GEOMETRICA Period 2002-2006 Jean-Daniel Boissonnat Evaluation - - PowerPoint PPT Presentation

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GEOMETRICA Period 2002-2006 Jean-Daniel Boissonnat Evaluation - - PowerPoint PPT Presentation

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The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges GEOMETRICA Period 2002-2006 Jean-Daniel Boissonnat Evaluation seminar November 14, 2006 Evaluation

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The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges

GEOMETRICA Period 2002-2006

Jean-Daniel Boissonnat Evaluation seminar November 14, 2006

Evaluation seminar GEOMETRICA Period 2002-2006

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The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges

The team

Permanent researchers

J-D. Boissonnat (head) F . Cazals

  • O. Devillers
  • M. Yvinec
  • M. Teillaud (GALAAD 2003-2005)

P . Alliez (2001)

  • S. Pion (2003)
  • D. Cohen-Steiner (2004)

Temporary employees

F . Chazal (U. Bourgogne) 11 Ph.D. Students, 3 Post Doc., 2 Engineers

Evaluation seminar GEOMETRICA Period 2002-2006

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The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges

Context and overall objectives 1/2

Context

◮ Processing (massive) geometric data is mandatory ◮ A theory for approximating geometric objects is lacking ◮ Complexity and robustness issues are critical

Focus: Effective Non-Linear Computational Geometry

◮ Geometric Algorithms ◮ Geometric Calculus ◮ Geometric Approximation

Evaluation seminar GEOMETRICA Period 2002-2006

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The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges

Context and overall objectives 2/2

Our approach

◮ Provide mathematical and algorithmic foundations ◮ Validate our theoretical advances through extensive

experimental research

◮ Develop the CGAL library and use it to disseminate our

results

Evaluation seminar GEOMETRICA Period 2002-2006

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The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges Sampling, meshing and reconstruction Ridges Topological persistence Delaunay triangulations and Voronoi diagrams Compact data structures

Some selected results

Sampling, meshing and reconstruction

◮ Sampling theorems and algorithms

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◮ Mesh generation

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Feature extraction

◮ Ridge extraction

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◮ Stability of geometric features

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Data Structures

◮ Delaunay triangulation and Voronoi diagrams

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◮ Compact data structures

zoom next Evaluation seminar GEOMETRICA Period 2002-2006

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The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges Sampling, meshing and reconstruction Ridges Topological persistence Delaunay triangulations and Voronoi diagrams Compact data structures

Sampling theory for compact sets in Rn

◮ Well-chosen offsets of sampled compact

subsets of Rn have the correct homotopy type if the sampling is dense enough.

◮ Automatic scale selection is possible using

the critical function of the point cloud.

◮ The sampling theory is based on

non-smooth analysis of distance functions.

SoCG 06

Evaluation seminar GEOMETRICA Period 2002-2006

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The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges Sampling, meshing and reconstruction Ridges Topological persistence Delaunay triangulations and Voronoi diagrams Compact data structures

Restricted Delaunay triangulation

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◮ If S is C1,1 or Lipschitz surface and E a

sufficiently dense sample of S, RDT(E) is a good approximation of S

◮ Can be turned into an approximation

algorithm, assuming knowledge about the reach of S (or k-Lipschitz radius)

  • Graph. Models 05, SOCG 06, S. Oudot’s PhD thesis

Evaluation seminar GEOMETRICA Period 2002-2006

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The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges Sampling, meshing and reconstruction Ridges Topological persistence Delaunay triangulations and Voronoi diagrams Compact data structures

Surface meshing by Delaunay refinement

◮ A simple and incremental algorithm for sampling and

meshing surfaces

◮ Produces a good approximation of the surface ◮ Produces sparse samples of optimal size ◮ Produces facets with good aspect ratio

◮ General (black box) model of surfaces

basic primitive: intersection line segment-surface

◮ Usable in various situations

implicit surfaces, remeshing, reconstruction simplification of large data sets

SGP 03, Graph. Models 05, SOCG 05-06, Oudot’s Ph.D. thesis

Evaluation seminar GEOMETRICA Period 2002-2006

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The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges Sampling, meshing and reconstruction Ridges Topological persistence Delaunay triangulations and Voronoi diagrams Compact data structures Evaluation seminar GEOMETRICA Period 2002-2006

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The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges Sampling, meshing and reconstruction Ridges Topological persistence Delaunay triangulations and Voronoi diagrams Compact data structures

Surface reconstruction

Evaluation seminar GEOMETRICA Period 2002-2006

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The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges Sampling, meshing and reconstruction Ridges Topological persistence Delaunay triangulations and Voronoi diagrams Compact data structures

Delaunay refinement for 3D meshes

◮ Coupling the surface mesher with a 3D Delaunay

refinement process

IMRT 05

◮ interleaved refinement of surfaces and volumes ◮ multi-volumes domain are meshed at once ◮ with optionally region-dependent granularity

Rineau’s Ph.D thesis

Evaluation seminar GEOMETRICA Period 2002-2006

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The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges Sampling, meshing and reconstruction Ridges Topological persistence Delaunay triangulations and Voronoi diagrams Compact data structures

Variational meshing

Minimize (locally) E =

  • j=1..k
  • x∈Rj

ρ(x)||x − xj||2dx Lloyd iteration alternates:

  • partition w.r.t. sites (Voronoi tiling)
  • move sites to centroids

Evaluation seminar GEOMETRICA Period 2002-2006

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The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges Sampling, meshing and reconstruction Ridges Topological persistence Delaunay triangulations and Voronoi diagrams Compact data structures

Variational meshing

back Evaluation seminar GEOMETRICA Period 2002-2006

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The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges Sampling, meshing and reconstruction Ridges Topological persistence Delaunay triangulations and Voronoi diagrams Compact data structures

Ridges

[SGP 03, CAGD 05-06, IJCGA 05, Pouget’s Ph.D. Thesis]

Def.: extrema of principal curvatures along curvature lines Difficulties :

◮ Differential quantities up to 4th order ◮ Singularities: umbilics + self-intersections ◮ Non global orientability of principal

directions Contributions :

◮ On meshes: guaranteed approximations ◮ On parametric surfaces: novel implicit

characterization as P(u, v) = 0

◮ On polynomial surfaces: certified

algorithms

Evaluation seminar GEOMETRICA Period 2002-2006

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The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges Sampling, meshing and reconstruction Ridges Topological persistence Delaunay triangulations and Voronoi diagrams Compact data structures

Extraction on parametric surfaces : illustration

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h(u, v) =116u4v4 − 200u4v3 + 108u4v2 − 24u4v − 312u3v4 + 592u3v3 − 360u3v2 + 80u3v + 252u2v4 − 504u2v3 + 324u2v2 − 72u2v − 56uv4 + 112uv3 − 72uv2 + 16uv.

  • P(u, v): bivariate polynomial of total

degree 84, of degree 43 in u, degree 43 in v with 1907 terms and coefficients with up to 53 digits.

  • Certified extraction < 10 minutes

Evaluation seminar GEOMETRICA Period 2002-2006

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The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges Sampling, meshing and reconstruction Ridges Topological persistence Delaunay triangulations and Voronoi diagrams Compact data structures

Topological persistence

How to distinguish “features” from “noise”?

◮ track the creation/destruction of components/loops... in

sub-level sets as “sea level” increases

◮ set of “persistence intervals” ≃ life-spans of features ◮ “features” yield long intervals, “noise” yields short ones

Evaluation seminar GEOMETRICA Period 2002-2006

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The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges Sampling, meshing and reconstruction Ridges Topological persistence Delaunay triangulations and Voronoi diagrams Compact data structures

Topological persistence

back

◮ The set of persistence intervals remains stable when the

function is perturbed.

SoCG 05a

◮ Applied to distance functions, the stability result gives a

way to estimate the homology groups of a compact subset

  • f a metric space from a point sample.

◮ Applied to height functions, it yields new approximation

theorems for the total mean curvature of surfaces and for the length of curves.

SoCG 05b

Evaluation seminar GEOMETRICA Period 2002-2006

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The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges Sampling, meshing and reconstruction Ridges Topological persistence Delaunay triangulations and Voronoi diagrams Compact data structures

Delaunay triangulations and Voronoi diagrams

Optimized Delaunay and regular triangulations

◮ Geometric filtering

ALENEX 03, IMACS 05

◮ Perturbation

SODA 03

◮ CGAL: 3D Delaunay of 106 points in <30 seconds

Complexity of DT of points on a manifold

◮ Polyhedral manifold

SMA 02, DCG 03,04

◮ Smooth generic surfaces

SOCG 03

Curved Voronoi diagrams

◮ see the gallery

SODA 032,07, ESA 02,03,05, ECG Book

Evaluation seminar GEOMETRICA Period 2002-2006

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The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges Sampling, meshing and reconstruction Ridges Topological persistence Delaunay triangulations and Voronoi diagrams Compact data structures

A gallery of Voronoi diagrams

back Evaluation seminar GEOMETRICA Period 2002-2006

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The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges Sampling, meshing and reconstruction Ridges Topological persistence Delaunay triangulations and Voronoi diagrams Compact data structures

Compact data structure for triangulations

Compression

◮ connectivity: 13n pointers → 13n log n → 416 bits per

vertex

◮ optimal compression: 3.25 bits per vertex

Compact data structures

◮ Triangulated surfaces: 2.17 bits per triangle ◮ Planar maps: 2 bits per edge

Castelli Aleardi’s Ph.D. Thesis

Evaluation seminar GEOMETRICA Period 2002-2006

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The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges Sampling, meshing and reconstruction Ridges Topological persistence Delaunay triangulations and Voronoi diagrams Compact data structures

Compact data structure for triangulations

back

Group triangles → use less pointers Refer to a catalog of tiny triangulations for details → share details

Evaluation seminar GEOMETRICA Period 2002-2006

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The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges

Strategy for CGAL

◮ Internally,

basis for most experiments made in the project collects new software for perennial use

◮ Externally,

main dissemination vehicle to industry and research

◮ Two Open Source licenses (LGPL or QPL)

+ a commercial license

◮ Startup company GeometryFactory (2003)

Evaluation seminar GEOMETRICA Period 2002-2006

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The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges

Diffusion

◮ 3 major public releases ◮ distributed on the CGAL web site

included in the Debian and Fedora Linux distributions

◮ GEOMETRICA is author of about 50% of CGAL

> 70% of the revenues of GeometryFactory

◮ Interfaces with SCILAB and PYTHON

Evaluation seminar GEOMETRICA Period 2002-2006

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The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges

Software development

◮ New packages contributed by GEOMETRICA during the

period

◮ 2D and 3D Interpolation ◮ dD Principal Component

Analysis

◮ 2D Placement of

Streamlines

◮ 2D Circular Kernel ◮ 2D & Surface Mesher ◮ Surface Parameterization

◮ Packages further developed

◮ 2D and 3D Kernel ◮ 3D Alpha Shapes ◮ 3D Triangulations Evaluation seminar GEOMETRICA Period 2002-2006

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The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges

Software development

Technology transfer

◮ 800 subscribers on the CGAL user mailing-list with an

intense activity

◮ > 30 CGAL licenses sold through GeometryFactory

(overall)

◮ Other industrial contracts : France T´

el´ ecom, Alcatel Alenia Space

◮ Web servers

◮ Surface reconstruction ◮ Modeling of interfaces in Structural Biology Evaluation seminar GEOMETRICA Period 2002-2006

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The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges

New challenges

Focus on two main research directions

◮ Mesh generation ◮ Geometric data processing

Proposals for two new actions

◮ CGAL ◮ Geometric modeling in structural biology

Evaluation seminar GEOMETRICA Period 2002-2006

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The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges

New challenges in mesh generation

Work program

◮ Make our meshing algorithms widely used ◮ Develop new meshing algorithms

◮ moving and deformable shapes ◮ anisotropic metric fields ◮ hexahedral meshes

Collaborations

◮ Collaborate with experts in scientic computing and users of

meshes, notably in the context of living sciences and medecine

Evaluation seminar GEOMETRICA Period 2002-2006

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The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges

New challenges in geometric data processing

Work program

◮ Data structures for massive data (theory and practice) ◮ Higher dimensions ◮ Noisy data

Collaborations

◮ Collaborate with experts in statistics, data analysis and

machine learning

◮ Consolidate the antenna in Saclay

Evaluation seminar GEOMETRICA Period 2002-2006

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The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges

A specific action for CGAL

Role

◮ CGAL: independent team with dedicated focus ◮ Consortium, relations to INRIA teams (providers & users)

Work program

◮ Software development management

◮ make CGAL the standard platform for geometry

◮ Efficient robust implementations of geometric algorithms

◮ numerical and algebraic issues, data structures ◮ software design Evaluation seminar GEOMETRICA Period 2002-2006

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The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges

A new action for Algorithms in Structural Biology

Algorithmic challenges

◮ Structure reconstruction from data (NMR, X-ray, Cryo-EM) ◮ Modeling docking and folding

Work program

◮ Modeling contacts and interfaces —Van der Waals models ◮ Modeling flexibility —parameter spaces, kinematics ◮ Structural alignments —3D shape matching ◮ Knowledge based potentials —data mining

Evaluation seminar GEOMETRICA Period 2002-2006

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The team Context and overall objectives A selection of scientific achievements Software development and technology transfer New challenges

More details on demand

◮ Mesh generation

[Alliez & Yvinec]

◮ Geometric data processing

[Cohen-Steiner]

◮ Compact data structures

[Devillers]

◮ CGAL

[Pion]

◮ Algorithms in Structural Biology

[Cazals]

Evaluation seminar GEOMETRICA Period 2002-2006