Dan Halperin School of Computer Science Tel Aviv University - - PowerPoint PPT Presentation

Exact, Rounded or Perturbed: How Would You Like Your Arrangements?

Dan Halperin School of Computer Science Tel Aviv University Heraklion, January 2013

Overview

 background  exact  rounded  perturbed  new ways  challenges and open problems

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Overview

 background  exact  rounded  perturbed  new ways  challenges and open problems

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given a collection of curves on a surface, the arrangement is the partition of the surface into vertices, edges and faces induced by the curves

Arrangements, 2D

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Arrangements in general

 an arrangement of a set S of geometric objects is

the subdivision of space where the objects reside induced by S

 possibly non-linear objects (circles), bounded

• bjects (segments), higher dimensions (planes,

simplices)

 numerous applications in robotics, molecular

biology,vision, graphics, CAD/CAM, statistics, GIS

Very brief history

 have been studied for decades - Matoušek (2002) cites

Steiner,1826; nowadays studied in combinatorial and computational geometry

 Edelsbrunner `87

arrgs of hyperplanes

<piano movers─see next slide>

 Sharir & Agarwal `95:

arrgs of curves and surfaces

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From linear to curved: The piano movers series by Schwartz and Sharir

 looking at the arrgs of critical (constraint) surfaces that

subdivide the configuration space into free vs. forbidden cells

 On the “piano movers” problem I. The case of a two-

dimensional rigid polygonal body moving amidst polygonal barriers (`83)

 On the “piano movers” problem II. General techniques

for computing topological properties of real algebraic manifolds (`82)

 … 7

[joe-ks.com]

Very brief history, cont’d

(more subjective)

 in recent years, new emphasis on effective

algorithms and implementation

 Boissonnat-Teillaud (Eds) `06:

ECG for curves and surfaces

 Fogel et al `12:

CGAL arrgs and their applications

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Characteristic features

 constructive algorithms (as opposed to

selective algorithms, e.g., CH)

 frequent use of non-linear objects

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What do we talk about when we talk about representation of arrgs

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faithfulness/ precision format form topology/ combinatorics geometry/ numerics

Overview

 background  exact  rounded  perturbed  new ways  challenges and open problems

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Exact arrangements

Pros

 the truth  algorithms can be easily transcribed, up to the general

position assumption (this is the CGAL approach)

 the only known general method to cope with the intricacies

• f geometric computing

Cons

 later

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CGAL arrangements package(s)

 constructs, maintains, modifies, traverses, queries, and

presents two-dimensional subdivisions

 robust and exact

 all inputs (including degenerate) are handled  exact number types are used to achieve exact results

 efficient  generic 13

[TAU, MPII]

Recent applications

 art gallery in practice x 2

[Braunschweig, Campinas]

 molecular structure determination by NMR  nano-particle membrane permeation

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[Kroeller et al] [Martin and Donald] [Angelikopoulos et al]

Excursions to higher dimensions (>2)

 envelopes in 3D (2.5 dim) [Meyerovitch-H `06]  arrgs of polyhedral surfaces in 3D [Shaul-H `02]:

efficient decomposition, efficient sweep

 arrgs of surfaces in 3D [Berberich-Kerber-

Sagraloff `09]: the piano movers technique revisited and implemented

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Exact arrangements

Pros



the truth



algorithms can be easily transcribed, up to the general position assumption (this is the CGAL approach)



the only known general method to cope with the intricacies of geometric computing

Cons

 requires special arithmetic  need to determine and handle all degeneracies  efficient algebraic machinery missing in higher dimensions

(>2)

 efficient format unknown/unclear in higher dimensions (>3)  exact numerical output may be huge, when at all possible 16

Users need: rounding (one more application of CGAL arrgs)

 Agilent: Electromagnetic field simulation

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[De Wielde] “Before switching to CGAL, Agilent used 6 different sweep line algorithms which used double-precision floating point coordinates. None of them operated correctly in all cases.”

Overview

 background  exact  rounded  perturbed  new ways  challenges and open problems

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Geometric rounding

 produce fixed precision geometric

representation with topological and geometric guarantees

 is rounding essential?

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Example: Arrgs of triangles w/ decomposition

 The coordinates (x,y,z) of every triangle corner

are each represented with a 16-bit over 16-bit rational

[http://object-e.blogspot.com/2009/11/sphere-packing-vs-sphere-growth.html]

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Geometric rounding planar polygonal maps: Best practice

 Snap Rounding

[Greene, unpublished manuscript], [Hobby 99]

 numerous variants and improved algorithms

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Snap rounding (SR) arrgs of segments, I/O

 input: n segments and a grid of pixels such that the

center points of the pixels have integer coordinates (the centers of pixels form the integer grid)

 reminder, the vertices of the arrangement: either

segment endpoints (2n) or intersection of segments (I, at most θ(n2)); in this context a.k.a.: critical points

 we call the input segments ursegments  output: an arrangement of segments where all the

vertices are centers of grid pixels; details follow

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Snap rounding arrgs of segments, definition

25  a pixel containing an arrg vertex is a hot pixel

Snap rounding arrgs of segments, definition

26  a pixel containing an arrg vertex is a hot pixel  for each ursegment s construct its approximating

polygonal chain s* by connecting the centers of the hot pixels that s crosses in the order of crossing

Snap rounding arrgs of segments, definition

27  a pixel containing an arrg vertex is a hot pixel  for each ursegment s construct its approximating

polygonal chain s* by connecting the centers of the hot pixels that s crosses in the order of crossing

Properties of snap rounding

 fixed precision representation

need to show: no new vertices are

 geometric proximity  topological similarity [Guibas-Marimont `98]:

transforming s into s* viewed as a continuous deformation process; features may collapse but a curve does not cross over a vertex

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Geometric rounding

Pros

 convenient numerical output  may not need to determine all degeneracies

(SR promotes degeneracies, but of a certain controllable type) Cons

 later

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Beyond polygons in the plane

 Geodesics on the sphere

[Kozorovitzky-H `10]

 Bezier in the plane

[Eigenwillig-Kettner-Wolpert `06]

 line segments in 3-space  polyhedral maps in 3-space [Fortune `99] (output precision depends on the input combinatorial size)

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Geometric rounding

Pros

 convenient numerical output  may not need to determine all degeneracies (SR

promotes degeneracies, but of a certain controllable type) Cons

 efficient consistent rounding (beyond planar

polygons) is hard

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Overview

 background  exact  rounded  perturbed  new ways  challenges and open problems

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Perturbing arrangements

 symbolic (infinitesimal) perturbation

 Simulation of Simplicity [Edelsbrunner and Mucke ‘90],

Symbolic Treatment of Geometric Degeneracies [Yap ‘90], [Emiris-Canny-Seidel ‘97], …

 shearing and the like (e.g., the internal works of CGAL

arrgs RIC point location)

part of exact computing

 actual perturbation

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Initial motivation: molecular modeling

 Pfizer project ~ 1995  exact computing for intersecting spheres inconceivable

back then

 accurate molecular surfaces [H-Shelton `97] 34

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Controlled perturbation, overview I

 you run your geometric-algorithm program with floating-point (fp)

arithmetic

 every algorithm predicate has a guard predicate, which is also an

fp-predicate

 when a guard detects a potential error in an algorithm predicate it

fires, and some perturbation needs to be applied to the input before continuing

 if no guard fired throughout an entire execution of the

algorithm, then the program has computed the desired output for the perturbed input [rigor]

 analysis and practice to minimize the perturbation magnitude

with good running time [effectiveness]

Terminology borrowed from Mehlhorn et al

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Controlled perturbation, overview II

 a fixed precision approximation scheme  resolution bound ε and perturbation bound δ (actual

perturbation), small backward error

 degeneracy := potential degeneracy  no degeneracy ⇒ no perturbation  otherwise identify and remove all degeneracies  predicates are accurately computed  trade-off between perturbation magnitude and

computation time

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CP vs. Infinitesimal Perturbation (IP)

 IP requires exact arithmetic, whereas CP uses

fixed precision (fp) arithmetic

 IP may require tailored post-processing,

whereas in CP no post-processing is needed

 <add your favorite difference here>

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Is it OK to perturb?

 in many scientific and industrial applications the

input is approximate to begin with (measurement and modeling errors)

 considerable slack for perturbation: often, the

maximum perturbation magnitude is well below the (in)accuracy of the model

H C N O F P S Cl 1.2 1.7 1.5 1.4 1.35 1.9 1.85 1.8

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Required analysis

 the only essential preprocessing: constructing

effective guards

 a guard checks that the algorithm predicate is not in

the uncertainty zone around a degeneracy

 difficulty: floating point analysis  more analysis (resolution bounds and volume of

forbidden regions) could give indication on the existence of a good perturbation bound, or indicate the need for more precision

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Alternative view of CP

 controlled perturbation moves the original input so

that if the algorithm is run on the perturbed input with fixed precision floating point filter, the filter will always succeed and will never have to resort to higher precision or exact computation

 indeed the analysis of guards uses similar tools for

fp error bounds

[Funke ’97] [Mehlhorn ‘10]

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Brief history and more results

 first steps: molecular surfaces, spherical

arrangements [H-Shelton `97]

 second application: approximate swept volumes,

arrangement of polyhedral surfaces [Raab-H `99]

 arrangements of polygons (topology preservation)

[Packer `02]

 first elaborate analysis: arrangement of circles [H-

Leiserowitz `03]

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Brief history and more results, cont’d

 different variant and analysis, more applications

[Funke-Klein-Mehlhorn-Schmitt `05]:

 total rerun  Convex Hulls and Delaunay Triangulations

 dynamic CP for 3D spherical arrangements

[Eyal-H `05]

 CP with smart processing order [Packer `06]

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Brief history and more results, cont’d

 general analysis of CP algorithms, take I

[Mehlhorn-Osbild-Sagraloff `06]

 evaluation of the above, arrgs of circles, Voronoi

diagram of line segments [Caroli `07]

 CLP, L for Linear [Milenkovic-Sacks-Kyung `10]  general analysis of CP algorithms, take II

[Mehlhorn-Osbild-Sagraloff `11]:

 analysis of all predicates defined by signs of polynomials  resolving the gap between perturbing in the space of real

numbers (theory) and the perturbation in fp (practice)

Controlled perturbation

Pros

 fixed precision scheme, no need for exact

computing

 removes degeneracies if needed

Cons

 requires special analysis, and in particular

requires determining all degeneracies

 preserving topology (even partially) is hard

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Interim summary

 exact  rounded  perturbed

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Overview

 background  exact  rounded  perturbed  new ways  challenges and open problems

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The piano movers problem revisited

 subdivisions with soft predicates [Yap, Chiang et

al]: resolution exact

 MMS: sampling-based with exact arrgs on

manifolds [Salzman-Hemmer-Raveh-H `11] caveat: application driven

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Sampling-based techniques in motion planning

 PRM (Probabilistic RoadMaps)

[Kavraki, Svestka, Latombe,Overmars 96]

 many variants followed, e.g.

RRT (Rapidly Exploring Random

trees), [LaValle-Kuffner 99,00]

Motion planning with manifold samples MMS

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[Salzman-Hemmer-Raveh-H `11 ]

MMS, cont’d

[Salzman-Hemmer-H `12]

 extension to higher dimensions  proof of probabilistic completeness of the

scheme

 the dimensionality of narrow passages ─

theoretical substantiation of the ascent of MMS

 implemented application to 6 dof system

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Experimental results (6D C-Space)

 Tightening the configuration space

20 20-fo fold ld speedup

MMS, further research

 approximation  single query  path quality  real system(s)

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Overview

 background  exact  rounded  perturbed  new ways  challenges and open problems

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Challenges and open problems

 exact arrgs of polyhedral surfaces in higher

dimensions (>2)

 consistent geometric rounding beyond planar

polygonal maps

 controlled perturbation: (i) automation, (ii)

incorporating topological guarantees

 add approximation to MMS

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THE END

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