Robust Repair of Polygonal ModelsRobust Repair of Polygonal ModelsRobust Repair of Polygonal ModelsRobust Repair of Polygonal Models

Tao JuRice University

Polygonal ModelsPolygonal Models

Closed ModelClosed Model

• Partitions the space into disjoint inside and outside volumes– Each polygon lies between inside and outside

Not Closed Closed ClosedNot Closed

Model RepairModel Repair

• Close polygonal models– Not just hole-filling

• Challenges:– Models may contain complex errors

– Models are often very big

– Geometry features need to be preserved

Previous WorkPrevious Work

• Mesh-based model repair– Zippering [Turk and Levoy 1994] – Stitching [Barequet and Kumar 1997] – Progressive boundary decimation [Borodin et al. 2002]– Hole filling with fairing [Liepa 2003]

• Scattered data reconstruction– Tangent plane estimation [Hoppe et al. 1992] – Level sets [Zhao and Osher 2002]– Radial basis functions [Turk and O’Brien 2002]– Partition-of-unity methods [Ohtake et al. 2003]– Moving least squares surfaces [Shen et al. 2004]– Context-based completion [Sharf et al. 2004]

Previous WorkPrevious Work

• Volumetric model repair

How to determine signs ? How to generate surface ?

Previous WorkPrevious Work

• Volumetric model repair – Sign generation– Adaptive signed distances [Frisken et al. 2000]

– Flood-filling [Oomes et al. 1997] [Andujar et al. 2002] – Space carving [Curless and Levoy 1996]– Volumetric diffusion [Davis et al. 2002]– Parity count and ray stabbing [Nooruddin and Turk

2003]

• Volumetric model repair – Contouring– Marching Cubes [Lorensen and Cline 1987]– Extended Marching Cubes [Kobbelt et al. 2001]– Dual Contouring [Ju et al. 2002]

Robust Model RepairRobust Model Repair

• Robust– Closes arbitrary polygon soups

• Efficient– Processes gigantic models on standard PCs

• Accurate– Preserves surface quality

• Simple !

Algorithm in a NutshellAlgorithm in a Nutshell

Contour

Patch

Closed Dual Surface

Scan-convert

“Dual Surface”

• They convert the input model to a volumetric form by constructing an octree grid that records edges intersecting the input model.

Scan-conversionScan-conversion

Scan-conversionScan-conversion

• They recursively walk down the octree, expanding nodes when necessary, until all the leaf cells at the bottom level of the tree that intersect the polygon are located .

• Then, cell edges that intersect the polygon are identified in those leaf cells and are marked as intersection edges (figure 4 (d)).

Scan-conversionScan-conversion

Scan-conversionScan-conversion

• Octree grid

• Memory-less octree construction

• Reliable and fast intersection tests

• Edges intersected with model

• Top-down creation

Fast intersection testsFast intersection tests

• a triangle and a cube are disjoint if their projections on any one of the following 13 vectors are disjoint:(1) the triangle face normal,(2) 3 cube face normals,(3) and 9 pair-wise cross-products of the 3 edges of

the cube and the 3 edges of the triangle.

Dual SurfaceDual Surface

• Each face dual to an intersected octree edge

Octree Edge Dual Face

Finding HolesFinding Holes

• Boundary edges– Odd-valence edges– Closed dual surface

No boundary edge

• Set of boundary edges partitioned into cycles– Each cycle encloses

a “hole”

Detect boundary cyclesDetect boundary cycles

•Detecting the boundary edges in (S) simply involves enumerating the cell faces on the primal grid containing an odd number of intersection edges.

• To form cycles, we introduce a bottom-up procedure detectProc[N] that returns all complete cycles B and incomplete cycles R inside the octree node N.

• At a leaf node, detectProc[N] returns B = 0 and R = 0.

• At an internal node,(1) Call detectProc[Ni] for every child node Ni,

which return cycles Bi and incomplete cycles Ri.

(2) Detect the boundary edges E crossing the faces between adjacent children nodes. (3) Connect Ri by E to form complete cycles B’ and incomplete cycles R. B is the union of B’ and the Bis.

Detect boundary cyclesDetect boundary cycles

Patch boundary cyclesPatch boundary cycles

• seek the surface of minimum area spanning a given curve in 3-D space.

Building a PatchBuilding a Patch

• Build one patch for each cycle– Each quad dual to an

octree edge– Patch boundary is

the cycle

• Divide-and-conquer!– Using octree

Integrating a PatchIntegrating a Patch

• Add a quad– If does not exist on

the dual surface

• Remove a quad– If already exists on

the dual surface

• Key: parity of edge valence

Integrating a PatchIntegrating a Patch

• Add a quad– If not already on the

dual surface

• Remove a quad– If already on the dual

surface

• Key: parity of edge valence

Sign GenerationSign Generation

• Sign changes across dual surface– Flood-filling

Dual Face Octree Signs

ContouringContouring

• Marching Cubes– Edge intersections– Rounded corners

• Dual Contouring– Hermite data– Sharp features

Examples – CAD ModelExamples – CAD Model

Input Model

Examples – CAD ModelExamples – CAD Model

Dual Surface

Examples – CAD ModelExamples – CAD Model

Closed Dual Surface

Examples – CAD ModelExamples – CAD Model

Output (Marching Cubes)

Examples – CAD ModelExamples – CAD Model

Output (Dual Contouring)

Examples - BunnyExamples - Bunny1. Input 2. DS

3. DS Closed

4. Output

Model Courtesy of the Stanford 3D Scanning Repository

Examples - HorseExamples - Horse1. Input 2. DS

3. DS Closed

4. Output

Model Courtesy of the Stanford 3D Scanning Repository

Examples – David (at 1mm)Examples – David (at 1mm)

Input Output

Model Courtesy of the Digital Michelangelo Project

PerformancePerformance

Model Triangles Grid Time Memory

Bunny 69,451 64 3.6 sec < 10 MB

Horse 80,805 128 6.0 sec < 10 MB

Dragon 871,414 256 45.2 sec 16 MB

Buddha 1,087,716 1024 1.3 min 28 MB

David (2mm) 8,254,150 4096 8.4 min 92 MB

David (1mm) 56,230,343 8192 53.2 min 417 MB

• On PC with 1.5GHz CPU and 2GB memory

ConclusionConclusion

• A simple, fast, and robust method to repair arbitrary polygon models

• Future work– Remove topological noise– Improve the quality of hole filling– Repair using an adaptive grid