3D Polyhedral Morphing
Reading Assignments
Interactive Collision Detection, by P. M. Hubbard, Proc. of IEEE Symp on Research Frontiers in Virtual Reality, 1993.
Evaluation of Collision Detection Methods for Virtual Reality Fly-Throughs, by Held, Klosowski and Mitchell, Proc. of Canadian Conf. on Computational Geometry 1995.
Efficient collision detection using bounding volume hierarchies of k-dops, by J. Klosowski, M. Held, J. S. B. Mitchell, H. Sowizral, and K. Zikan, IEEE Trans. on Visualization and Computer Graphics, 4(1):21–37, 1998.
Collision Detection between Geometric Models: A Survey, by M. Lin and S. Gottschalk, Proc. of IMA Conference on Mathematics of Surfaces 1998.
Reading Assignments
OBB-Tree: A Hierarchical Structure for Rapid Interference Detection, by S. Gottschalk, M. Lin and D. Manocha, Proc. of ACM Siggraph, 1996.
Rapid and Accurate Contact Determination between Spline Models using ShellTrees, by S. Krishnan, M. Gopi, M. Lin, D. Manocha and A. Pattekar, Proc. of Eurographics 1998.
Fast Proximity Queries with Swept Sphere Volumes, by Eric Larsen, Stefan Gottschalk, Ming C. Lin, Dinesh Manocha, Technical report TR99-018, UNC-CH, CS Dept, 1999. (Part of the paper in Proc. of IEEE ICRA’2000)
Methods for General Models
Decompose into convex pieces, and take minimum over all pairs of pieces:
Optimal (minimal) model decomposition is NP-hard.
Approximation algorithms exist for closed solids, but what about a list of triangles?
Collection of triangles/polygons:
n*m pairs of triangles – brute force expensive
Hierarchical representations used to accelerate minimum finding
Hierarchical Representations
Two Common Types:
Bounding volume hierarchies – trees of spheres, ellipses, cubes, axis-aligned bounding boxes (AABBs), oriented bounding boxes (OBBs), K-dop, SSV, etc.
Spatial decomposition – BSP, K-d trees, octrees, MSP tree, R-trees, grids/cells, space-time bounds, etc.
Do very well in “rejection tests”, when objects are far apart
Performance may slow down, when the two objects are in close proximity and can have multiple contacts
BVH vs. Spatial Partitioning
BVH: SP:
– Object centric – Space centric
– Spatial redundancy – Object redundancy
BVH vs. Spatial Partitioning
BVH: SP:
– Object centric – Space centric
– Spatial redundancy – Object redundancy
BVH vs. Spatial Partitioning
BVH: SP:
– Object centric – Space centric
– Spatial redundancy – Object redundancy
BVH vs. Spatial Partitioning
BVH: SP:
– Object centric – Space centric
– Spatial redundancy – Object redundancy
Spatial Data Structures & Subdivision
Many others……
(see the lecture notes)
Uniform Spatial Sub
Quadtree/Octree
kd-tree
BSP-tree
Uniform Spatial Subdivision
Decompose the objects (the entire simulated environment) into identical cells arranged in a fixed, regular grids (equal size boxes or voxels)
To represent an object, only need to decide which cells are occupied. To perform collision detection, check if any cell is occupied by two object
Storage: to represent an object at resolution of n voxels per dimension requires upto n3 cells
Accuracy: solids can only be “approximated”
Octrees
Quadtree is derived by subdividing a 2D-plane in both dimensions to form quadrants
Octrees are a 3D-extension of quadtree
Use divide-and-conquer
Reduce storage requirements (in comparison to grids/voxels)
Bounding Volume Hierarchies
Model Hierarchy:
each node has a simple volume that bounds a set of triangles
children contain volumes that each bound a different portion of the parent’s triangles
The leaves of the hierarchy usually contain individual triangles
A binary bounding volume hierarchy:
Type of Bounding Volumes
Spheres
Ellipsoids
Axis-Aligned Bounding Boxes (AABB)
Oriented Bounding Boxes (OBBs)
Convex Hulls
k-Discrete Orientation Polytopes (k-dop)
Spherical Shells
Swept-Sphere Volumes (SSVs)
Point Swetp Spheres (PSS)
Line Swept Spheres (LSS)
Rectangle Swept Spheres (RSS)
Triangle Swept Spheres (TSS)
BVH-Based Collision Detection
Collision Detection using BVH
1. Check for collision between two parent nodes (starting from the roots of two given trees)
2. If there is no interference between two parents,
3. Then stop and report “no collision”
4. Else All children of one parent node are checked
against all children of the other node
5. If there is a collision between the children
6. Then If at leave nodes
7. Then report “collision”
8. Else go to Step 4
9. Else stop and report “no collision”
Evaluating Bounding Volume Hierarchies
Cost Function:
F = Nu x Cu + Nbv x Cbv + Np x Cp
F: total cost function for interference detection
Nu: no. of bounding volumes updated
Cu: cost of updating a bounding volume,
Nbv: no. of bounding volume pair overlap tests
Cbv: cost of overlap test between 2 bounding volumes
Np: no. of primitive pairs tested for interference
Cp: cost of testing 2 primitives for interference
Designing Bounding Volume Hierarchies
The choice governed by these constraints:
It should fit the original model as tightly as possible (to lower Nbv and Np)
Testing two such volumes for overlap should be as fast as possible (to lower Cbv)
It should require the BV updates as infrequently as possible (to lower Nu)
Observations
Simple primitives (spheres, AABBs, etc.) do very well with respect to the second constraint. But they cannot fit some long skinny primitives tightly.
More complex primitives (minimal ellipsoids, OBBs, etc.) provide tight fits, but checking for overlap between them is relatively expensive.
Cost of BV updates needs to be considered.
Trade-off in Choosing BV’s
increasing complexity & tightness of fit
decreasing cost of (overlap tests + BV update)
AABB
OBB
Sphere
Convex Hull
6-dop
Building Hierarchies
Choices of Bounding Volumes
cost function & constraints
Top-Down vs. Bottum-up
speed vs. fitting
Depth vs. breadth
branching factors
Splitting factors
where & how
Sphere-Trees
A sphere-tree is a hierarchy of sets of spheres, used to approximate an object
Advantages:
Simplicity in checking overlaps between two bounding spheres
Invariant to rotations and can apply the same transformation to the centers, if objects are rigid
Shortcomings:
Not always the best approximation (esp bad for long, skinny objects)
Lack of good methods on building sphere-trees
Methods for Building Sphere-Trees
“Tile” the triangles and build the tree bottom-up
Covering each vertex with a sphere and group them together
Start with an octree and “tweak”
Compute the medial axis and use it as a skeleton for multi-res sphere-covering
Others……
k-DOP’s
k-dop: k-discrete orientation polytope a convex polytope whose facets are determined by half-spaces whose outward normals come from a small fixed set of k orientations
For example:
In 2D, an 8-dop is determined by the orientation at +/- {45,90,135,180} degrees
In 3D, an AABB is a 6-dop with orientation vectors determined by the +/-coordinate axes.
Choices of k-dops in 3D
6-dop: defined by coordinate axes
14-dop: defined by the vectors (1,0,0), (0,1,0), (0,0,1), (1,1,1), (1,-1,1), (1,1,-1) and (1,-1,-1)
18-dop: defined by the vectors (1,0,0), (0,1,0), (0,0,1), (1,1,0), (1,0,1), (0,1,1), (1,-1,0), (1,0,-1) and (0,1,-1)
26-dop: defined by the vectors (1,0,0), (0,1,0), (0,0,1), (1,1,1), (1,-1,1), (1,1,-1), (1,-1,-1), (1,1,0), (1,0,1), (0,1,1), (1,-1,0), (1,0,-1) and (0,1,-1)
Building Trees of k-dops
The major issue is updating the k-dops:
Use Hill Climbing (as proposed in I-Collide) to update the min/max along each k/2 directions by comparing with the neighboring vertices
But, the object may not be convex…… Use the approximation (convex hull vs. another k-dop)
Building an OBBTree
Recursive top-down construction:
partition and refit
Building an OBB Tree
Given some polygons,
consider their vertices…
Building an OBB Tree
… and an arbitrary line
Building an OBB Tree
Project onto the line
Consider variance of
distribution on the line
Building an OBB Tree
Different line,
different variance
Building an OBB Tree
Maximum Variance
Building an OBB Tree
Minimal Variance
Building an OBB Tree
Given by eigenvectors
of covariance matrix
of coordinates
of original points
Building an OBB Tree
Choose bounding box
oriented this way
Building an OBB Tree: Fitting
Covariance matrix of
point coordinates describes
statistical spread of cloud.
OBB is aligned with directions of
greatest and least spread
(which are guaranteed to be orthogonal).
292.unknown
Fitting OBBs
Let the vertices of the i’th triangle be the points ai, bi, and ci, then the mean µ and covariance matrix C can be expressed in vector notation as:
where n is the number of triangles, and
Building an OBB Tree
Good Box
Building an OBB Tree
Add points:
worse Box
Building an OBB Tree
More points:
terrible box
Building an OBB Tree
Compute with extremal points only
Building an OBB Tree
“Even” distribution:
good box
Building an OBB Tree
“Uneven” distribution:
bad box
Building an OBB Tree
Fix: Compute facets of convex hull…
Building an OBB Tree
Better: Integrate over facets
Building an OBB Tree
… and sample them uniformly
Building an OBB Tree: Summary
OBB Fitting algorithm:
covariance-based
use of convex hull
not foiled by extreme distributions
O(n log n) fitting time for single BV
O(n log2 n) fitting time for entire tree
Tree Traversal
Disjoint bounding volumes:
No possible collision
Tree Traversal
Overlapping bounding volumes:
split one box into children
test children against other box
Tree Traversal
Tree Traversal
Hierarchy of tests
Separating Axis Theorem
L is a separating axis for OBBs A & B, since A & B become disjoint intervals under projection onto L
Separating Axis Theorem
Two polytopes A and B are disjoint iff there
exists a separating axis which is:
perpendicular to a face from either
or
perpedicular to an edge from each
Implications of Theorem
Given two generic polytopes, each with E edges and F faces, number of candidate axes to test is:
2F + E2
OBBs have only E = 3 distinct edge directions, and only F = 3 distinct face normals. OBBs need at most 15 axis tests.
Because edge directions and normals each form orthogonal frames, the axis tests are rather simple.
OBB Overlap Test: An Axis Test
s
h
a
b
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L
h
b
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L is a separating axis iff:
OBB Overlap Test: Axis Test Details
Box centers project to interval midpoints, so
midpoint separation is length of vector T’s image.
293.unknown
OBB Overlap Test: Axis Test Details
Half-length of interval is sum of box axis images.
295.unknown
OBB Overlap Test
Typical axis test for 3-space.
Up to 15 tests required.
s = fabs(T2 * R11 – T1 * R21);
ha = a1 * Rf21 + a2 * Rf11;
hb = b0 * Rf02 + b2 * Rf00;
if (s > (ha + hb)) return 0;
OBB Overlap Test
Strengths of this overlap test:
89 to 252 arithmetic operations per box overlap test
Simple guard against arithmetic error
No special cases for parallel/coincident faces, edges, or vertices
No special cases for degenerate boxes
No conditioning problems
Good candidate for micro-coding
OBB Overlap Tests: Comparison
Benchmarks performed on SGI Max Impact,
250 MHz MIPS R4400 CPU, MIPS R4000 FPU
Parallel Close Proximity
Q: How does the number of BV tests increase as the gap size decreases?
Two models are in parallel close proximity when every point on each model is a given fixed distance (e) from the other model.
In order to quantify the performance of the various BV types, we consider an easily parameterized configuration: parallel close proximity. When the models are concentric, every point on one model is a fixed distance (epsilon) from the other model, and vice-versa. This is our definition of transverse contact.
This means the two models are offset surfaces of one another, and the offset is the gap size (epsilon).
As epsilon decreases, we will have to descend the hierarchies more and more deeply in order to bound the models apart. How far will we have to go? That depends on the specific shape of the bounding volumes.
298.unknown
Parallel Close Proximity: Convergence
1
Consider two bounding volumes covering opposing parallel segments. We will call the distance between the segments one unit.
Parallel Close Proximity: Convergence
1
2
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When the separation is cut in half, we have to BVs half as big, which means we have to use twice as many. On each successive halving of the gap, we double the number of BV required. At each successive level, each sphere covers half the length of segment as the level above, and each sphere sticks out half as far away from the segment. Because the length covered and the amount extended from the segment diminish proportionally, we say spheres have linear convergence.
Parallel Close Proximity: Convergence
1
For AABBs, the reasoning is the same. Consider the two line segments, and the AABBs which bound them. If the segments were themselves axis-aligned, then the AABBs would fit perfectly. Let us assume the segments are tilted some arbitrary amount. Then, let us define the closest distance they can get without the AABBs touching to be one unit.
Parallel Close Proximity: Convergence
1
2
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2
1
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/
Now, keeping the tilt angle the same, but cutting the gap size in half results in a doubling of the number of required AABBs. This is true regardless of the angle of the line. As with spheres, with each successive level of the hierarchy, the amount the AABB extends outward from the segment diminishes at the same rate as does the amount of segment covered, so AABBs have linear convergence.
Parallel Close Proximity: Convergence
1
OBBs are different. First of all, any straight segment of any tilt angle would be bounded perfectly by a well-fitted OBB, so we will assume we are working with parallel arcs of low curvature. Given the arcs of specific curvature, position them as closely as possible without letting the OBBs touch.
Note, the longer the radius of curvature relative to the size of the OBBs, the tighter the fit will be.
Parallel Close Proximity: Convergence
1
4
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1
16
/
If the curvature is low, then will be able to reduce the gap by approximately a factor of four, and still bound the arcs apart by only doubling the number of OBBs. Each additional factor of four reduction in gap size doubles the number of OBBs.
One way of thinking about this is that the radius of curvature stays fixed, but since the OBBs are getting smaller, the radius curvature is increasing relative to the OBBs, and hence, the fit is getting progressively tighter.
The amount the OBB extends away from the arc is proportional to the SQUARE of the arc length covered.
Parallel Close Proximity: Convergence
1
4
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1
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Now, notice what quartering the gap does with each BV type. It quadruples the number of BVs required by spheres and AABBs, but it only doubles the number required by OBBs. This means that for every quartering of the gap, we descend additional two levels of the sphere and AABB trees, but only one additional level of the OBB tree.
Performance: Overlap Tests
k
O(n)
2k
O(n2)
OBBs
Spheres & AABBs
OBBs asymptotically outperform AABBs and spheres
Log-log plot
Gap Size (e)
Number of BV tests
Parallel Close Proximity: Experiment
We verify this in the plot of BV tests versus gap size. This log-log plot shows gap size for concentric spheres on the x-axis, and number of BV tests for that configuration — the experiment was repeated for each BV type. For extremely large gap sizes, on the right, we have a unit size sphere model inside a very large sphere model — and there is some minimum number of BV tests required to determine non contact. Then, as we reduce the gap size, we start seeing an increase in the number of BV tests needed — but the rate of increase of the AABB and sphere tests is greater than the rate of increase of the OBB tests. In fact, the slopes are different — the AABB and sphere slopes are approximately twice that of the OBB slope. On a log-log plot, only monomial functions are straight lines, and the slope is the degree of the monomial. This means the slope -2 of the AABBs and sphere indicates v = O(1/e^2) and the -2 slope for OBBs means v = O(1/e). So indeed, the former is the square of the latter.
Example: AABB’s vs. OBB’s
Approximation
of a Torus
Implementation: RAPID
Available at: http://www.cs.unc.edu/~geom/OBB
Part of V-COLLIDE: http://www.cs.unc.edu/~geom/V_COLLIDE
Thousands of users have ftp’ed the code
Used for virtual prototyping, dynamic simulation, robotics & computer animation
Hybrid Hierarchy of
Swept Sphere Volumes
PSS LSS RSS
[LGLM99]
Swept Sphere Volumes (S-topes)
PSS LSS RSS
299.bin
SSV Fitting
Use OBB’s code based upon Principle Component Analysis
For PSS, use the largest dimension as the radius
For LSS, use the two largest dimensions as the length and radius
For RSS, use all three dimensions
Overlap Test
One routine that can perform overlap tests between all possible combination of CORE primitives of SSV(s).
The routine is a specialized test based on Voronoi regions and OBB overlap test.
It is faster than GJK.
Hybrid BVH’s Based on SSVs
Use a simpler BV when it prunes search equally well – benefit from lower cost of BV overlap tests
Overlap test (based on Lin-Canny & OBB overlap test) between all pairs of BV’s in a BV family is unified
Complications
deciding which BV to use either dynamically or statically
PQP: Implementation
Library written in C++
Good for any proximity query
5-20x speed-up in distance computation over prior methods
Available at http://www.cs.unc.edu/~geom/SSV/
Test Method
Speed(us)
Separating Axis
6.26
GJK
66.30
LP
217.00
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Test Method
Speed(us)
Separating Axis
6.26
GJK
66.30
LP
217.00
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