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Isosurfaces Over Simplicial Partitions of Multiresolution Grids Josiah Manson and Scott Schaefer Texas A&M University

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Motivation: Uses of Isosurfaces

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Motivation: Goals Sharp features Thin features Arbitrary octrees Manifold / Intersection-free

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Motivation: Goals Sharp features Thin features Arbitrary octrees Manifold / Intersection-free

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Motivation: Goals Sharp features Thin features Arbitrary octrees Manifold / Intersection-free Octree Textures on the GPU [Lefebvre et al. 2005]

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Motivation: Goals Sharp features Thin features Arbitrary octrees Manifold / Intersection-free

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Related Work Dual Contouring [Ju et al. 2002] Intersection-free Contouring on an Octree Grid [Ju 2006] Dual Marching Cubes [Schaefer and Warren 2004] Cubical Marching Squares [Ho et al. 2005] Unconstrained Isosurface Extraction on Arbitrary Octrees [Kazhdan et al. 2007]

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Dual Contouring + - +++ + +++ + + -- +

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+ - +++ + +++ + + -- +

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+ - +++ + +++ + + -- +

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Dual Contouring [Ju et al. 2002]Our method

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Dual Contouring Dual Contouring [Ju et al. 2002]Our method

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Related Work Dual Contouring [Ju et al. 2002] Intersection-free Contouring on an Octree Grid [Ju 2006] Dual Marching Cubes [Schaefer and Warren 2004] Cubical Marching Squares [Ho et al. 2005] Unconstrained Isosurface Extraction on Arbitrary Octrees [Kazhdan et al. 2007]

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Dual Marching Cubes

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+ - + + + - -

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+ - + + + - -

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+ - + + + - -

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[Schaefer and Warren 2004] Our method

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Dual Marching Cubes

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Related Work Dual Contouring [Ju et al. 2002] Intersection-free Contouring on an Octree Grid [Ju 2006] Dual Marching Cubes [Schaefer and Warren 2004] Cubical Marching Squares [Ho et al. 2005] Unconstrained Isosurface Extraction on Arbitrary Octrees [Kazhdan et al. 2007]

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Our Method Overview Create vertices dual to every minimal edge, face, and cube Partition octree into 1-to-1 covering of tetrahedra Marching tetrahedra creates manifold surfaces Improve triangulation while preserving topology

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Terminology Cells in Octree – Vertices are 0-cells – Edges are 1-cells – Faces are 2-cells – Cubes are 3-cells Dual Vertices – Vertex dual to each m-cell – Constrained to interior of cell

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Terminology Cells in Octree – Vertices are 0-cells – Edges are 1-cells – Faces are 2-cells – Cubes are 3-cells Dual Vertices – Vertex dual to each m-cell – Constrained to interior of cell

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Terminology Cells in Octree – Vertices are 0-cells – Edges are 1-cells – Faces are 2-cells – Cubes are 3-cells Dual Vertices – Vertex dual to each m-cell – Constrained to interior of cell

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Terminology Cells in Octree – Vertices are 0-cells – Edges are 1-cells – Faces are 2-cells – Cubes are 3-cells Dual Vertices – Vertex dual to each m-cell – Constrained to interior of cell

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Terminology Cells in Octree – Vertices are 0-cells – Edges are 1-cells – Faces are 2-cells – Cubes are 3-cells Dual Vertices – Vertex dual to each m-cell – Constrained to interior of cell

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Our Partitioning of Space Start with vertex

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Our Partitioning of Space Build edges

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Our Partitioning of Space Build faces

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Our Partitioning of Space Build cubes

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Minimal Edge (1-Cell)

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Building Simplices

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Traversing Tetrahedra

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Octree Traversal from DC [Ju et al. 2002]

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Finding Features Minimize distances to planes

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Surfaces from Tetrahedra

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Manifold Property Vertices are constrained to their dual m-cells Simplices are guaranteed to not fold back Tetrahedra share faces Freedom to move allows reproducing features

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Finding Features

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Improving Triangulation

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Possible Problem: Face BeforeAfter

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Possible Problem: Edge BeforeAfter

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Preserving Topology Only move vertex to surface if there is a single contour. Count connected components.

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Preserving Topology Only move vertex to surface if there is a single contour. Count connected components.

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Improving Triangulation Before After

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Results

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Times Armadillo ManMechanical PartLensTank Depth89108 Ours2.58s4.81s9.72s8.78s Ours (Improved Triangles) 2.69s6.80s10.35s8.19s Dual Marching Cubes1.85s3.54s6.42s5.29s Dual Contouring1.35s2.97s5.99s3.78s

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Conclusions Calculate isosurfaces over piecewise smooth functions Guarantee manifold surfaces Reproduce sharp and thin features Improved triangulation

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Contour Refinement & Error Metric

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Finding Features

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Limitations Imperfect detection of sharp edges with multiple features Cube corners cannot move, limiting triangulation improvements Speed

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Times Time Breakdown of Tank OursOurs (Improved Triangles)DMCDC Total Time8.78s8.19s5.29s3.78s Time Tree4.02s6.13s3.41s2.69s Time Extract4.77s2.06s1.88s1.09s Triangles3.63M1.13M1.16M727k

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Traversing Edges

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