G.I. With quadtree Mesh 64×64 59.2 s 0.140 s 1.547 s 108.5 s 0.328 s 1.563 s 128×128 1932.2 s 0.906 s 4.453 s 58.3 0.078 s 5.234 s 2008-04-17 ICST, Peking University

ICST, Peking University Summary An efficient Boolean operation method Free-form solid models represented by geometry images Benefits from regular data organization Accelerated intersection by using hierarchical quadtrees Independent of resolutions of geometry images Future works To improve the robustness of intersection computation To achieve better performance of Boolean operation by utilizing graphics hardware 2008-04-17 ICST, Peking University

Thank you for your attention! Questions? Contact: zhoubingfeng@icst.pku.edu.cn 2008-04-17 ICST, Peking University