Interactive Boolean Operations on SurfelBounded Solids Bart Adams

Interactive Boolean Operations on Surfel-Bounded Solids Bart Adams Philip Dutré Katholieke Universiteit Leuven

Goal: CSG on free-form solids • union, difference and intersection • on free-form solids • at interactive rates A B A-B A B

“Bond of Union” 90 k surfels 94 k surfels 8 FPS

Related work Kristjansson et al. [2001] subdivision surfaces Museth et al. [2002] level set framework Pauly et al. [2003] points and moving least squares surface ACM SIGGRAPH

Surfels are considered as small disks • position x • normal n • radius r n x r surfels shown at half size

Three categories of surfels head surfels that lie completely helix surfels with the other solid’s surface helix surfels that lie completely head

Algorithm overview Preprocess inside-outside partitioning classification of surfels Interactive loop keep appropriate set of surfels resampling operator

For each solid an octree is constructed depth = 1 depth = 2 depth = 3

Classification of empty leaf cell as interior s ns ns points away from the empty cell

Classification of empty leaf cell as exterior ns s ns points towards the empty cell

Three types of leaf cells in the resulting octree Interior Boundary Exterior

Partitioning of boundary cells using parallel planes P 1 P 2 All surfels between P 1 and P 2

The empty half-space of P 1 is interior P 2 P 1 ns s ns points away from the empty space

The empty half-space of P 2 is exterior P 1 P 2 ns s ns points towards the empty space

In 3 D, the boundary cell is partitioned in three volumes Interior Boundary Exterior

Algorithm overview Preprocess inside-outside partitioning classification of surfels Interactive loop keep appropriate set of surfels resampling operator

Classification of surfels s 1 • s 1 outside octree s 5 s 3 s 4 s 2 outside • s 2 in exterior cell outside • s 3 in interior cell inside

Classification of surfels s 4 s 5 boundary cell • surfels s 4 and s 5 in empty half-spaces: s 4 outside s 5 inside

Classification of surfels • surfel s between parallel planes: t s boundary cell – search for nearest neighbor surfel t – test disks of s and t for intersection

Use octree as acceleration structure solid B root node intersects with boundary cells test children solid A

Use octree as acceleration structure solid B child node only intersects with exterior cells all surfels outside solid A

Use octree as acceleration structure solid B child node intersects with boundary cells test children solid A

Use octree as acceleration structure solid B leaf node only intersects with exterior cells all surfels outside solid A

Use octree as acceleration structure solid B leaf node still intersects with boundary cells test surfels individually solid A

Inside-outside classification 200 k surfels classified in group 47% individual surfels not in boundary cells 26% surfels in empty space of boundary cells 22% surfels between parallel planes 5%

Inside-outside classification 200 k surfels classified in group 47% individual surfels not in boundary cells 26% surfels in empty space of boundary cells 22% surfels between parallel planes 5% NN search for only 5%, 80% of this 5% is intersecting

Algorithm overview Preprocess inside-outside partitioning classification of surfels Interactive loop keep appropriate set of surfels resampling operator

Resampling operator: motivation Intersection of two spheres Include surfels Exclude surfels Resample surfels

Resample surfel by smaller surfels • clip surfels: – irregularly shaped surfels • our method: resample surfels: – surfel replaced by smaller surfels – only one type of primitive

Close-up of resampled surfels shown at half size

Resampling results in sharp edges and corners 350 k surfels 230 k surfels 46 k surfels 4. 4 FPS

Local smoothing eliminates sharp creases no smoothing 340 k surfels 250 k surfels smoothing 650 k surfels 3. 3 FPS

“Bond of Union” 350 k surfels 370 k surfels 2 FPS

Discussion Advantages • • fast classification sharp edges and corners only one type of primitive octree is easy to update Limitations • partitioning might fail in case of incorrectly oriented surfels • choice of parallel planes is not always optimal

Conclusion Contributions • fast inside-outside test • resampling operator Future work • CSG operations on mixed polygon -surfel models • implementation on GPU

Acknowledgements • graphics group K. U. Leuven • the reviewers • aspirant F. W. O. -Vlaanderen

Remark: surfels are considered being disks, not just points enlarge bounding boxes for overlap tests

Remark: surfels are considered being disks, not just points translate planes in boundary cells

Choice of parallel planes is not always optimal

Comparison with the technique of Pauly et al. Surface Points and MLS surface representation Kd-tree NN search Intersection Points on curve Rendering Adams and Dutré Disks Octree + TINN Disks intersect on curve Surface splatting Rendering of disks extended with (any surfel rendering clipping algorithm)

Surfels close to surface of other solid Inside with our technique, outside with the technique of Pauly et al. Solutions: denser sampling differential points (Kalaiah and Varshney)