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Efficient MaxNorm Distance Computation and Reliable Voxelization

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Happy Buddha. Voxelization. Close-up. Results: Voxelization ... Bunny. 256x256x256. 128x128x128. 64x64x64. Results: ADF Generation. CAD benchmarks ... – PowerPoint PPT presentation

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Title: Efficient MaxNorm Distance Computation and Reliable Voxelization


1
Efficient Max-Norm Distance Computation and
Reliable Voxelization
  • Gokul Varadhan
  • Young J. Kim
  • Dinesh Manocha
  • University of North Carolina at Chapel Hill

Shankar Krishnan Suhas Diggavi ATT Labs
2
Max-Norm (l8)
  • Distance under max-norm

D8(x) max (x, y, z)
  • Applications
  • Markov decision processes Tsitsiklis et al. 96,
    Guestrin et al. 2001
  • Discrete objects in supercover model Andres et
    al. 96
  • Image analysis Lindquist 99
  • Dynamics and control systems Zhou et al. 96,
    Ghaoui 99
  • Tolerance analysis and NC machining Requicha 93,
    Farin et al. 2002
  • Volume graphics Wang Kaufman 94, Sramek
    Kaufman 99

3
Motivation
  • Voxelization
  • Represent a scene by a discrete set of voxels
  • Reduce to max-norm distance computation
  • Applications
  • Volume rendering Lorensen Cline 87, Wang
    Kaufman 94
  • Ray tracing Wang Kaufman 93
  • Implicit modeling Kobbelt et al. 2001, Ju et al.
    2002
  • Shape representation Frisken et al. 2000
  • Model repair Nooruddin Turk 2003

4
Previous Work
  • Distance Computation
  • Distance Fields and Voxelization

5
Previous Work Distance Computation
  • Euclidean
  • Survey in Lin et al. 98
  • Max-norm
  • Voronoi diagram complexity Boissonnat et al. 95,
    Koltun Sharir 2002
  • Skeleton computation Aichholzer Aurenhammer
    96
  • 2D Voronoi diagram and VLSI manufacturing
    Papadopoulou Lee 2001

6
Previous Work Distance Fields and Voxelization
  • Curvature Gibson 98, Shekhar et al. 96
  • Adaptive Distance Fields Frisken et al. 2000,
    Perry Frisken 2001
  • Tri-linear interpolation
  • Alias-free voxelization Sramek Kaufman 99

7
Outline
  • l8 Distance Computation
  • Problem
  • Optimization Framework
  • Specialized Algorithms
  • Convex Primitives
  • Algebraic Primitives
  • Triangulated Models
  • Complex Models
  • Bounding Volume Hierarchy
  • Graphics Hardware Approach
  • Reliable Voxelization
  • Adaptive Distance Field (ADF) Generation
  • Results

8
Problem
l8 Distance between a point and a primitive
Distance Cube size/2
9
Optimization Framework
D8(x) max (x, y)
10
Optimization Framework
x lies within R
P
R - x dominating region
11
Outline
  • l8 Distance Computation
  • Problem
  • Optimization Framework
  • Specialized Algorithms
  • Convex Primitives
  • Algebraic Primitives
  • Triangulated Models
  • Complex Models
  • Bounding Volume Hierarchy
  • Graphics Hardware Approach
  • Reliable Voxelization
  • Adaptive Distance Field (ADF) Generation
  • Results

12
Convex Primitives
  • Non-linear optimization reduces to convex
    optimization
  • Simpler solution when the query point is inside
    the primitive

13
Convex Primitives Point inside
  • Closest point Q has to lie on the vertex of the
    distance cube
  • Compute directed distance along 4 directions

Q
P
14
Convex Primitives Point outside
  • Perform convex optimization
  • Interior point methods Nesterov Nemirovsky 94
  • Quadrics
  • Second-order cone program
  • Convex polytopes
  • Make A 0
  • Linear program

15
Outline
  • l8 Distance Computation
  • Problem
  • Optimization Framework
  • Specialized Algorithms
  • Convex Primitives
  • Algebraic Primitives
  • Triangulated Models
  • Complex Models
  • Bounding Volume Hierarchy
  • Graphics Hardware Approach
  • Reliable Voxelization
  • Adaptive Distance Field (ADF) Generation
  • Results

16
Algebraic Primitives
  • Equation Solving approach
  • Convex as well as non-convex
  • Solve for the closest point

17
Vertex Case
  • Directed distance query
  • One of 8 directions ( 1/v3, 1/v3, 1/v3)

f(x) 0 x y 0 x z 0
18
Edge Case
  • Local minima/maxima
  • Partial derivative is zero
  • Lies on a partitioning plane

f(x) 0 df/dz(x) 0 x y 0
19
Face Case
  • Two partial derivatives are zero

f(x) 0 df/dy(x) 0 df/dz (x) 0
20
Equation Solving
  • Solve above equations for each vertex, edge and
    face
  • Solution set is finite in general
  • Obtain a set X of feasible values for the closest
    point
  • Calculate min x-p8 x ? X

21
Equation Solving
  • Quadrics
  • Quadratic Equation
  • Torus
  • Symmetry
  • Degree 8 polynomial

22
Outline
  • l8 Distance Computation
  • Problem
  • Optimization Framework
  • Specialized Algorithms
  • Convex Primitives
  • Algebraic Primitives
  • Triangulated Models
  • Complex Models
  • Bounding Volume Hierarchy
  • Graphics Hardware Approach
  • Reliable Voxelization
  • Adaptive Distance Field (ADF) Generation
  • Results

23
Distance Computation for a Triangle
12 partitioning triangles
24
Bounding Volume Hierarchy
  • Large polyhedral model
  • Naïve algorithm
  • Minimum over distance to each triangle
  • Speed it up using a precomputed bounding volume
    hierarchy

25
Outline
  • l8 Distance Computation
  • Problem
  • Optimization Framework
  • Specialized Algorithms
  • Convex Primitives
  • Algebraic Primitives
  • Triangulated Models
  • Complex Models
  • Bounding Volume Hierarchy
  • Graphics Hardware Approach
  • Reliable Voxelization
  • Adaptive Distance Field (ADF) Generation
  • Results

26
Graphics Hardware Approach
  • Approach similar to Hoff et al. 1999
  • Render 3D polygonal mesh approximation to the
    distance functions
  • Z-buffer holds the distance field

27
Distance Field Generation
28
Distance Functions
  • Non-linear in Euclidean case
  • Hyperboloid
  • Elliptical cone
  • Linear in case of l8
  • Collection of polygons

29
Distance Function Point
Z
Y
X
30
Distance Function Point
Z
Y
X
31
Distance Function Point
Z
Y
X
32
Distance Function Point
Z
Y
X
33
Distance Function Point
Z
Y
X
34
Distance Function Edge
Z
Y
X
35
Distance Function Edge
Z
Y
X
36
Distance Function Triangle
Z
Y
X
37
Distance Function Triangle
Z
Y
X
38
Non-convex Implicit Primitives
  • Triangulate the primitive
  • Obtain an estimate using graphics hardware
  • Refine estimate using a local non-linear
    optimization tool like LOQO.

39
Sources of Error
  • Total Error
  • Tessellation Error
  • Approximating a non-convex implicit or curved
    primitive by a polygonal mesh
  • Hardware Precision Error
  • Conservative estimates on the distance by
    offsetting the distance functions

40
Outline
  • l8 Distance Computation
  • Problem
  • Optimization Framework
  • Specialized Algorithms
  • Convex Primitives
  • Algebraic Primitives
  • Triangulated Models
  • Complex Models
  • Bounding Volume Hierarchy
  • Graphics Hardware Approach
  • Reliable Voxelization
  • Adaptive Distance Field (ADF) Generation
  • Results

41
Reliable Voxelization
  • Marching Cubes (MC)
  • Accuracy depends on resolution of the grid
  • missed components
  • unwanted handles, components
  • Unreliable voxel-intersection test
  • Detects surface by checking sign change
    (inside/outside status)
  • Complex voxels

MC
42
Voxel-Intersection Test
  • Reduce to max-norm distance computation
  • l8 distance at center of voxel lt voxel size/2

43
Voxel-Intersection Test
  • Exact test
  • Non-uniform voxels
  • use weighted norm
  • x8 max (w1 x, w2 y, w3 z)
  • where wi 1/a, 1/b, 1/c, i 1,2,3
  • and a, b, c are dimensions of the voxel along
    the three coordinate axes

44
Outline
  • l8 Distance Computation
  • Problem
  • Optimization Framework
  • Specialized Algorithms
  • Convex Primitives
  • Algebraic Primitives
  • Triangulated Models
  • Complex Models
  • Bounding Volume Hierarchy
  • Graphics Hardware Approach
  • Reliable Voxelization
  • Adaptive Distance Field (ADF) Generation
  • Results

45
Adaptive Distance Field (ADF) Generation
  • Goal
  • Bounded two-sided Hausdorff distance
  • Cannot provide such a bound if complex voxels
    exist
  • Given a Hausdorff distance bound e,
  • Check if voxel is intersecting using the
    voxel-intersection test
  • if no intersection
  • STOP
  • if complex voxel or voxel size gt e
  • SUBDIVIDE else STOP
  • Early termination in Step 2

46
Timings Algebraic primitives
  • Equation solving approach
  • l8 distance query
  • 40-50 usec for quadrics
  • 1-1.2 msec for torus
  • Need to solve few degree 8 polynomials

47
Timings Triangulated Models
  • 10 usec per triangle
  • Hierarchical Approach
  • Query time varied from 0.6 6.14 msec

48
Timings Graphics Hardware
  • SIMD-like capability
  • 128 x 128 x 128 uniform grid
  • 8 , 2.7, 5.6 secs for wrinkled torus, cup and
    spoon

49
Results Voxelization
Happy Buddha
Voxelization
Close-up
50
Results Voxelization
Dragon
Close-up
Voxelization
51
Results Voxelization
  • Localized Distance Computation
  • Given a point p and distance bound B,
  • Cull away primitives whose bounding boxes do not
    intersect the cube of length 2B centered at p

52
Results ADF Generation
  • CAD benchmarks
  • 1-5 solids defined using 3-5 Boolean operations
  • Generated an adaptive grid based on l8 distance
    computation
  • 15 secs per solid for Hausdorff distance bound e
    1/128
  • Reconstructed a boundary representation using an
    extended dual contouring algorithm Varadhan et
    al. 2003

53
Prior Voxel-Intersection Tests
  • Implicit surfaces
  • Lipschitz condition Kalra Barr 1989
  • Interval arithmetic Snyder 1992
  • Conservative
  • Euclidean distance Perry Frisken 2001
  • Surface intersects a voxel if
  • Euclidean distance lt half diagonal length
  • Surface intersects the bounding sphere of the
    voxel
  • Conservative

54
Future Work
  • Object-object l8 distance computation
  • Other applications of max-norm distance

55
Ongoing Work CSG Reconstruction
  • G. Varadhan, S. Krishnan, Y. Kim, D. Manocha,
    Feature-Sensitive Subdivision and Isosurface
    Reconstruction, accepted at IEEE Visualization
    2003

http//gamma.cs.unc.edu/recons
56
Acknowledgements
  • ARO Contract DAAD 19-99-1-0162
  • NSF Awards ACI-9876914, ACI-0118743
  • ONR Contracts N00014-01-1-0067
  • Intel Corporation
  • Kenny Hoff for his HAVOC software
  • Joe Warren and Scot Schaefer for dual contouring
    code

57
Thank You!
58
Optimization Framework
  • Assume f(x) 0 represents the interior of the
    primitive
  • Without loss of generality, assume the query
    point P is outside the primitive

59
Non-linear Optimization
hTx
Minimize
f(x) 0
subject to
gjTx 0, j1,2, …, 2(d-1)
and
60
Convex Decomposition
  • Leaf node in hierarchy obtained by
  • Decomposing surface into convex patches
  • Convex hull of each patch
  • Hierarchy constructed by merging children nodes
    and computing their convex hull
  • Convex hull computation
  • Real, Extraneous (virtual) triangles

61
Traversal
  • Maintain three values
  • U Upper bound to the distance from p to the
    polyhedron
  • Ub Upper bound to the distance from p to the
    currently visited node
  • Minimum over distance to all the real triangles
  • Lb Lower bound to the distance from p to the
    currently visited node
  • Minimum over distance to all triangles (real
    virtual)

62
Traversal
  • At each node
  • if Ub lt U
  • U ? Ub
  • As we go down the hierarchy, U will decrease and
    converge to the actual distance
  • Culling
  • if Lb gt U
  • return from the node

63
Degenerate Systems
  • Degenerate configurations (put an image)
  • Solution set not finite
  • Special case handling
  • Extraneous Solutions
  • Back substitution
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