Data Structures for 3D Searching
partly based on:
chapter 12 in
Fundamentals of Computer Graphics, 3rd ed.
(Shirley & Marschner)
Slides by Marc van Kreveld
1
Data structures
• Data structures for representation
– geometry (coordinates)
– topology (connectivity)
– attributes (anything else like material, …)
• Data structures for searching
– use the geometry to reduce the number objects that need
be tested from all of them to some small subset
– windowing queries, ray tracing, nearest neighbors
2
Data structures for searching
• Several steps in the algorithms we have seen until
now can be sped up using data structures for efficient
searching
– Find the k nearest neighbors to each point
(straightforward in O(n2) time; linear time per point)
– Find the support of a plane tried in RANSAC
(straightforward in O(n) time)
– Test whether a mesh-changing operator causes selfintersections of the manifold (straightforward in O(n) time)
– Also: for ray tracing, windowing
3
Data structures for searching
•
•
•
•
•
Grid structure
Quadtree and Octree
Bounding Volume Hierarchy, R-tree
Kd-tree
BSP tree (Binary Space Partition)
4
Simple grid structure
• Also: cubical grid, cubic partitioning
• Tile the space with equal-size cubes and store each
object with every cube it intersects
• The 3D grid is a 3D array of lists; list elements are
pointers to the objects
• Small grid cells: much storage overhead, but good
for query time
• Big grid cells: little storage overhead, worse query
time
5
Simple grid structure
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Simple grid structure
ray tracing query
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Simple grid structure
k nearest neighbors query
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Simple grid structure
RANSAC line support computation
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Simple grid structure
windowing query
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Simple grid structure
• A hierarchical version of a grid may give more efficient
querying: one emptiness test at a higher level square
may make
several tests
at lower level
squares
unnecessary
11
Quadtrees and Octrees
• Tree version of the (hierarchical) square/cube grid
• Storage requirements better than grid when large
parts of the grid contain no objects
• In a quadtree
– the root corresponds to a bounding square
– children correspond to four subsquares : NW, NE, SW, SE
– nodes with 0 or 1 objects in their square are leaves
• In an octree
– the root corresponds to a bounding cube
– children correspond to eight subcubes
– nodes with 0 or 1 objects in their cube are leaves
12
Quadtrees and Octrees
NW
NE
NW NE SW SE
root
leaf
leaf
leaf storing one object
SW
SE
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Quadtrees and Octrees
NW
NE
NW NE SW SE
root
leaf
leaf
leaf storing one object
SW
SE
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Quadtrees and Octrees
NW
NE
This ray tracing query
visits 10 nodes in the
quadtree (12 squares
in the original grid)
SW
SE
15
Quadtrees and Octrees
• Other queries are performed in the straightforward
way
– windowing query: easy
– RANSAC support of line: determine all cells intersected by
the strip, and test points in those cells only for inclusion in
the strip, right normal, and therefore support
– k nearest neighbors: explore cells further and further away
from the query point until we know for sure that we have
the k nearest neighbors
 explore all cells that intersect the circle centered at the
query point and through the k-th closest point
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17
Octree
18
Quadtrees and Octrees
• Often a quadtree or octree is not made deeper than a
desired level of precision
• E.g., given an object of 1 x 1 x 1 meters stored in a
triangle mesh, store the triangles in an octree up to a
cell size of 4 x 4 x 4 millimeters
• Leaves that contain more than one object store them
in a list (it is assumed that there would only be O(1) of
them at this detail level)
• Also possible: split a square/cube until at most some
constant c objects or points intersect a cell
19
Bounding Volume Hierarchies
• Tree structure where internal nodes store bounding
shapes of all objects in the subtree
• If a query object intersects the bounding shape, it
may intersect some object in that subtree, otherwise
certainly not
• Bounding shapes can be spheres, axis-parallel
bounding boxes, or arbitrarily oriented bounding
boxes; axis-parallel BB is most common
20
Bounding Volume Hierarchies
• In graphics usually binary trees
• For huge data sets where the data must be on disk,
usually higher-degree trees: R-tree
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R-trees
• 2-dimensional version of the B-tree:
B-tree of maximum degree 8; degree between 3 and 8
Internal nodes with k children have k – 1 split values
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R-trees
• Can store:
–
–
–
–
a set of polygons (regions of a subdivision)
a set of polygonal lines (or boundaries)
a set of points
a mix of the above
• Stored objects may overlap
23
R-trees
• Originally by Guttman, 1984
• Dozens of variations and optimizations since
• Suitable for windowing, point location, intersection
queries, and ray tracing
• Heuristic structure, no order bounds ( O(..) )
• Example of a bounding volume hierarchy
24
R-trees
• Every internal node contains
entries (rectangle, pointer to
child node)
• All leaves contain entries
(rectangle, pointer to object)
in database or file
• Rectangles are minimal axisparallel bounding rectangles
• The root has  2 and  M
entries
• All other nodes have at
least m and at most M
entries
• All leaves have the same
depth
• m > 1 and M > 2m
(e.g. m = 200; M = 1000)
25
R-trees
• M is chosen so that a full node still (just) fits in a
single block of disk memory
• m is chosen depending on a trade-off in search time
and update time
– larger m: faster queries, slower updates
– smaller m: slower queries, faster updates
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Object descriptions
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Grouping of objects
Windowing query: the fewer
rectangles intersected, the fewer
subtrees to descend into
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Grouping of objects
• Objects close together in same leaves  small
rectangles  queries descend in only few subtrees
• Group the child nodes under a parent node such that
small rectangles arise
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Heuristics for fast queries
•
•
•
•
Small area of rectangles
Small perimeter of rectangles
Little overlap among rectangles
Well-filled nodes (tree less deep  fewer disk
accesses on each search path)
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Example R-tree
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34
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Object descriptions
36
point
containment
query
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point
containment
query
38
Searching in an R-tree
• Q is query object (point, window, object); we search
for intersections with stored objects
• For each rectangle R in the current node,
if Q and R intersect,
– search recursively in the subtree under the pointer at R
(at an internal node)
– get the object corresponding to R and test for intersection
with R (at a leaf)
39
Nearest neighbor queries
• An R-tree can be used for nearest neighbor queries
• The idea is to perform a DFS, maintain the closest
object so far and use the distance for pruning
pruned
closest
object so far
queried
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1
4
5
2
3
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Inserting in an R-tree
• Determine minimal bounding rectangle of new object
• When not yet at a leaf (choose subtree):
– determine rectangle whose area increment after insertion
of R is smallest
– increase this rectangle if necessary and insert R
• At a leaf:
– if there is space, insert, otherwise Split Node
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Split Node
• Divide the M+1 rectangles into two groups, each with
at least m and at most M rectangles
• Make a node for each group, with the rectangles and
corresponding subtrees as entries
• Hang the two new nodes under the parent node in
the place of the overfull node; determine the new
bounding rectangles (if the root was overfull, make a
new root with two child nodes)
• If the parent has M+1 children, repeat Split Node with
this parent
48
Split Node, example
new bounding
rectangles
49
Strategies for Split Node
• Determine R1 and R2 with largest bounding rectangle:
the seeds for sets S1 and S2
• While |S1| , |S2| < M – m and not all rectangles
distributed:
– Take not yet distributed rectangle Rj , add to the set whose
bounding rectangle increases least
Linear R-tree of Guttman, 1984
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Example Split Node
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Strategies for Split Node
• If the total x-extent is larger than the total y-extent,
then sort the rectangles by x-coordinate of center,
otherwise by y-coordinate of center
• Split halfway in this sorted order
• Alternatively: choose a split close to halfway if this
gives smaller resulting bounding box overlap
52
Example Split Node
x-extent is larger
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Example Split Node
split in the middle
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Example Split Node
split close to the middle
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Deletion from an R-tree
• Find the leaf (node) and delete object; determine new
(possibly smaller) bounding rectangle
• If the node is too empty (< m entries):
– delete the node recursively at its parent
– insert all entries of the deleted node into the R-tree
• Note: Insertion of entries/subtrees always occurs at
the level where it came from
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60
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62
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Insert as rectangle
on middle level
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Insert in a leaf
object
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67
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72
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75
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Bounding Volume Hierarchy
• Other examples of bounding volume hierarchies are
often binary trees
• Not all leaves will be on the same depth
• Different balancing schemes are needed if insertions
and deletions occur
• All bounding volume hierarchies try to group objects
suitably in their subtrees
Kd-trees
• Binary search tree that splits a point set through the
middle, alternating on x- and y- (3D: on x-, y- and z-)
coordinate
split on x, vertical line
split on y, horizontal line
leaves, each with one point
78
Kd-trees
79
Kd-trees
80
Kd-trees in 3D
• The point set is split on x-, y- and z-coordinate in the
topmost three levels, and this is repeated
• Geometrically, this is splitting by axis-parallel planes
x = c, y = c, or z = c
81
Kd-trees
• Every node corresponds to a region of the plane that
is a rectangle (possibly unbounded)
• The points in the subtree below that node are exactly
the points of the stored set in that region
• Denote the region of a node  by Region()
– Region(root) = R2 (the whole plane)
– Region(child of root) = half-plane left/right of vertical line
– “most” nodes : Region() = some bounded rectangle
82
Kd-trees, windowing query
• Windowing query (report all points in the window) in
a kd-tree with window W
– [ at node , initially the root ]
– if  is a leaf, then report the stored point if it is in W
– otherwise, if Region() intersects W then recursively query
further in both subtrees
Note: if Region() does not intersect W, then we return
from recursion (we do nothing at this node, nor its subtree)
83
Kd-trees, window query
• For a set of n points in the plane, a kd-tree that stores
them allows windowing queries with an axis-parallel
rectangular window takes O(n + k) time in the worst
case, where k is the number of answers reported
3 2
• In 3D this is O(n + k) = O(n2/3 + k) time
• For circular windowing queries similar bounds hold in
practice, but these are not provable worst-case
bounds
• For queries with small windows the observed bounds
are better
84
Kd-trees, other objects, other queries
• kd-trees can store triangles, etc., where objects are
split by the vertical and horizontal lines
 need different splitting rule
 storage may become large
• Ray tracing query can be performed easily
• k-nearest neighbor query can be done similar to such
a query in a quadtree
85
3D kd-tree
BSP-trees
• BSP = Binary Space Partition
• Similar to a kd-tree that stores
triangles, but with arbitrarily
oriented splitting lines/planes
• Often lines/planes are chosen
through edges/triangles of the
set to be stored
• For query answering and for
producing depth orders for
the painter’s algorithm
87
BSP-trees
• BSP-trees were used in Doom and in Quake
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BSP-trees
89
BSP-trees
90
BSP-trees
B
E
A
D
B
D
A
C
E
F
G
H
H
F
C
G
91
BSP-trees
B
E
A
D
B
C
P
D
A
P
R
F
H
E
F
G
Q H
P
Q
Q
R
Q
C
G
92
BSP-trees
• BSP-trees are not necessarily balanced
– no problem for the painter’s algorithm
– unbalanced is not good for querying
• BSP-trees split objects; such fragmentation is
undesirable because it costs memory space and
lowers efficiency
93
BSP-trees, painter’s algorithm
94
BSP-trees, painter’s algorithm
B
A
E
B
D
D
A
C
E
F
G
H
H
F
C
G
Given a viewpoint, draw
the objects back to front,
“overpainting” what was
drawn before
95
BSP-trees, painter’s algorithm
B
E
Given a viewpoint, draw
the objects back to front,
“overpainting” what was
drawn before
D
A
H
For each split line, all
object parts “behind” it
are drawn first, and then
all objects in front of it
(w.r.t. viewpoint)
F
C
G
96
BSP-trees, painter’s algorithm
B
E seventh
Given a viewpoint, draw
second fifth
the objects back to front,
“overpainting” what was
sixth
eighth
drawn before
first
D
third
A
fourth
tenth
ninth
H
For each split line, all
object parts “behind” it
are drawn first, and then
all objects in front of it
(w.r.t. viewpoint)
F
eleventh C
G
97
BSP-trees, painter’s algorithm
• In 3D it works similarly: choose splitting planes that
contain triangles (and may cut other triangles)
• First draw stuff behind the plane, then the triangle
in the plane, then the stuff in front of the plane
• Cutting may be necessary: cyclic overlap
98
BSP-trees, painter’s algorithm
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BSP-trees, painter’s algorithm
100
Summary data structures
• There are two types of data structures: those for
representation and those for efficient searching
• Data structures can partition the underlying space, or
partition the objects it stores
• Choosing a data structure:
–
–
–
–
small data set: not worthwhile
medium size data set: choose a simple structure, it will help
large data set: orders of magnitude in efficiency are gained
huge data sets: use a structure suitable for disk storage
101
Questions
1. How would you implement ray tracing in an R-tree?
2. How many levels does the octree of slide 19 have, at most?
3. How many cyclic
overlaps do you
see? ;-)
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