11 - Spatial Data Structures
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1 11 - Spatial Data Structures cknowledgement: Marco Tarini
2 Types of Queries Graphic applications often require spatial queries Find the k points closer to a specific point p (k-nearest Neighbours, knn) Is object X intersection with object Y? (Intersection) What is the volume of the intersection between two objects? rute force search is expensive. Instead, you can solve these queries with an initial preprocessing that creates a data structure which supports efficient queries The data structure to use is application-specific
3 Two Main Ideas 1. You can explicitly index the space itself (Spatial Index) 2. You can sort the primitives in the scene, which implicitly induces a partition of the space (ounding Volume Hierarchies)
4 Spatial Indexing Structures
5 Spatial Indexing Structures Data structures to accelerate queries of the kind: I m here. Which object is around me? Tasks: (1) construction / update for static parts of the scene, a preprocessing. for moving parts of the scene, an update. (2) access / usage as fast as possible The most common structures are: Regular Grid kd-tree Oct-Tree/Quad-Tree SP Tree
6 Regular Grid (aka lattice) Image opyright: Marco Tarini
7 Regular Grid (aka lattice) Image opyright: Marco Tarini
8 Regular Grid (aka lattice) a b g h i j c d e f k l m n o p q r s Image opyright: Marco Tarini
9 a b c d e f g h i j k l m n o p q r s Regular Grid (aka lattice) a b c d e f g h i j k l m n o p q r s Image opyright: Marco Tarini
10 a b c d e f g h i j k l m n o p q r s Regular Grid (aka lattice) a b c d e f g h i j k l m n o p q r s Image opyright: Marco Tarini
11 a b c d e f g h i j k l m n o p q r s Regular Grid (aka lattice) a b c d e f g h i j k l m n o p q r s Image opyright: Marco Tarini
12 a b c d e f g h i j k l m n o p q r s Regular Grid (aka lattice) a b c d e f g h i j k l m n o p q r s Image opyright: Marco Tarini
13 a b c d e f g h i j k l m n o p q r s Regular Grid (aka lattice) a b c d e f g h i j k l m n o p q r s Image opyright: Marco Tarini
14 a b c d e f g h i j k l m n o p q r s Regular Grid (aka lattice) a b c d e f g h i j k l m n o p q r s Image opyright: Marco Tarini
15 Regular Grid (or: lattice) rray 3D of cells (same size) each cell: a list of pointers to colliding objects Indexing function: Point3D! cell index, (constant time!) onstruction: ( scatter approach) for each object [ i ] find the cells [ j ] which it touches add a pointer in [ j ] to [ i ] Queries: ( gather approach) given a point to test p, find cell [ j ], test all objects linked to it Problem: cell size too small: memory occupancy too large quadratic with inverse of cell size! too big: too many objects in one cell sometimes, no cell size is good enough
16 kd-tree Image opyright: Marco Tarini
17 kd-tree Image opyright: Marco Tarini
18 kd-tree Image opyright: Marco Tarini
19 kd-tree D E D E Image opyright: Marco Tarini
20 kd-tree D E D E Image opyright: Marco Tarini
21 kd-tree D E D E Image opyright: Marco Tarini
22 kd-tree D E F G D F G E Image opyright: Marco Tarini
23 kd-tree D E F G D F G E Image opyright: Marco Tarini
24 kd-tree D E F G D H H I F G E I Image opyright: Marco Tarini
25 kd-tree D E F G D H H I F G E I Image opyright: Marco Tarini
26 kd-tree D E F G D H H I J K F G J E I K Image opyright: Marco Tarini
27 kd-tree D E F G D H H I J K F G J E I K Image opyright: Marco Tarini
28 kd-tree D E F G D H H I J K F G L J L M M E I K Image opyright: Marco Tarini
29 kd-tree D E F G D H H I J K F G L J L M M E I K Image opyright: Marco Tarini
30 kd-tree D E D E F G F G I H H I K J J K L M L M N O N O D E F H K M Image opyright: Marco Tarini
31 kd-tree D E D E F G F G I H H I K J J K L M L M N O N O D E F H K M N Image opyright: Marco Tarini
32 kd-tree D E D E F G F G I H H I K J J K L M L M N O N O D E F H K M N O Image opyright: Marco Tarini
33 kd-trees Hierarchical structure: a tree each node: a subpart of the 3D space root: all the world child nodes: partitions of the father objects linked to leaves kd-tree: binary tree each node: split over one dimension (in 3D: X,Y,Z) variant: each node optimizes (and stores) which dimension, or always same order: e.g. X then Y then Z variant: each node optimizes the split point, or always in the middle
34 Quad-Tree (2D) Image opyright: Marco Tarini
35 Quad-Tree (2D) Image opyright: Marco Tarini
36 Quad-Tree (2D) Image opyright: Marco Tarini
37 Quad-Tree (2D) Image opyright: Marco Tarini
38 Quad-Tree (2D) Image opyright: Marco Tarini
39 Quad-Tree (2D) Image opyright: Marco Tarini
40 Quad-Tree (2D) Image opyright: Marco Tarini
41 Oc-Tree (3D)
42 Quad trees (in 2D) Oct trees (in 3D) Similar to kd-trees, but: tree: branching factor: 4 (2D) or 8 (3D) each node: splits into all dimensions at once, (in the middle) onstruction (just as kd-trees): continue splitting until a end nodes has few enough objects (or limit level reached)
43 SP-tree inary Spatial Partitioning tree Image opyright: Marco Tarini
44 SP-trees for the oncave Polyhedron proxy Image opyright: Marco Tarini
45 SP-trees for Inside-Outside Test Image opyright: Marco Tarini
46 SP-trees for Inside-Outside Test Image opyright: Marco Tarini
47 SP-trees for Inside-Outside Test Image opyright: Marco Tarini
48 SP-trees for Inside-Outside Test OUT Image opyright: Marco Tarini
49 SP-trees for Inside-Outside Test OUT Image opyright: Marco Tarini
50 SP-trees for Inside-Outside Test OUT Image opyright: Marco Tarini
51 SP-trees for Inside-Outside Test OUT Image opyright: Marco Tarini
52 SP-trees for Inside-Outside Test OUT Image opyright: Marco Tarini
53 SP-trees for Inside-Outside Test OUT OUT Image opyright: Marco Tarini
54 SP-trees for Inside-Outside Test OUT D OUT Image opyright: Marco Tarini
55 SP-trees for Inside-Outside Test OUT D OUT D Image opyright: Marco Tarini
56 SP-trees for Inside-Outside Test OUT D OUT D OUT IN Image opyright: Marco Tarini
57 SP-trees for Inside-Outside Test E OUT D OUT D OUT IN Image opyright: Marco Tarini
58 SP-trees for Inside-Outside Test E OUT E D OUT D OUT IN Image opyright: Marco Tarini
59 SP-trees for Inside-Outside Test E OUT E D OUT D OUT OUT IN Image opyright: Marco Tarini
60 SP-trees for Inside-Outside Test E OUT F E D OUT D OUT OUT IN Image opyright: Marco Tarini
61 SP-trees for Inside-Outside Test E OUT F E D OUT D OUT F OUT IN Image opyright: Marco Tarini
62 SP-trees for Inside-Outside Test E OUT F E D OUT D OUT F OUT IN OUT IN Image opyright: Marco Tarini
63 SP-tree inary Spatial Partitioning tree nother variant a binary tree (like the kd-tree) root = all scene (like kd-tree) but, each node is split by an arbitrary plane (or a line, in 2D) plane is stored at node, as (nx, ny, nz, k) planes can be optimized for a given scene e.g. to go for a 50%-50% object split at each node nother use: to test (Generic) Polyhedron proxy: note: with planes defined in its object space each leaf: inside or outside (no need to store them: left-child = in, right-child = out) tree precomputed for a given object
64 Primitive Sorting Structures
65 ounding Volume Hierarchies (VH) Image opyright: Marco Tarini
66 ounding Volume Hierarchies (VH) D E F D F E Image opyright: Marco Tarini
67 ounding Volume Hierarchies (VH) D E F D F E Image opyright: Marco Tarini
68 ounding Volume Hierarchies (VH) G D G E F D F E Image opyright: Marco Tarini
69 ounding Volume Hierarchies (VH) G H D G H E F D F E Image opyright: Marco Tarini
70 ounding Volume Hierarchies (VH) J G H J D G H E F D F E Image opyright: Marco Tarini
71 ounding Volume Hierarchies (VH) J G H J K D G H E F D F E K Image opyright: Marco Tarini
72 ounding Volume Hierarchies (VH) M J M G H J K D G H E F D F E K Image opyright: Marco Tarini
73 VH ounding Volume Hierarchy Idea: use the scene hierarchy given by the scene graph (instead of a spatial derived one) associate a ounding Volume to each node rule: a V of a node bounds all objects in the subtree construction / update is fast bottom-up: recursive using it: top-down: visit note: not a single root to leaf path may need to follow multiple children of a node (in a SP-tree: only one)
74 Spatial Indexing Structures Regular Grid the most parallelizable (to update / construct / use) constant time access (best!) quadratic / cubic space (2D, 3D) kd-tree, Oct-tree, Quad-tree compact simple non constant accessing time (still logarithmic on average) SP-tree optimized splits! best performance when accessed optimized splits! more complex construction / update ideal for static parts of the scene (also, used for generic polyhedron inside/outside test) VH simplest construction non necessarily very efficient to access may need to traverse multiple children if you do not have a scene-graph you need to create one ideal for dynamic parts of the scene
75 Intersection cceleration Data Structures
76 ollision Detection It is easy to do, the challenge is to do it efficiently n observation: most pair of objects do not intersect each other in a scene, collisions are rare optimizing the intersections directly is important but not sufficient, we need to optimize the detection of non intersecting pairs ( early rejects )
77 Geometric Proxies Idea: use a geometric proxy to approximate the objects in the scene
78 Geometric Proxy Extremely coarse approximation Used as a: ounding Volume the entire object must be contained inside exact result, you need to do more work if you detect a collision ollision Object (or hit-box ) approximation of the object no need to do anything else if an approximation is ok for your use case
79 Example: Fighting Games Street Fighter lpha, POM 1995
80 Extremely ommon
81 Extremely ommon Physic engine collision detection collision response
82 Extremely ommon Physic engine collision detection collision response Rendering view frustum culling occlusion culling
83 Extremely ommon Physic engine collision detection collision response Rendering view frustum culling occlusion culling I visibility test
84 Extremely ommon Physic engine collision detection collision response Rendering view frustum culling occlusion culling I visibility test GUI picking
85 Properties of Geometric Proxies 1. How expensive are they to compute/update? 2. How much space do you need? 3. re they invariant to the transformations applied on the object? 4. How good is the approximation? 5. How expensive are the collision queries with the other objects in the scene?
86 Geometry Proxies: Sphere Easy to compute and update ompact (center, radius) Very efficient collision tests an only be transformed rigidly The quality of the approximation is low
87 Geometry Proxies: apsule Def: Sphere == set of all points with dist from a point < radius apsule == set of all points with dist from a segment < radius i.e. a cylinder ended with two half-spheres (all same radius) Stored with: a segment (two end-points) a radius (a scalar) Popular option, compact to store, easy to construct, easy to detect intersections, good approximation
88 Geometry Proxies: Half Space Trivial, but useful e.g. for a flat terrain, or a wall Storage: (nx, ny, nz, k) a normal, a distance from the origin Tests are trivial
89 Geometry Proxies: xis-ligned ounding ox () Easy to update ompact (three intervals) Trivial to test It can only be translated or scaled, rotations are not supported
90 Geometry Proxies: ox Similar to, but not axisaligned More expensive to compute and store You need intervals and a rotation Still not a great approximation, but it is invariant to rotations and it is fast to compute and use
91 Geometry Proxies (in 2D): onvex Polygon Intersection of half-planes each delimited by a line Stored as: a collection of (oriented) lines Test: a point is inside iff it is in each half-plane Good approximation Moderate complexity
92 Geometry Proxies (in 3D): onvex Polyhedron Intersection of half-spaces Similar as previous, but in 3D Stored as a collection of planes Each plane is a normal + distance from origin Test: inside proxy iff inside each half-space
93 Geometry Proxies (in 3D): (General) Polyhedron Luxury Hit-oxes :) The most accurate approximations The most expensive tests / storage Specific algorithms to test for collisions requiring some preprocessing and data structures (SP-trees) reation (as meshes): sometimes, with automatic simplification often, hand made (low poly modelling)
94 3D Meshes as Hit-oxes These are often NOT the meshes that you use for rendering much lower resolution (~ O(10 2 ) ) no attributes (no uv-mapping, no col, etc) closed, water-tight (inside!= outside) often convex only can be polygonal (as long as the faces are flat)
95 3D Meshes as Hit-oxes mesh for rendering (~600 tri faces) (in wireframe) ollision object: 10 (polygonal) faces
96 3D Meshes as Hit-oxes mesh for rendering (~300 tri faces) (in wireframe) ollision object: 12 (polygonal) faces
97 Geometry Proxies: omposite Hit-oxes Union of Hit-oxes inside iff inside of any sub Hit-ox Flexible union of convex Hit-oxes ==> concave Hit-ox shape partially defined by a sphere, partially by a box ==> better approximation reation: typically by hand (remember: hit-boxes are usually assets)
98 How To hoose The Proxy? pplication dependent Note: # of intersection tests to be implemented quadratic wrt # of types supported VS Type Type Type Point Ray Type algorithm algorithm algorithm algorithm algorithm Type algorithm algorithm algorithm algorithm useful, e.g. Type algorithm algorithm algorithm for visibility
99 ollision Detection Strategies Static ollision detection ( a posteriori, discrete ) approximated simple + quick Frame 1 Frame 2 Dynamic ollision detection ( a priori, continuous ) accurate demanding
100 ollision Detection Strategies Static ollision detection ( a posteriori, discrete ) approximated simple + quick Frame 1 Frame 2 Dynamic ollision detection ( a priori, continuous ) accurate demanding
101 ollision Detection Strategies Static ollision detection ( a posteriori, discrete ) approximated simple + quick Frame 1 Frame 2 Dynamic ollision detection ( a priori, continuous ) accurate demanding
102 ollision Detection Strategies Static ollision detection ( a posteriori, discrete ) approximated simple + quick Frame 1 Frame 2 Dynamic ollision detection ( a priori, continuous ) accurate demanding
103 ollision Detection Strategies Static ollision detection ( a posteriori, discrete ) approximated simple + quick Frame 1 Frame 2 Dynamic ollision detection ( a priori, continuous ) accurate demanding???
104 Existing Implementations Intel Embree - VH Tree - Nori - VH - pproximate knn - Intersections -
105 References Foundations of Multidimensional and Metric Data Structures Hanan Samet Polygon Mesh Processing Mario otsch, Leif Kobbelt, Mark Pauly, Pierre lliez, runo Levy Fundamentals of omputer Graphics, Fourth Edition 4th Edition by Steve Marschner, Peter Shirley hapter 12
Geometry proxies (in 2D): a Convex Polygon
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Collision detection: strategies
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