# CS计算机代考程序代写 mips c++ B tree data structure algorithm Reading Assignments

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.

UNC Chapel Hill

M. C. Lin

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)

UNC Chapel Hill

M. C. Lin

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

UNC Chapel Hill

M. C. Lin

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

UNC Chapel Hill

M. C. Lin

BVH vs. Spatial Partitioning

BVH:

– Object centric

– Spatial redundancy

SP:

– Space centric

– Object redundancy

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M. C. Lin

BVH vs. Spatial Partitioning

BVH:

– Object centric

– Spatial redundancy

SP:

– Space centric

– Object redundancy

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M. C. Lin

BVH vs. Spatial Partitioning

BVH:

– Object centric

– Spatial redundancy

SP:

– Space centric

– Object redundancy

UNC Chapel Hill

M. C. Lin

BVH:

– Object centric

– Spatial redundancy

SP:

– Space centric

– Object redundancy

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M. C. Lin

Spatial Data Structures & Subdivision

Uniform Spatial Sub

Quadtree/Octree

kd-tree BSP-tree

Many others……

(see the lecture notes)

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M. C. Lin

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”

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M. C. Lin

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)

UNC Chapel Hill

M. C. Lin

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:

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M. C. Lin

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)

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BVH-Based Collision Detection

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Collision Detection using BVH

1.

2. 3. 4.

5. 6. 7. 8. 9.

Check for collision between two parent nodes (starting from the roots of two given trees)

If there is no interference between two parents, Then stop and report “no collision”

Else All children of one parent node are checked

against all children of the other node If there is a collision between the children

Then If at leave nodes

Then report “collision”

Else go to Step 4

Else stop and report “no collision”

UNC Chapel Hill

M. C. Lin

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

UNC Chapel Hill

M. C. Lin

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)

UNC Chapel Hill

M. C. Lin

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. UNC Chapel Hill

M. C. Lin

Trade-off in Choosing BV’s

Sphere AABB OBB 6-dop increasing complexity & tightness of fit

decreasing cost of (overlap tests + BV update)

Convex Hull

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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

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M. C. Lin

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

UNC Chapel Hill

M. C. Lin

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……

UNC Chapel Hill

M. C. Lin

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.

UNC Chapel Hill

M. C. Lin

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)

UNC Chapel Hill

M. C. Lin

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)

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Building an OBBTree

Recursive top-down construction: partition and refit

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Building an OBB Tree

Given some polygons, consider their vertices…

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Building an OBB Tree

… and an arbitrary line

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Building an OBB Tree

Project onto the line

Consider variance of distribution on the line

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Building an OBB Tree

Different line, different variance

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Building an OBB Tree

Maximum Variance

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Building an OBB Tree

Minimal Variance

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Building an OBB Tree

Given by eigenvectors of covariance matrix of coordinates

of original points

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Building an OBB Tree

Choose bounding box oriented this way

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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).

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M. C. Lin

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

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M. C. Lin

Building an OBB Tree

Good Box

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Building an OBB Tree

Add points: worse Box

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M. C. Lin

Building an OBB Tree

More points: terrible box

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M. C. Lin

Building an OBB Tree

Compute with extremal points only

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Building an OBB Tree

“Even” distribution: good box

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Building an OBB Tree

“Uneven” distribution: bad box

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Building an OBB Tree

Fix: Compute facets of convex hull…

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Building an OBB Tree

Better: Integrate over facets

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Building an OBB Tree

… and sample them uniformly

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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

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M. C. Lin

Tree Traversal

Disjoint bounding volumes: No possible collision

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Tree Traversal

Overlapping bounding volumes:

• split one box into children

• test children against other box

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Tree Traversal

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Tree Traversal

Hierarchy of tests

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Separating Axis Theorem

• L is a separating axis for OBBs A & B, since A & B become disjoint intervals under projection onto L

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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

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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.

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OBB Overlap Test: An Axis Test

L

s ha

hb

L is a separating axis iff:

s >h+h ab

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M. C. Lin

OBB Overlap Test: Axis Test Details

Box centers project to interval midpoints, so midpoint separation is length of vector T’s image.

B A TB

T

s

s = (TA −TB )•n

TA

n

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M. C. Lin

OBB Overlap Test: Axis Test Details

Half-length of interval is sum of box axis images. B

rB

n

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r =bRB•n+bRB•n+bRB•n B11 22 33

OBB Overlap Test

Typical axis test for 3-space.

s=fabs(T2*R11 – T1*R21); ha=a1*Rf21 + a2*Rf11; hb=b0*Rf02 + b2*Rf00;

if (s > (ha + hb)) return 0;

Up to 15 tests required. UNC Chapel Hill

M. C. Lin

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

UNC Chapel Hill

M. C. Lin

OBB Overlap Tests: Comparison

Test Method

Speed(us)

Separating Axis

GJK LP

6.26

66.30 217.00

Benchmarks performed on SGI Max Impact, 250 MHz MIPS R4400 CPU, MIPS R4000 FPU

UNC Chapel Hill

M. C. Lin

Parallel Close Proximity

1ε 1ε

Two models are in parallel close proximity when every point on each model is a given fixed distance (ε) from the other model.

Q: How does the number of BV tests increase

as the gap size decreases?

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Parallel Close Proximity: Convergence

1

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Parallel Close Proximity: Convergence

1 /2

1 /4

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Parallel Close Proximity: Convergence

1

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Parallel Close Proximity: Convergence

1 1/

/2 4

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M. C. Lin

Parallel Close Proximity: Convergence

1

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Parallel Close Proximity: Convergence

1 /4

1 /

16

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M. C. Lin

Parallel Close Proximity: Convergence

1 /4

1 1/

/4 4

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Performance: Overlap Tests

k

O(n)

OBBs

2k

O(n2)

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Spheres & AABBs

Parallel Close Proximity: Experiment

3 2

106 5 3

2

105 5 3

2

104 6

43 2

103 5 3

2

102 6

43 2

Log-log plot

101

10-4 2 3 456710-3 2 3 456710-2 2 3 456710-1 2 3 4567100 2 3 4567101

Gap Size (ε)

OBBs asymptotically outperform AABBs and spheres

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M. C. Lin

Number of BV tests

Example: AABB’s vs. OBB’s

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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

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Hybrid Hierarchy of Swept Sphere Volumes

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PSS LSS RSS

[LGLM99]

Swept Sphere Volumes (S-topes)

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PSS LSS RSS

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 UNC Chapel Hill

M. C. Lin

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.

UNC Chapel Hill

M. C. Lin

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

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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/

UNC Chapel Hill

M. C. Lin