1. Sara McMains UC Berkeley
Computer-Aided Design and Manufacturing Laboratory: GPU-accelerated Hausdorff Distance Computation for Freeforms
Sara McMains
UC Berkeley

2. Outline Numerical Iteration Algorithm Bounding-Box Algorithm
Introduction
Numerical Iteration Algorithm
Results
16 April 2019

3. Directed Hausdorff Distance
Measure of similarity from one geometric object to another
Asymmetric
Can be extended to 3D surfaces and meshes [Alt et al., Barton et al., Elber & Grandine, Tang et al., Guthe et al.]
Computes maximum deviation from one set to another
Applications include shape matching, penetration depth, mesh optimization, etc.
From A to B
16 April 2019

4. Outline Numerical Iteration Algorithm Bounding-Box Algorithm
Introduction
Numerical Iteration Algorithm
Results
16 April 2019

5. BB Algorithm Overview Surface bounding-boxes
Build bounding-box hierarchies
Bottom-up
Traverse hierarchy top-down & cull bounding-box pairs
Compute min-max distance ranges
Perform culling of bounding-box pairs at each level
Compute HD for unculled sub-patches
Finest level of hierarchy SA SB
16 April 2019

6. Min-Max Distance Ranges Between BB-Pairs
Surface SB
= Distance Range ∞
Max
Min
Surface SA
Min/Max distance ranges
16 April 2019

7. Culling Tests Surface SB Min/Max distance matrix Surface SA
Perform culling of bounding-box pairs based on the distance ranges
Surface SB
Surface SA
In our CUDA implementation, we store it as a 2D array to increase the maximum number of BBox pairs
Min/Max
distance matrix
16 April 2019

8. Bounding Box Hierarchy B Bounding Box Hierarchy A
Hierarchy Traversal
Extend culling tests to each level of AABB hierarchy E F I J G H K L M N Q R O P S T A B A B C D
Efficiently group together the non-culled bounding-box pairs for GPU execution C D
Example starting at base level
Bounding Box Hierarchy B
Create next level address array 4 5 8 9 6 7 10 11 12 13 16 17 14 18 19 1 1 2 3 2 3
Bounding Box Hierarchy A 4F 4S 5E 5F 5T 9F 7R
12H
13G
17F
14F
18G
17J
17N
18J
18N
19P 0A 0B 0C 0D 1A 1B 1C 1D 2A 2B 2C 2D 3A 3B 3C 3D 0A 0B 0C 0D 1A 1B 1C 1D 2A 2B 2C 2D 3A 3B 3C 3D 0A 0D 1A 2A 2C 3A 3B 3C
Cull Bounding-box Pairs
Address Array
16 April 2019

9. GPU Indirection int arrayIndex = arrayIndexData[index];
float arrayVal;
if (arrayIndex >= 0) {
arrayVal = arrayData[arrayIndex]; }
//// process arrayVal ////
int arrayIndex = arrayIndexData[index];
bool arrayValInit = true;
if (arrayIndex < 0) {
arrayIndex = 0;
arrayValInit = false; }
float arrayVal = arrayData[arrayIndex];
if (arrayValInit)
//// process arrayVal ////
16 April 2019

10. Outline Numerical Iteration Algorithm Bounding-Box Algorithm
Introduction
Numerical Iteration Algorithm
Results
16 April 2019

11. Numerical Iteration Algorithm
Split surface SB into Bezier patches (less branching)
Sample points on surface SA
Surface SB
From each point on surface SA
For each Bezier patch
Find the closest point in the patch using numerical iteration
Ping-pong Bezier patches to find closest point on surface SB
Thrust reduction to find Hausdorff (max) distance
Surface SA
16 April 2019

12. Projection and Numerical Iteration
Find closest control point
Evaluate (u,v) point corresponding to this control point
Find parameterized tangent plane
Find projection of the point to the tangent plane
Use the parameterization of the projection point to find new (u,v) point
Stop iteration when distance between two consecutive projection points is less than the stopping distance k du
kdu
16 April 2019

13. Outline Numerical Iteration Algorithm Bounding-Box Algorithm
Introduction
Numerical Iteration Algorithm
Results
16 April 2019

14. Bounding-Box Method Timing
Position
Position
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15. Timing Comparison
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16. Bounding-Box Method Failure Cases
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17. Future Work Bounding-box method Numerical Iteration method
Time dependent on positions of the surfaces
Slow or fails for certain cases such as offset surfaces
Numerical Iteration method
Often slower particularly for surfaces with many knots
Constant computation time for different positions
Future directions
Hybrid method for Hausdorff distance computations if offset surfaces are detected
Multi-level method for Hausdorff distances of objects consisting of multiple surfaces
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18. Conclusions Interactively compute Hausdorff distances
Dynamic deformable surfaces
Novel hierarchical culling tests
Novel iterative solver
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19. Acknowledgements
Adarsh Krishnamurthy
Iddo Hanniel
SolidWorks
NSF

20. Questions?
16 April 2019

21. 16 April 2019

22. Accuracy
16 April 2019

23. Theoretical Bounds
Based on K, curvature based maximum deviation for each surface
Curved Surface
Linear Approximation
h(A,ΔB) a1 a3 a2 b1 b3 b2
h(A,B)
Bounds between the actual HD and the HD between the triangular linear approximation of the surface
HD bounded by KA +KB
h(ΔA, ΔB)
h(A, ΔB) ΔB b1 b2 a1 a2 ΔA KA
16 April 2019

24. Outline Numerical Iteration Algorithm Bounding-Box Algorithm
Introduction
Numerical Iteration Algorithm
Results
16 April 2019