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

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Problem
l8 Distance between a point and a primitive
Distance Cube size/2
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Optimization Framework
D8(x) max (x, y)
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Optimization Framework
x lies within R
P
R - x dominating region
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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
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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
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Edge Case

• Local minima/maxima
• Partial derivative is zero
• Lies on a partitioning plane

f(x) 0 df/dz(x) 0 x y 0
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Face Case

• Two partial derivatives are zero

f(x) 0 df/dy(x) 0 df/dz (x) 0
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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

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Distance Computation for a Triangle
12 partitioning triangles
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Bounding Volume Hierarchy

• Large polyhedral model
• Naïve algorithm
• Minimum over distance to each triangle
• Speed it up using a precomputed bounding volume
  hierarchy

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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
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Distance Functions

• Non-linear in Euclidean case
• Hyperboloid
• Elliptical cone
• Linear in case of l8
• Collection of polygons

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Distance Function Point
Z
Y
X
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Distance Function Point
Z
Y
X
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Distance Function Point
Z
Y
X
32
Distance Function Point
Z
Y
X
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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
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Voxel-Intersection Test

• Reduce to max-norm distance computation
• l8 distance at center of voxel lt voxel size/2

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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

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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

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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

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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

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Results Voxelization
Happy Buddha
Voxelization
Close-up
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Results Voxelization
Dragon
Close-up
Voxelization
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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

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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

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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

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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
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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
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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

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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)

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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

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Degenerate Systems

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