Advanced 3D-Data Structures - Semantic Scholar...Objects can be combined by appending the ......

Advanced 3D-Data Structures

Eduard Gröller, Martin Haidacher

Institute of Computer Graphics and Algorithms

Vienna University of Technology

Motivation

For different data sources and applications different representations are necessaryExamples:

3D scanner: produces a set of spatial points which are not connected to each otherComputer game: Scenes and characters are usually represented as surface model consisting of many polygons

A data structure for a certain application should be able to fulfill the necessary requirements

2Gröller, Theußl, Haidacher

Properties

Representation of general objectsExact representation of objectsCombinations of objectsLinear transformationInteractionFast spatial searchesMemory consumptionFast rendering


3D-Data Structures: Overview

Point CloudWire-frame ModelBoundary RepresentationBinary Space Partitioning TreekD TreeOctreeConstructive Solid Geometry TreeBintreeGrid


Point Cloud

Object = set (list) of pointse.g. from a digitizer or 3D scanner

Exact representationif >=1 points/pixel

More efficient than1 pixel sizedpolygons

Gröller, Theußl, Haidacher 5

Application Example (1/2)

3D model of Machu Picchu (http://www.cast.uark.edu/home/research/archaeology-and-historic-preservation/archaeological-geomatics/archaeological-laser-scanning/laser-scanning-at-machu-picchu.html)


Application Example (2/2)

Team from TU Wien scanned Southern train station of Viennahttp://www.tuwien.ac.at/aktuelles/news_detail/article/6203/


Operations with Point Clouds

TransformationsMultiply the points in the point list with linear transformation matrices

CombinationsObjects can be combined by appending the point lists to each other

RenderingProject and draw the points onto the image plane


Properties of Point Clouds

AdvantagesFast renderingExact representation & rendering possibleFast transformations

DisadvantagesMany points (curved obj., exact representation)High memory consumptionLimited combination operations


Surfels (SURface ELementS)

http://www.merl.com/publications/TR2000-10Movies: cab, wasp, salamander with holes, salamander corrected


QSplat (1/2) http://graphics.stanford.edu

/software/qsplat/ 3D scan of 2.7 meter statue

of St. Matthew at 0.25 mm 102.868.637 points Preprocessing time: 1 hour Demo on laptop (PII 366, 128

MB), no 3D graphics hardware


QSplat (2/2)

Interactive (8 frames/sec) High quality (8 sec)


Wire-Frame Model

Object is simplified to 3D lines, each edge of the object is represented by a line in the model


17 I

2

3

4

5

6

8 2A

BCD

EF G

HJ

KL x1

y1z1

x8y8z8

x5y5z5

x4y4z4

x3y3z3

x2y2z2

........

point list

LB........

edge listCA

1 2 3 4 5 8

AEdge list

Vertex list

Operations with Wire-Frame Model

TransformationsMultiply the points in the point list with linear transformation matrices

CombinationsObjects can be combined by appending the point and edge lists to each other

RenderingProjection of all points onto image plane and drawing of edges in between


Properties of Wire-Frame Models

AdvantagesQuick renderingEasy and quick transformationsGeneration of models via digitization

DisadvantagesInexact (no surfaces, no occlusion)Restricted combination possibilitiesCurves are approximated by straight lines


Boundary Representation (B-Rep)

V1

V2

V3 V4

V5E1

E2

E3E4

E5

E6

S1 S2

vertex list edge list face listV1: x1 y1 z1 E1: V1 V2 S1: E1 E2 E3V2: x2 y2 z2 E2: V2 V3 S2: E2 E4 E5 E6V3: x3 y3 z3 E3: V3 V1V4: x4 y4 z4 E4: V3 V4V5: x5 y5 z5 E5: V4 V5

E6: V5 V218Gröller, Theußl, Haidacher

Lists for B-Reps (1/4)Face list S1 S2


Lists for B-Reps (2/4)Face list S1 S2

E1 E2 E3 E4 E5 E6


Lists for B-Reps (3/4)Face list

x y z x y z x y z x y z x y zV1 V2 V3 V4 V5

S1 S2


Lists for B-Reps (4/4)Face list

Edgelist

Vertexlist

x y z x y z x y z x y z x y zV1 V2 V3 V4 V5

S1 S2


Winged Edge Data Structure

Alternative for normal hierarchical B-RepHere the central element is the edge:

Pend Pstartpred_cwsucc_cw

pred_ccw

face_cw

face_ccw

succ_ccw

edge


PstartPend

face_cwface_ccwpred_cw

succ_ cwpred_ccwsucc_ccw

edge list

faces xyz1st edge

points

1st edge

Operations with B-Reps (1/2)

TransformationsAll points are transformed as with wire-frame model, additionally surface equations or normal vectors can be transformed

RenderingHidden surface or hidden line algorithms can be used because the surfaces of the objects are known, so that the visibility can be calculated


Properties of B-Reps

AdvantagesGeneral representationGeneration of models via digitizationTransformations are easy and fast

DisadvantagesHigh memory requirementCombinations are relatively costlyCurved objects must be approximated


Partitioning of Object Surfaces

Necessary to approximate curved surfacesSurfaces that can be parameterized:

E.g. free form surfaces, quadrics, superquadricspartitioning of parameter space, one patch for every 2D parameter interval

Surfaces that cannot be parameterized:E.g. implicit surfaces, "bent" polygons tesselation, subdivision surfaces


Tesselation

Divide polygons in smaller polygons (triangles) until the approximation is exact enoughNormal vector criterion as termination condition:

Normal vectors of neighboring polygons are similar:

N12 1–

N1N2 Objekt

Approximation


Operations with B-Reps (2/2)

Combinations1. Split the polygons of object A at the intersections with the polygons of object B2. Split the polygons of object B at ... of A3. Classify all polygons of A as "in B", "outside B" or "on the surface of B“4. Classify all polygons of B in the same way5. Remove the redundant polygons of A and B according to the operator and combine the remaining polygons of A and B


Combinations of B-Reps (1/4)

Every polygon has a box enclosure simple test if polygons can intersectUse only convex polygons and produce only convex polygons as results simple intersection tests

A B A B A B

1: 2:



A ray is traced in the direction of the normal vector of the polygon to be classified:

Ray hits no polygon of B "outside B"First polygon of B hit from front "outside B"First polygon of B hit from back "in B"

AB

AB

"outside B" "in B"



Improvement: points of A, which lie on the surface of B, are marked as border points during the dividing process (and vice versa) only very few polygons have to be classified with the complex method

in B outside A

border point

border point

AB



Polygons can be removed according to tables:

op.A or B

A and BA sub B

in B

yesnoyes

outside B

noyes no

NV equalnonoyes

differentyesyesno

on B (coplanar)For

polygonsof A

op.A or B

A and BA sub B

in A

yesnono

outside A

noyes yes

NV equalyesyesyes

differentyesyesyes

on A (coplanar)For

polygonsof B


Requirements on B-Reps for this Alg.No open (non-closed) objectsOnly convex polygonsNo double pointsAdditional links in the vertex list between neighbor points with equal classification

........ xnynzn

x4y4z4

x5y5z5

x5y5z5

x3y3z3

x2y2z2

x1y1z1

vertex list


Binary Space Partitioning Tree

Special B-Rep for quick rendering with visibilityEspecially of static scenes


x1y1z1

xnynzn

x5y5z5

x4y4z4

x3y3z3

x2y2z2

........

point list

........

........

.... .... ....

........

.... .... .... ....

........

........

........

........

........ ........

........

............ ........

polygon nodes withsurface equation and normal vector polygon vertices

Binary Space Partitioning Tree

The base plane of the polygon in a node partitions space in two halves:

In front of and behind the polygonLeft subtree of the node: contains only polygons that are in front of the basis planeRight subtree of the node: contains only polygons that are behind the basis planePolygons that lie in both halves are divided by the base plane into two parts


Generation of BSP Trees

Generate BSP Tree:1. Find the polygon who's plane intersects the fewest other polygons and cut these in two2. Divide the polygon list in two sets:in front of that plane / behind that plane3. The polygon found in 1. is the root of the BSP tree, the left and the right subtrees can be generated recursively (from two "halves")


More BSP Examples

12

34

5

6

3

4

1

2

5

6 or1

2

3

4

1a1b

2

3

4

1 1a

1b2

34


BSP Trees as Solids

Left empty trees represent outside spaceRight empty trees represent inside volumes

1

2

34 1

2

3

4

1a1b

2

3

4

inout in

out

out

out

out out

out

in in

in

out

BSP treeor


Generation of BSP Trees

Convex objects:BSP tree is alinear list


1

2

34

Object

BSP-TreePolygon 1Polygon 2

Polygon 3Polygon 4

Properties of BSP Trees

AdvantagesFast renderingFast transformationCombinations faster than for B-RepsGeneral representationGeneration of models via digitizationTree structure (fast search)


Properties of BSP Trees

DisadvantagesCurved objects must be approximatedOnly convex polygonsHigh memory cost


Operations with BSP Trees (1/2)

RenderingBSP trees are very good for fast renderingPainter´s Algorithm:

IF eye is in front of a (in A+)THEN BEGIN draw all polygons of A-;

draw a;draw all polygons of A+ END

ELSE BEGIN draw all polygons of A+;(draw a);draw all polygons of A- END;


Operations with BSP Trees (2/2)

TransformationsPoints, plane equation and normal vector have to be transformed

CombinationsPerform combination with B-Rep, then generate BSP treeCombine BSP trees directly (faster)


Combination of BSP Trees

The structure of one tree has to act as structure for the result one tree has to be included into the other

B

A


BSP Algorithm for A op B = C:

A or B homogeneous (full or empty) simple rulesElse:

1. Divide root polygon a of A at object B in ain, aout

2. Root node c of C: if op=”and" then c:=ainelse c:=aout (with its plane)3. Divide B at plane of a in Bin, Bout

4. Recursive evaluation of the subtrees: Cleft=Aout op Bout Cright=Ain op Bin


Combination of BSP Trees:

aAb

cd

1B2

34

1inBin

3in

4

1outBout

23out

aoutC

Aout Bout

( = Bout )Ain Bin

Aout Ain

a aout

A1in

2

4

ain3out

Bout

Bin

bc

d

B1out

3in


Simple BSP Node Combination Rules

opor

and

sub

Ainhom.inhom.

fullemptyinhom.inhom.


fullempty

Bfull

emptyinhom.inhom.



A op BfullA

fullBA

emptyB

emptyempty

A –B

empty48Gröller, Theußl, Haidacher


aAb

cd

1B2

34

1out

23out

aoutC

a aout

A1in

2

4

ain3out

Bout

Bin

bc

d

B1out

b

cdout

3in

4out3in



aAb

cd

1B2

34

1inBin

3in

4

1outBout

23out

ainC

Aout Bout

( = {} )Ain Bin

Aout Ain

a aout

A1in

2

4

ain3out

Bout

Bin

bc

d

B1out

3in



aAb

cd

1B2

34

1in

4in

ainC

a aout

A1in

2

4

ain3Bout

Bin

bc

d

B1out

(bin and cin are empty)

din


kD Tree

Special case of BSP Tree Only axes-aligned partitioning planes =>

specified by one value Partitioning direction specified either implicitly

(pre-defined order) or explicitly 1D Tree binary tree


kD Tree Example: 2-D Tree


Octree

Used to represent solid volumetric objectsEach node is subdivided in 8 subspaces Each subspace is either empty, full or further dividedThe subdivision stops when an object can be represented accurate enough


Octree Example


Operations with Octrees

TransformationsHard to implement; easy: rotations of 90°

CombinationsCan easily be done by logical operations; both octrees must be adapted to each other to have the same depth in each subspace

RenderingThe octree is rendered depending on the view direction starting with the subspace farthest away from the viewer


Properties of Octrees

AdvantagesCombinations are easy to implementSpatial search is fast due to the tree structureRendering algortihm is fast

DisadvantagesHigh storage consumption for approximated objectsTransformations are not trivial in generalGeneral objects cannot be represented exactly


Extended Octrees

Additional node types:Face nodes: contain a surfaceEdge nodes: contain an edgeVertex nodes: contain a corner point

Vertex-Node

Face-Node

Edge-Node


Generation of Extended Octrees

1. Generate B-Rep2. Divide point and surface list at the subdivision planes into 8 sets3. For each octant:

Point and surface lists empty full or emptyOnly one vertexvertex nodeOnly one surface face node Only two surfaces edge node Else: subdivide recursively


Octree as Spatial Directory

Octree as search structure for objects in other representationsE.g. for B-Reps:

octree of low depth is sufficient

........

........

object list

face list

search octree


Constructive Solid Geometry Tree

A Constructive Solid Geometry (CSG) Tree consists of simple primitives, transformations and logical operationsUseful to describe complex objects with a small number of primitivesExamples for primitives

CubeSphereCylinder


CSG Tree Example


Operations with CSG Trees

TransformationsAn object is transformed by adding the transformation to the transformation of each primitive

CombinationsTwo objects are simple combined by adding them as children in a new tree

RenderingNeeds to be converted into a B-Rep or it is rendered with raytracing


Properties of CSG Trees

AdvantagesMinimal storage consumptionCombinations and transformations are simpleObjects can be represented exactlyTree structure (fast search)

DisadvantagesCannot be rendered directly; slow renderingModel generation cannot be done through digitization of real objects


Bintree

3D TreeSubdivision order xyzxyz... Choose separation plane for optimized (irregular) subdivisionFewer nodes than octree


Grid

Regular subdivisionDirectly addresses cellsSimple neighborhood finding O(1)

E.g. for ray traversalProblem:

Too few/many cells Hierarchical grid


Advanced 3D-Data Structures - Semantic Scholar...Objects can be combined by appending the ......

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