the size of the 3d visibility skeleton: analysis and

The Size of the 3D Visibility Skeleton:Analysis and Application

Ph.D. thesis proposal

Linqiao Zhang

[email protected]

School of Computer Science, McGill University

March 20, 2008

thesis proposal: The Size of the 3D Visibility Skeleton: Analysis and Application – p. 1/41

Overview

the visibility problem and its related literature

experimental study of the size of the visibility skeletonin 3D

a succinct 3D visibility skeleton

a new application to motion planning

summary of my thesis status


The Visibility Problem

Given a set of input objects, what an object can see?

application: computer graphics, robotics, computer vision

two types of problems:

easy: enquire visibility information from a fixed direction.

solutions: ray shooting ...

harder: enquire visibility information from all directions

solutions: discretization; visibility skeleton data structure ...


Visibility Skeleton

a graph that contains vertices and arcs

vertex: corresponds to a maximal free line segmentthat has 0-degrees of freedom

arc: corresponds to a set of maximal free linesegments that have 1-degree of freedom, and formone connected component

in 2D:

1

4

3

2AB

4

1 2

3


Visibility Skeleton

a graph that contains vertices and arcs

vertex: corresponds to a maximal free line segmentthat has 0-degrees of freedom

arc: corresponds to a set of maximal free linesegments that have 1-degree of freedom, and formone connected component

in 3D:

1

2

3

4

55

2 1

3

4


Visibility Skeleton in 2D

only one type of skeleton vertex in 2D:

1

4

3

2AB A B

4

2

3

1



only one type of skeleton vertex in 2D:

1

4

3

2AB A B

4

2

3

1

only one type of skeleton arc in 2D:

1

4

3

2AB A B

4

2

3

1



eight types of skeleton vertices in 3D:

ee2

e

3

1

e

4

v

e

e

3

1e

2

e

4

e

e

3

4

2

e1

ef

v2

v1

e

e

e

e3

4

1

2

EEEE VEE FEE VV

f

2

1

e

4

e3

f

e

e

1

2

1

e

e e2

e3

4

v

fe

f

v

v

f

FF FvE FE FVV



four types of skeleton arcs in 3D:

e 1

e2

e 3

e

v

ef

f

v

v

v

EEE VE FE FVE


Literature

visibility skeleton (complex):

2D: theory [G. Vegter, M. Pocchiola, 93]; implementation [P.Angelier, M. Pocchiola, 03]

3D: theory [F. Durand, G. Drettakis, C. Puech, 97]; brute forceimplementation [F. Durand, 97]

size of the visibility skeleton: when k is the number of inputobjects (e.g. polygons, polytopes, discs, spheres) and n is the totalnumber of edges

2D: worst case: Θ(k2); experimental evidence: Θ(k) [F.Cho, D. Forsyth, 99]

3D: worst case: Θ(n4); experimental evidence: Θ(n2)[F. Durand et al. 99]


Literature – cont.

more on size of the 3D visibility skeleton:

worst case: Θ(n2k2), when inputs are k polytopes withtotal complexity n. [H. Bronnimann, O. Devillers, V. Dujmovic,H. Everett, M. Glisse, X. Goaoc, S. Lazard, H.-S. Na, and S.Whitesides, 07]

worst case: O(nk2√

nk), when input polytopes haveconstant complexity. [M. Glisse, 07]

expected size: O(k), when inputs are randomlydistributed unit spheres. [O. Devillers, V. Dujmovic, H. Everett,X. Goaoc, S. Lazard, H.-S. Na, and S.Petitjean, 03]


Goals of the Thesis

provide an efficient and robust implementation toenable experimental studies (in 3D)

experimentally study the size of the visibility skeleton inpractice

seek a succinct visibility skeleton

find new applications of the visibility skeleton


3D Experimental Study: Software

implemented a sweep plane algorithm:

running time complexity: O(n2k2 log k) versus O(n5)

input:

disjoint convex polytopes in general position

output:

EEEE, VEE, FEE vertices


3D Experimental Study: Software– cont.

efficiency:

designed certain predicates, e.g. ordering a pair ofsweep planes

robustness:

used filtered_exact number type

software verification:

used geomview to visualize the intermediate stepsand the output of the software [socg07 video]

compared the output with a brute forceimplementation: tested on 20 input scenes


3D Experimental Study: the Set Up

The universe: a great sphere with radius R

The objects k: ‘randomly’ distributed disjoint convex polytopes

The scene density µ: calculated using the outer tangentspheres that generated the polytopes

R = 3

√

k/µ


3D Experimental Study: the Set Up– cont.

number type: doublethe observed failure rate less than 0.1%

double is four times faster than filtered_exact

medium performance machine: i686 with Pentium 2.8 GHzCPU and 2 GB main memory

measure:

the number of EEEE, VEE, FEE verticesrunning time


3D Experimental Study: Parameters

three parameters:

n/k: complexity of a polytope

k: number of polytopes

µ: scene density

three suites of experiments:

suite I: fix µ, choose n/k ≈ 7.5, 40, 85, and vary ksystematically

suite II: fix µ, choose k = 30, 60, 90, and vary n/ksystematically

suite III: repeat suite I, and vary µ systematically


3D Experimental Study: Parameters– cont.

sample input of suite I:

fix µ, choose n/k ≈ 7.5, 40, 85, and k = 50

n/k ≈ 7.5 n/k ≈ 40 n/k ≈ 85


3D Experimental Study:Suite I Results

size of skeleton versus total number of edges n

0 1000 2000 3000 4000 5000 6000 7000 8000 90000

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

5

n

Num

ber

of A

ll V

ertic

es

n / k = 7.5n / k = 40n / k = 85


3D Experimental Study:Suite I Results – cont.

size of skeleton versus k2√

n/k

0 2 4 6 8 10 12 14

x 104

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

5

sqrt(n / k) × k2

Num

ber

of A

ll V

ertic

es

n / k = 7.5n / k = 40n / k = 85


3D Experimental Study:Interpretation of k2

√

n/k

given a universe U of radius R, the expected size of thevisibility skeleton is [O. Devillers, V. Dujmovic, H. Everett, X. Goaoc, S. Lazard,

H.-S. Na, and S.Petitjean]:

Θ(k) O(k2) when U consists of k randomly distributedpolytopes that are

inside U near the boundary of U

bounded aspect ratio

constant complexity



√

n/k – cont.

given a universe U of radius R that consists polytopes ofany given density

polytopes that are inside U : Θ(R3)

polytopes that are near the boundary of U : Θ(R2)

assume polytopes have constant complexity

number of visibility skeleton vertices with supportpolytopes are

inside U : Θ(R3)

near the boundary of U : O(R4)



√

n/k – cont.

experimental evidence:

0 2 4 6 8 10 12 14

x 104

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

5

sqrt(n / k) × k2

Num

ber

of A

ll V

ertic

es

n / k = 7.5n / k = 40n / k = 85



√

n/k – cont.

experimental evidence:

0 2 4 6 8 10 12 14

x 104

0

0.5

1

1.5

2

2.5

3

3.5x 10

5

sqrt(n / k) × k2

VE

E V

ersu

s O

ther

Typ

es o

f Ver

tices

VEE

others

n / k = 7.5n / k = 40n / k = 85



√

n/k – cont.

theoretical evidence:



√

n/k – cont.

theoretical evidence:

Conjecture: The expected number of type VEE vertices islinearly related to the expected silhouette size of thepolytopes.

proving of this conjecture would generalize the results of [O.

Devillers, V. Dujmovic, H. Everett, X. Goaoc, S. Lazard, H.-S. Na, and S.Petitjean]


A Succinct 3D Visibility Skeleton

recall the eight types of visibility skeleton vertices:

ee2

e

3

1

e

4

v

e

e

3

1e

2

e

4

e

e

3

4

2

e1

ef

v2

v1

e

e

e

e3

4

1

2

EEEE VEE FEE VV

f

2

1

e

4

e3

f

e

e

1

2

1

e

e e2

e3

4

v

fe

f

v

v

f

FF FvE FE FVVthesis proposal: The Size of the 3D Visibility Skeleton: Analysis and Application – p. 23/41


consists of type EEEE, VEE, FEE and VV vertices only

impact:

reduces the size of the visibility skeleton

simplifies the computation procedure, and reduces thecomputation time

gives a simpler data structure

preserves the visibility information



designed an algorithm to update the succinct visibilityskeleton incrementally

can maintain the visibility skeleton in dynamic scenesat low cost

e 1

e2

e 3

e

e

e

i

j

k

a b c d e


Finding a Relatively Shortest Path ina Scene of Polytope Obstacles

literature:

computing the exact shortest path in 3D polytopeobstacle space is NP-hard [John Canny 1987]

finding the supporting edge sequence of the shortestpath is NP-hard [John Canny 1987]



literature:

computing the exact shortest path in 3D polytopeobstacle space is NP-hard [John Canny 1987]

finding the supporting edge sequence of the shortestpath is NP-hard [John Canny 1987]

various polynomial algorithms to compute the(approximated) shortest paths on one or two polytopes

exponential algorithms to compute the shortest pathson more than two polytopes [M. Sharir and A. Schorr 1986],[K. Senevirantne and S. Earles 1993]



my approach: use computed visibility skeleton vertices tofind a relatively shortest path in polynomial time



algorithm:step one: construct a search graph

represent each polytope as a vertex in the search graph

represent each visibility skeleton vertex as an edge of the searchgraph

edge length is the length of the segment that is represented by thevisibility skeleton vertex



algorithm:step one: construct a search graph

represent each polytope as a vertex in the search graph

represent each visibility skeleton vertex as an edge of the searchgraph

edge length is the length of the segment that is represented by thevisibility skeleton vertex

step two: compute the initial shortest path from the searchgraph

use Dijkstra’s algorithm



algorithm:step three: optimize the “shortest” path



algorithm:step three: optimize the “shortest” path

step four: obtain a computed relatively shortest path



running time: O(n2k2logk)

n is the total number of edges, k is the total number of polytopes

quality of the approximation

theoretically: open problem

experimentally: future work


Intended Contributions of my Thesis

an efficient and robust implementation [SoCG07 video]

experimental study of the size of the 3D visibilityskeleton and the related theoretical results

a succinct 3D visibility skeleton

finding a relatively shortest path in a scene of polytopeobstacles

the algebraic degree of the predicates [Computational Geometry: Theory

and Application, accepted]

experimental study of the 2D visibility skeleton [International Journal of

Computational Geometry and Applications, 2007]


Experimental Study: in 2D

when k is the total number of inputsworst case: Θ(k2) [V. Gegter, M. Pocchiola]

expected size:

theoretically: O(k) (proof is similar to the 3D case)inputs are randomly distributed discs or polygonsinputs are with bounded aspect ratio

experimentally:use the available software [P. Angelier02]



model:

The universe: a large disc

The objects: n randomly distributed disjoint unit discs withvarying density µ

density = 0.55 density = 0.1



setting:

vary: scene density µ & number of discs n

measure: size of the 2D visibility skeleton

report: mean of the 10 experiments, and omit thestandard derivations since the observed values aresmall

slow machine: i686 with AMD Athlon 1.73 GHz CPU and 1 GBmain memory



results:

0

200 000

400 000

600 000

800 000

1e+06

1.2e+06

1.4e+06

1.6e+06

1.8e+06

2e+06

500 1 000 1 500 2 000 2 500 3 000

number of unit discs

scene density: 0.025

bitangentsmemory

time

0

200 000

400 000

600 000

800 000

1e+06

1.2e+06

1.4e+06

1.6e+06

1.8e+06

0 100 200 300 400 500 600 700 800 900 1 000



bitangentsmemory

time

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

500 1000 1500 2000 2500 3000


scene density: 0.55

bitangentsmemory

time

0

200 000

400 000

600 000

800 000

1e+06

1.2e+06

1.4e+06

0 100 200 300 400 500 600 700 800 900 1 000



bitangentsmemory

time



results:

17.07

µ∗ n −

63.6

µ2−

2232

µ+ 5110 for n >

22.15

µ+ 61

no = 101

Nbts = 10 152

n : 300 µ : 0.55



√

n/k



Θ(k) when U consists of k randomly distributed unitspheres

Θ(k) when U consists of k randomly distributedpolytopes that are

inside U


constant complexity



√

n/k



Θ(k) when U consists of k randomly distributed unitspheres

Θ(k) O(k2) when U consists of k randomly distributedpolytopes that are

inside U near the boundary of U


constant complexity


3D Experimental Study:Running Time

running time versus n1.5k log k

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 108

0

0.5

1

1.5

2

2.5

3x 10

4

sqrt(n3) × k × log(k)

Run

ning

Tim

e

n / k = 7.5n / k = 40n / k = 85


3D Experimental Study:Suite II Results

size of skeleton versus k2√

n/k

0 1 2 3 4 5 6 7 8 9

x 104

0

0.5

1

1.5

2

2.5

3x 10

5

sqrt(n / k) × k2

Num

ber

of A

ll V

ertic

es

k = 30k = 60k = 90


3D Experimental Study:Suite II Results – cont.

size of skeleton versus√

n/k

2 3 4 5 6 7 8 9 10 110

0.5

1

1.5

2

2.5

3x 10

5

sqrt(n / k)

Num

ber

of A

ll V

ertic

es

k = 30k = 60k = 90


Remaining Work of my Thesis

experimental study of the size of the 3D visibilityskeleton in terms of varying scene density: one month

write up in detail the succinct 3D visibility skeleton datastructure: one month

write up the proposal of the algorithm for computingthe relatively shortest collision free path: two weeks

prepare the thesis manuscript: six months


Intended Main Chapters of my Thesis

Experimental Study of the 2D Visibility Skeleton

Design Aspects of the Implementation

The Algebraic Degree of the Predicates

Experimental Study of the 3D Visibility Skeleton


Finding a Relatively Shortest Path in a Scene ofPolytope Obstacles


the size of the 3d visibility skeleton: analysis and

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