Download Hierarchical data structures for gigantic geometric models

Hierarchical data structures for
gigantic geometric models
Master in Computing
Pere Brunet
2010
Basic geometric problems
•  Geometric search (by position) -> Exists ? – Information
•  Geometric search (by region): region clipping
•  Geometric sorting, topological sorting, ordering (*)
•  Neighbour finding
Data and complexity
Model complexity vs scene
complexity
Points
Point sets
Meshes
General objects
(*)
Sorting = classifying (screws)
Ordering = sorting according to an ordering
Data and complexity
Model complexity versus scene complexity
Elements
Points
(surfels)
Points
Polygon
Soups
Polygons
Meshes
Polygons
Instances
Objects
Geometry
Appearance
Structure
Semantics
Data and complexity
Scene Example
Elements
Geometry
Appearance
Structure
Semantics
Points
(surfels)
Points
X
X
X
Polygon
Soups
Polygons
Meshes
Polygons
X
X
Instances
Objects
X
X
X
Space division
(uniform)
Uniform grid
Buckets with data
Memory requirements
Geometric search by position
Geometric search by region
Topological sorting
Neighbour finding
Type of data:
•  Points
•  One object or mesh
•  Set of many objects
Parameters:
•  Size of cells
•  Size of buckets
Hierarchical (recursive) space subdivision
Bintrees
Space subdivision
Image space
Tree nodes
boxes
Tree structure: binary
Nodes: G, T (W,B,L)
Tree depth
G=T-1
Universe
Subdivision criterion: split, 50%
Alternate directions (or not; flag)
Leaf criterion
Hierarchical (recursive) space subdivision
Unrestricted quadtree: 2D
Space subdivision
Image space
Tree nodes
squares
Tree structure: quaternary
Nodes: G, T (W,B,L)
Tree depth
G=(T-1)/3.
Subdivision criterion: split in four
Leaf criterion
•  Points
•  Mesh
•  Objects
Hierarchical (recursive) space subdivision
Point quadtrees: 2D
Space subdivision
Object space
Tree nodes
rectangles
Tree structure: quaternary
Nodes: G, T
Information on G nodes
G=(T-1)/3.
Subdivision criterion: split in four
Leaf criterion
•  Points
Hierarchical (recursive) space subdivision
Unrestricted octree: 3D
Space subdivision
Image space
Tree nodes
cubes
Tree structure: octal
Nodes: G, T (W,B,L)
Tree depth
G=(T-1)/7.
Subdivision criterion: split in eight
Leaf criterion
•  Points
•  Mesh
•  Objects
Hierarchical (recursive) space subdivision
Kd-trees
Space subdivision
Object space
Tree nodes
boxes
Tree structure: binary
Nodes: G, T (W,B,L)
Tree depth
G=(T-1)
Subdivision criterion: orthogonal
Alternate / Flagged direction
Leaf criterion
•  Points
•  Mesh
•  Objects
Hierarchical (recursive) space subdivision
Vector octrees (and quadtrees)
Space subdivision
Image space
Tree nodes
cubes
Tree structure: octal
Nodes: G, T (W,B,F,E,V,TG)
Tree depth
G=(T-1)/7.
Subdivision criterion: orthogonal
Leaf criterion
•  Polyhedral Objects
Hierarchical (recursive) space subdivision
Binary space partition (BSP) trees
Space subdivision
Object space
Tree nodes
polyhedra
Tree structure: binary
Nodes: G, T (W,B)
Grey nodes: Plane
G=(T-1)
Subdivision criterion: general
Search of the best plane
Leaf criterion
•  Polyhedral Objects
•  Sets of objects, scenes
Hierarchical (recursive) space subdivision
Binary space partition (BSP) trees
Space subdivision
Object space
Tree nodes
polyhedra
Tree structure: binary
Nodes: G, T (W,B)
Grey nodes: Plane
G=(T-1)
Subdivision criterion: general
Search of the best plane
Leaf criterion
•  Polyhedral Objects
•  Sets of objects, scenes
Hierarchical space subdivision - Unrestricted quadtree
Hierarchical space subdivision - Restricted quadtrees
1-restricted quadtree
(edges)
1-restricted quadtree
(edges and vertices)
1-restricted quadtree
(edges)
1-restricted quadtree
(edges and vertices)
Summary on Space Subdivision
y
Number of son
nodes
2
4
8
Bintrees
Quadtrees
Octrees
Kd-trees
PointQuadtrees
PointOctrees
BSP
-
-
x
50 % ortho split
Orthogonal split
General split
Depth,
N=8.000.000.000
32
16
11
s=Number_Sons
s1=s-1
logs ((s1*N+1)/s)
Tetra Puzzles
SQRT(2) Subdivision
Pyramid of geometry
Subdivision ex.
a
Avoid T-junctions
Deeper trees - slower
b
a
b
SQRT(2) Subdivision
(a 2D example)
Pyramid of geometry
2D and 3D Diamonds
Split all tetras in the
same diamond
a) Diamond-external
vertices are locked
b) Diamond-internal
vertices are free.
Vertices alternate
between a) and b)
Subdivision ex.
SQRT(2) Subdivision
(a 2D example)
Pyramid of geometry
2D and 3D Diamonds
Split all tetras in the
same diamond
a) Diamond-external
vertices are locked
b) Diamond-internal
vertices are free.
Vertices alternate
between a) and b)
Subdivision ex.
SQRT(2) Subdivision
(a 2D example)
Pyramid of geometry
2D and 3D Diamonds
Split all tetras in the
same diamond
a) Diamond-external
vertices are locked
b) Diamond-internal
vertices are free.
Vertices alternate
between a) and b)
Subdivision ex.
SQRT(2) Subdivision
(a 2D example)
Diamond constraints
No T-junctions
Pyramid of geometry
2D and 3D Diamonds
Split all tetras in the
same diamond
a) Diamond-external
vertices are locked
b) Diamond-internal
vertices are free.
Vertices alternate
between a) and b)
Subdivision ex.
SQRT(2) Subdivision
(a 2D example)
Diamond constraints
No T-junctions
Pyramid of geometry
2D and 3D Diamonds
Split all tetras in the
same diamond
a) Diamond-external
vertices are locked
b) Diamond-internal
vertices are free.
Vertices alternate
between a) and b)
Subdivision ex.
Hierarchical (recursive) scene subdivision
R-trees
Scene subdivision
Object space
Tree nodes
set + box
Tree structure: binary
Nodes: G, T
Top-down construction
Overlapping boxes
Hierarchical (recursive) scene subdivision (or grouping)
Box-trees
Scene subdivision
Object space
Tree nodes
set + box
Tree structure: binary
Nodes: G, T
Bottom-up construction
Overlapping boxes
AABB trees
OBB trees
Sphere trees
BV-trees: k-dops
etc
Hierarchical data structures: space vs scene subdivision
Implementation of hierarchical data structures
Pointer trees (octrees)
Linear octrees
DF- Representation:
Operations on hierarchical data structures
•  Point searching/classification
•  Region clipping
•  Neighbour finding
•  Spatial sorting
Hierarchical data structures: some problems
(revisited: algorithms and discussion)
Hierarchical frustum culling – region queries
Surface simplification and multiresolution trees
Octal search (extension of binary search)
Spatial ordering – visibility, transparencies
Neighbour finding and traversal
Surface subdivision
Collision detection
Hierarchical data structures: some problems
Hierarchical frustum culling – region queries
Exact
Conservative
Hierarchical data structures:
Conservative hierarchical frustum
culling
Requirements:
Non- overlapping node regions (Space
subdivision)
Regular-shaped regions (cubes better
than general boxes)
Preorder tree traversal with early
prunning
Node region vs. Frustum test
Stop if
Node region is outside Frustum
Node is empty
Hierarchical data structures:
Conservative hierarchical frustum
culling
Requirements:
Non- overlapping node regions (Space
subdivision)
Regular-shaped regions (cubes better
than general boxes)
Preorder tree traversal with early
prunning
Node region vs. Frustum test
Stop if
Node region is outside Frustum
Node is empty
Problem: objects in several regions
Hierarchical data structures: some examples
Conservative hierarchical frustum culling: boxes vs spheres
Convex polyhedra intersection test:
Vertex of A in B
Vertex of B in A
Two edges:
P of A.edge in B
Hierarchical data structures: some examples
Conservative hierarchical frustum culling: spheres
Good and bad
region shapes:
Hierarchical data structures: some examples
Conservative hierarchical frustum culling
Sphere test:
Still more conservative: centre point vs. enlarged Frustum
Hierarchical data structures: some problems
Surface simplification and multiresolution trees
Bottom-up simplification. For each level, it’s good to have preserved borders
changing position (in next iteration they become internal AND simplified)
R. Scopigno et al.
Hierarchical data structures: some problems
Surface simplification and multiresolution trees
Requirements:
•  Space subdivision tree
•  Grey nodes with information: simplified regions
•  Minimum of T-junctions: restricted trees or sqrt(2)
Algorithm:
Bottom-up simplification. For each level, it’s good to have
preserved borders changing position (in next iteration they
become internal AND simplified)
Hierarchical data structures: some problems
Octal search (extension of binary search)
Example: locate (and query) an object
Search key: spatial location – coordinates or region
Requirements:
•  Space subdivision
•  Balanced tree
Algorithm:
•  Preorder traversal
•  Complexity = O(Depth(tree))
Hierarchical data structures: some problems
Spatial ordering – visibility, transparencies
Omni-directional 3D
ordering
•  In 1D, ordering is well-defined
•  In nD, sorting is not unique
•  Dominant point required
•  Closest point problem, or
•  Visible objects from Obs
•  Avoid sorting at every
frame !!
Obs
Hierarchical data structures: some problems
Spatial ordering – visibility, transparencies
Omni-directional 3D ordering
Requirements:
• Hierarchical space subdivision tree - BSP
Algorithm:
•  Test Obs – Root
•  if Obs in a- then
•  Traverse a+
•  Output a
•  Traverse a•  else …
Obs
•  Complexity: O(Num_Nodes)
Hierarchical data structures: some problems
Neighbour finding and traversal
Example: Find Eastern neighbour
Requirements:
•  Space subdivision
•  Octree or quadtree
Algorithm:
•  Climb until common ancestor
•  Descend on symmetric path
Hierarchical data structures: some problems
Neighbour finding and traversal
Example: Find Eastern neighbour
Algorithm:
•  Climb until common ancestor
•  Descend on symmetric path
East
Hierarchical data structures: some problems
Surface subdivision – requirements and algorithm
•  Curvature metrics
•  Subdivision in the parametric space (u,v)
•  Mapping of the subdivision onto the 3D space
•  Crack avoidance
•  High quality triangles or polygons Min_Angle
Max:
Hierarchical data structures: some problems
Collision detection
Requirements:
•  Scene subdivision, boxes adapted to objects
•  Overlaps: traversal of multiple branches
Algorithm:
•  Simultaneous preorder traversal of trees A, B
•  Box (or sphere) tests
A
B
Hierarchical data structures: Conclusions
•  The “zoo” of Hierarchical Data Structures
•  No free lunch
•  Extremely powerful search data structures
•  Logarithmic depth
•  Problem-adapted