comparison of collision detection algorithms tony young m.math candidate july 19th, 2004
TRANSCRIPT
Comparison of Collision Detection Algorithms
Tony Young
M.Math Candidate
July 19th, 2004
Outline• Collision Detection Background
– What is collision detection?– Applications and Importance
• Detection Algorithms– Bounding Boxes– Bounding Spheres– BSP Trees– Hubbard
What is Collision Detection?• Collision detection is the processing of
two object’s bounds to determine if those bounds intersect at any time, t
• Collision detection is a difficult problem to solve in constant or linear time
• Collision detection schemes are used in many applications
Applications and Importance
• Marine, Land and Air navigation– Detecting if/when collisions between two
vehicles will take place– Aiding in avoiding collisions with alerts to
captains, drivers and pilots
Applications and Importance
• Accident Alerts– Automatically notifying police of collisions
at intersections, on highways, etc.– Aids in response time and provides details
as to the speed of collision, etc.– Potentially saves lives!
Applications and Importance
• Graphics Rendering– Important for rendering a scene– We want realistic things to happen
Applications and Importance
• Simulation– Simulations should be as accurate to life
as possible– Objects move according to impacts
Applications and Importance
• Animation– In order to properly render animations we
must know when fabrics are pulled tight against bodies, items are resting on tables, etc.
– Lifelike animation requires lifelike modeling of collisions, and this requires collision detection
Applications and Importance
• Computer Games– In order to determine when a player is hit,
or how an object should move in a scene, collisions must be detected
Applications and Importance
• Overriding themes– Safety– Entertainment– Research– Three opposites, but both very important to
our current value system in North America
Bounding Boxes• Place a box around an object
– Box should completely enclose the object and be of minimal volume
– Fairly simple to construct
• Test intersections between the boxes to find intersections
Bounding Boxes• Each box has 6 faces (planes) in 3D
– Simple algebra to test for intersections between planes
– If one of the planes intersects another, the objects are said to collide
Bounding Boxes• Example bounding boxes
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Bounding Boxes• Space complexity
– Each object must store 8 points representing the bounding box
– Therefore, space is O(8) and Ω(8)
Bounding Boxes• Time complexity
– Each face of each object must be tested against each face of each other object
– Therefore, O((6n)2) = O(n2)• n is the number of objects
Bounding Boxes• Pro
– Very easy to implement– Very little extra space needed
• Con– Very coarse detection– Very slow with many objects in the scene
Bounding Spheres• Similar to bounding boxes, but instead
we use spheres– Must decide on a “center” point for the
object that minimizes the radius– Can be tough to find such a sphere that
minimizes in all directions– Spheres could leave a lot of extra space
around the object!
Bounding Spheres• Each sphere has a center point and a
radius– Can build an equation for the circle– Simple algebra to test for intersection
between the two circles
Bounding Spheres• Example bounding spheres
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Bounding Spheres• Space complexity
– Each object must store 2 values - center and radius - to represent the sphere
– Therefore, space is O(2) and Ω(2)– Space is slightly less than bounding boxes
Bounding Spheres• Time complexity
– Each object must test it’s bounding sphere for intersection with each other bounding sphere in the scene
– Therefore, O(n2)• n is the number of objects
– Significantly fewer calculations than bounding boxes!
Bounding Spheres• Pro
– Even easier to implement (than bounding boxes)
– Even less space needed (than bounding boxes)
• Con– Still fairly coarse detection– Still fairly slow with many objects
BSP Trees• BSP (Binary Space Partitioning) trees
are used to break a 3D object into pieces for easier comparison– Object is recursively broken into pieces
and pieces are inserted into the tree– Intersection between pieces of two object’s
spaces is tested
BSP Trees• We refine our BSP trees by recursively
defining the children to contain a subset of the objects of the parent– Stop refining on one of a few cases:
• Case 1: We have reached a minimum physical size for the section (ie: one pixel, ten pixels, etc)
• Case 2: We have reached a maximum tree depth (ie: 6 levels, 10 levels, etc)
• Case 3: We have placed each polygon in a leaf node• Etc… - Depends on the implementer
BSP Trees• Example BSP Tree
BSP Trees• Example BSP Tree
BSP Trees• Example BSP Tree
BSP Trees• Example BSP Tree
BSP Trees• Collision Detection
– Recursively travel through both BSP trees checking if successive levels of the tree actually intersect
• If the sections of the trees that are being tested have polygons in them:
– If inner level of tree, assume that an intersection occurs and recurse
– If lowest level of tree, test actual intersection of polygons
• If one of the sections of the trees that are being tested does not have polygons in it, we can surmise that no intersection occurs
BSP Trees• Collision Detection Example
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BSP Trees• Collision Detection Example
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BSP Trees• Collision Detection Example
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BSP Trees• Collision Detection Example
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BSP Trees• Collision Detection Example
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BSP Trees• Space Complexity
– Each object must store a BSP tree with links to children
– Leaf nodes are polygons with geometries as integer coordinates
– Therefore, space depends on number of levels of tree, h, and number of polygons (assume convex triangles - most common), n
– Therefore, space is O(4h + 3n) and Ω(4h + 3n)
BSP Trees• Time Complexity
– Each object must be tested against every other object - n2
– If intersection at level 0, must go through the tree - O(h)
• Assume all trees of same height
– Depends on number of intersections, m– Therefore, O(n2 + m*h) and Ω(n2)
BSP Trees• Pros
– Fairly fine grain detection
• Cons– Complex to implement– Still fairly slow– Requires lots of space
Hubbard• Makes use of Sphere Trees and Space-
time Bounds to iteratively refine it’s accuracy
• Two phases– Broad Phase: constructs space-time
bounds and does coarse comparisons– Narrow Phase: run only if broad phase
detects a collision; refines detection to determine if there really was a collision
Hubbard - Broad Phase• Constructs space-time bounds
– 4D structure giving conservative estimate of where objects might be in the future
• Finds intersections between space-time bounds at time ti for objects O1 & O2
– Can stop processing until ti arrives
Hubbard - Broad Phase– At ti, checks O1 & O2’s bounding spheres to
determine if an intersection occurs– If yes, refers objects to narrow phase– If no, recalculate space-time bounds and
start again
Hubbard - Narrow Phase• Progressively refines intersection test
accuracy– Uses better approximations to O1 and O2
through the constructed sphere trees– Each level of a sphere tree fits object more
tightly– Can recursively check for intersections on
small parts of the tree
Hubbard - Narrow Phase• Allows system to interrupt algorithm after
each iteration– System gets as accuracy proportional to amount
of computing time it can spare
• Narrow phase terminates when– Case 1: System interrupts: current result is final– Case 2: Finds no intersection at a sphere tree
level– Case 3: Hits sphere tree leaves: Intersection
Hubbard - Example• Start System
– Construct Sphere Trees– Run scene
• Broad Phase on Frame 1– Calculates space-time bounds– Finds intersection between O1 and O2 at ti
= 5
• Run scene to Frame 5
Hubbard - Example• Broad Phase on Frame 5
– Detects no collision on bounding spheres– Calculates space-time bounds
– Finds intersection between O2 and O3 at ti = 7
• Run scene to Frame 7
Hubbard - Example• Broad Phase on Frame 7
– Detects collision on bounding spheres
• Narrow Phase on Frame 7 for O2 & O3
– Detects collision on level 1 spheres– Detects collision on level 2 spheres– Detects no collision on level 3 spheres
• Exit case 2
Hubbard - Example• Broad Phase on Frame 8
– Calculates space-time bounds
– Finds intersection between O3 and O4 at ti = 10
• Run scene to Frame 10
Hubbard - Example• Broad Phase on Frame 10
– Detects collision on bounding spheres
• Narrow Phase on Frame 10 for O3 and O4
– Detects collision on level 1 spheres– Detects collision on level 2 spheres– Detects collision on level 3 spheres– No more levels - collision detected
• Exit case 3
Hubbard - Example• Broad Phase on Frame 11
– Calculates space-time bounds
– Finds intersection between O4 and O5 at ti = 12
• Run scene to Frame 12
Hubbard - Example• Broad Phase on Frame 12
– Detects collision on bounding spheres
• Narrow Phase on Frame 12 for O4 and O5
– Detects collision on level 1 spheres– Detects collision on level 2 spheres– Detects collision on level 3 spheres– Interrupted by system - collision currently detected
• Exit case 1
Hubbard - Problem!• Collision may not be present further
down the tree– Collision is reported anyway because
system interrupted without narrow phase completing
– Leads to false positives
Hubbard - Space-Time Bounds
• Space-time bounds are 4D structures– represent object’s possible location in 3D
over time
• System knows– Object’s position - p(x,y,z)– Object’s velocity - v(x,y,z)– Object’s acceleration - a(x,y,z)– Value, M, such that | a(t) | ≤ M until t = t j
Hubbard - Space-Time Bounds
• Then, we know that p is subject to the inequality
| p(t) - [p(0) + v(0)t] | ≤ (M / 2)t2
• Thus, object’s position is inside a sphere of radius (M / 2)t2 centered at p(0) + v(0)t
Hubbard - Space-Time Bounds - Example
• Position of p at t = 0
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Hubbard - Space-Time Bounds - Example
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• Possible position of p at t = 0.1
Hubbard - Space-Time Bounds - Example
• Possible position of p at t = 0.2
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Hubbard - Space-Time Bounds - Example
• Possible position of p at 0 ≤ t ≤ 1
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Hubbard - Space-Time Bounds - Example
• Possible enclosing hypertrapezoid
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Hubbard - Space-Time Bounds
• This approximation results in large amounts of conservatism– Hypertrapezoid bounds areas that object
could never occupy
• If we know that acceleration is limited to a certain directional vector, d(t), we can generate a cutting plane behind object’s position
Hubbard - Space-Time Bounds
• Cutting plane– allows us to reduce the size of the
hypertrapezoid– Allows us to make more accurate broad
phase detections– Requires application to provide d(t)
Hubbard - Space-Time Bounds
• A complete space-time bound, B, requires– Application provided M– Application provided d(t)– Calculated hypertrapezoid - T– Calculated cutting plane - P
Hubbard - Space-Time Bounds - Example
• Possible complete space-time bound
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Hubbard - Using Bounds• How do we use space-time bounds?
– Recall: If they intersect, we have detected a coarse collision
• Intersection of bound B1 and B2 can happen in three ways:– (1) A face, f1, of T1 intersects a face, f2, of
T2
Hubbard - Using Bounds• Intersection of bound B1 and B2 can
happen in three ways:– (2) A face, f, of a T intersects a cutting
plane, P, of a T• Object can never be behind the cutting plane• Only care about part of f in front of P• To get in front of P, f must intersect another
face• Only care about intersection of two faces
Hubbard - Using Bounds• Intersection of bound B1 and B2 can
happen in three ways:– (3) A cutting plane, P1, of T1 intersects a
cutting plane, P2, of T2
• Object can never be behind the cutting plane• Only care about part of P1 in front of P2 and vs.• To get in front of P1, a face from T1 must
intersect a face from T2
• Only care about intersection of two faces
Hubbard - Using Bounds• Reduced three cases of intersection to
one that we care about
• How do we find intersections?– Projection– Subdivision
Hubbard - Finding Intersections
• Relies on three axioms (proved in papers, not here)– Each face f of a hypertrapezoid T is
“normal” to one of the axis of the coordinate system
– Each face f is included in a face set,Fa = f | f is normal to axis a, a in x, y, z
– Each intersection takes place between two faces in the same Fa
Hubbard - Finding Intersections
• Algorithm can test each combination in each face set
Hubbard - Finding Intersections: Projection
• Step 1: Project each face in a face set, Fa, onto the a-t plane.
– Faces will appear as 2D lines on the plane– Intersection of these lines is necessary but
not sufficient for an intersection of two faces
Hubbard - Finding Intersections: Projection
• Step 2: Find intersections between 2D line segments– Trivial algebra and mathematics– Keep track of each intersection, I, as a set
of intersecting faces, and the point on the t plane where they intersect
Hubbard - Finding Intersections: Projection
• Step 3: Check intersection of cube cross-sections of the hypertrapezoids that the two faces in I belong to at time t– Trivial algebra and mathematics– If intersection no longer present, drop
faces– If intersection still present, keep faces
Hubbard - Finding Intersections: Projection
• Step 4: Determine if point of intersection is behind a cutting plane for either hypertrapezoid– If so, intersection is discarded– If not, intersection is real and is reported
• Remember: We are only interested in the EARLIEST intersection!
Hubbard - Projection Example
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Hubbard - Projection Example
• Faces in the 3D plane
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Hubbard - Projection Example
• Faces projected onto t-z plane
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Hubbard - Projection Example
• Projections intersect at t=1
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Hubbard - Projection Example
• Cube cross-sections intersect
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Hubbard - Projection Example
• Intersection below cutting plane
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Hubbard - Finding Intersections: Subdivision
• Recoursively divide hypertrapezoids into 3D cubes along the t-axis– Test for intersections between any pair of cube
faces in 3D– If there is an intersection, subdivide the cubes and
check again
• Aim is to find time t at which intersection takes place– Base case is an application-provided ∆t
Hubbard - Subdivision Example
• Check first section
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Hubbard - Subdivision Example
• Intersection: subdivide
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Hubbard - Subdivision Example
• Intersection: subdivide again
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Hubbard - Subdivision Example
• ∆t reached with intersection at t=1
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Hubbard - Projection vs. Subdivision
• Projection is more difficult to implement• Subdivision might suffer from false positives
due to bad ∆t• Projection executed faster during empirical
tests and is thus used in the final version of the algorithm– Tests used 200 sets of randomly distributed
hypertrapezoids of varying parameters (ie: position, velocity, etc.)
Hubbard• Space-time bounds are nice, but are
only half the equation
• We need sphere trees for each object in the scene in order to run our narrow phase
Hubbard - Sphere Trees• Sphere trees are a constant refinement
of a bounding sphere for an object– An object is represented as a set of
overlapping spheres– The overlapping spheres are represented
as a set of tighter overlapping spheres, etc. until we reach the application’s requested accuracy level
Hubbard - Sphere Trees– Progressively more spheres are used at
each level of the tree in order to approximate the object closer
– The spheres at level i + 1 more tightly cover the area that the spheres at level i cover
– Spheres are very easy to compare for intersection
Hubbard - Sphere Trees - Example
• Level 0: the root
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Hubbard - Sphere Trees - Example
• Level 1
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Hubbard - Sphere Trees - Example
• Level 2: etc…
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Hubbard - Sphere Trees• Accuracy of sphere tree, or “tightness of
fit”, is important to ensure that we have proper accuracy– Algorithm could stop narrow phase at any
point– Want to make sure that we have the most
accurate detection possible to that point
Hubbard - Sphere Trees• Construction of sphere tree is a form of
multiresolution modeling– Very complex task– Tough to automate efficiently
• Many algorithms– We will look at Medial-Axis Surface method
Hubbard - Sphere Trees• Medial-axis surface
– Corresponds to the “skeleton” of an object– Difficult to build for a 3D polyhedron
• Algorithm works backwards– Covers the object with tightly fitting spheres first– Combine spheres into larger ones at next step– Sphere tree is constructed bottom-up
Hubbard - Sphere Trees• Start by making the smallest sphere that
covers an area of the object and touches it at eight points– Repeat until the entire object is enclosed in
these spheres
• Continue by combining adjacent spheres into larger ones– Stop when we combine all remaining
spheres into one large sphere
Hubbard - Sphere Trees - Example
• Possible first computation of medial-axis surface algorithm
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Hubbard - Problem• This process takes a HUGE amount of
time!– An object with 626 triangles took 12.4
minutes to generate a sphere tree!– Note that sphere trees are also constructed
prior to running the scene
Hubbard - Putting it all together
• Take scene as input– Object models– Values of M and d(t)
• Generate sphere tree for each object• Calculate first space-time bound
– Calculate hypertrapezoid and cutting plane
– Calculate ti where first intersection takes place
• Start scene and broad/narrow phase testing
Hubbard - Example• Sample scene
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Hubbard - Example• Sphere trees
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Hubbard - Example• Initial space-time bound
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Hubbard - Example• Initial intersection
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Hubbard - Example• Initial intersection above cut plane: ti=5
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Hubbard - Example• Run scene to t=5
• Recalculate space-time bounds– Still intersects
• Run narrow phase at t=5
Hubbard - Example• Sphere trees intersect at level 0
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Hubbard - Example• Sphere trees intersect at level 0
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Hubbard - Example• Sphere trees intersect at level 1
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Hubbard - Example• Sphere trees intersect at level 2
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Hubbard - Example• No levels left - intersection reported• Could have exited if application
interrupted us– Since we had an intersection at each level,
we would have reported an intersection
• Could have exited if we found no intersection at a lower level– We had no lower levels to resort to
Hubbard - Evaluation• Tested against Turk’s method which tests for
intersections between spheres stored in BSP trees (no space-time bounds)– If detections are found very early in the simulation
(before 0.25 sec), Hubbard is slower than Turk– Otherwise, Hubbard experiences a detection
speed boost of approximately 10 times over Turk
Hubbard - Evaluation• Speed-up in % over Turk’s BSP trees
– Mean speedup is %– Level is tree level (their 1 = example 0)– Number is of cases to reach that level
Hubbard - Evaluation• Space Complexity
– Each object has• One Hypertrapezoid - 4 D = 16 points• One Sphere Tree - 2h links with 2h spheres (2h*(2 points))• One set of direction and acceleration vectors - 2 points• One set of bounding values (M and d(t)) - 2 values
– Object’s structures do not depend on each other– Therefore, space is O(2h + 4h + 20) and Ω(2h + 4h
+ 20)
Hubbard - Evaluation• Time Complexity
– Broad Phase• Must compare each face in each set to each other face
in each set - O(3(2n)2) = O(n2) - and - Ω(1)– n is the number of objects
– Narrow Phase• Must compare spheres in the sphere tree for the two
colliding objects - O(m) - and - Ω(1)– m is the number of levels of sphere tree compared
– Therefore, time is Ω(1) and O(n2 + m) per run of detection algorithm
Hubbard - Conclusions• Algorithm allows average-case real-time
collision detection• Faster than Turk’s algorithm (previously
thought to be the best)• May have false positives if application
interrupts processing too early in narrow phase (doesn’t get far enough down the tree)
• Constructing sphere trees is slow
Summary• Collision detection is a difficult problem
– Many different strategies– Most based on object models– Tough to do in real time
• Collision detection strategies have not progressed significantly– Some small advances in speed and space– No major new developments
• Most focus is now on collision detection through image processing
Summary• Bounding Boxes
– Space but not time efficient– Very inaccurate detection– Not real-time
• Bounding Spheres– Space but not time efficient– Relatively inaccurate detection– Not real-time
Summary• BSP Trees
– Inefficient in time and space– Quite accurate detection– Not real-time
• Hubbard– Relatively efficient in time but not space– Highly accurate detection– Not real-time (but can be close)
Summary• The clear winner depends on the application
– Hubbard performs best for most “real world” applications where objects are known ahead of time (ie: games, animation, ATC, etc.)
– BSP trees are good for objects that are not known ahead of time (ie: auto collision
– Bounding boxes / spheres are good for objects of that approximate shape, or for very fast detection (ie: primitive games, primitive scene animation)
Questions?