Shape Reconstruction From Planar Cross Sections CAI Hongjie | May 28, 2008.

Shape Reconstruction From Planar Cross Sections

CAI Hongjie | May 28, 2008

Motivation

S. Hahmann, et al. CAGD 2000, GMP 2008

Reconstruction From Cross Sections

• Triangulated tiling

Reconstruction From Cross Sections

• Smooth skinning

Papers List• T.W. Sederberg, K.S. Klimaszewski, et al

Triangulation of Branching Contours using Area Minimization ,1997

• Jean-Daniel Boissonnat

Shape Reconstruction From Planar Cross-Sections , 1988• L.G. Nonato, A.J. Cuadros-Vargas, et al

Beta-Connection: Generating a Family of Models from Planar Cross Sections, 2005

• N.C. Gabrielides, A.I. Ginnis, et al

G1-Smooth Branching Surface Construction From Cross Sections, 2007

Triangulation of Branching Contours using Area Minimization

T.W. Sederberg, K.S. Klimaszewski Mu Hong, Kazufumi Kaneda

International Journal of Computational Geometry & Applications, 1997

Thomas W. Sederberg

• Main Post Professor of Computer Science,

Brigham Young University

• Research Interests Computer aided geometric design Computer graphics Image morphing

• Publications SIGGRAPH 11; CAGD 26; CAD 6; …

Ambiguous Triangulation

Previous Works

• Graph representation

• Some heuristics Volume maximized (E. Keppel, 1975) Faces area minimized (H. Fuchs et al, 1977)

Branching and Link-Edge

Bad Case and Remedy

Polygonal Bridge & Area Minimization

More Results

Shape Reconstruction From Planar Cross-Sections

Jean-Daneil Boissonat

INRIA

Computer Vision, Graphics, and Image Processing, 1988

Jean-Daneil Boissonat

• Main Posts Research Director at

INRIA Sophia-Antipolis Head of the Geometrica project Chair of the Evaluation Board of INRIA

• Research Interests Discrete and Computational Geometry Voronoi diagrams & Delaunay triangulation Surface reconstruction

3D Delaunay Triangulation

A

B

CD

F

E

a

b

c

Computing 3D Delaunay Triangulation

Subdivision of Contours

Result

Beta-Connection: Generating a Family of Models from

Planar Cross Sections

L.G. Nonato, A.J. Cuadros-Vargas

R. Minghim, M.C. F.DE Oliveira

ACM Transactions on Graphics, 2005

Different Correspondence

β-Connection

Tetrahedron Types

• Internal tetrahedron

• External tetrahedron

• Redundant tetrahedron

• Reverse tetrahedron

Graph Representation

• Graph G from Delaunay triangulation

Sphere node: region Cylinder node: redundant tetrahedron Cone node: external tetrahedron

β-Components

• Definition 1: a,b region nodes in G , dG(a,b) is the length of the shortest path connecting a and b.

• Definition 2: ,region nodes a and b are said to be β-connected, denoted , if

G, where ,such that dG(σi, σi+1) ≤ β, i=1,…,k-1.

• Lemma 1: is an equivalence relation.• Definition 3: β-components are defined by the equi

valence class generated by .

a b

1 { ,..., }k region nodes

1 , ka b

Algorithm

• 3D Delaunay triangulation

• Create graph representation and determine

β-components

• Remove cylinder and cone nodes connecting different β-components

• For each β-component, tackle with the singularity

Examples

More Example

G1-Smooth Branching Surface Construction From Cross Sections

N.C Gabrielides, A.I. Ginnis,

P.D. Kaklis, M.I. Karavelas

Computer-Aided Design, 2007

Menelaos Karavelas

• Curriculum Vitae Ph.D. at Stanford University Post-doc at INRIA Sophia-Antipolis Assistant professor at the Department of

Applied Mathematics of the University of Crete

• Research Interests Voronoi diagrams and Delaunay triangulations Shape-preserving interpolation Shape reconstruction

Transfinite Interpolation Surface

• Given parametric surface f(u,v),0≤u,v ≤1.Let u- and v-isoparametric boundaries be 0v,1v;u0,u1. Then

are transfinite interpolation surfaces.

T1 1

T2 2

1 2

P P ( , ) (1 , )( , ) ,

P P ( , ) ( , )(1 , ) ,

1 1P P ( , ) (1 , ) ( , ) (1 , ) ,

u v u u

u v v v

v vu v u u u u

v v

0 1

0 1

0 00 010 1

1 10 11

f f v v

f f u u

vf u u

v

Coons, MIT Project MAC-TR-41,1967

Coons Bi-cubic Blending Surface

• Let boundaries hodograph of f(u,v) be 0vu,1vu; u0v,u1v (0vu=∂ f(u,v)/ ∂u|u=0), then Coons surface is

where

1 2 1 2 1 2P P ( , ) (P P P P ) ( , )u v u v f f

T1 0 1 0 1

T2 0 1 0 1

0

1 2 0 1 0 1

P ( , ) ( ( ), ( ), ( ), ( ))( , , , ) ,

P ( , ) ( , , , )( ( ), ( ), ( ), ( )) ,

P P ( , ) ( ( ), ( ), ( ), ( ))

u u

v v

v v

v v

u u uv uv

u u uv uv

u v F u F u G u G u

u v F v F v G v G v

F

u v F u F u G u G u

0 1 0 1

0 1 0 1

00 01 00 01

10 11 10 11

00 01 00 01

10 11 10 11

f v v v v

f u u u u

f 1

0

1

( )

( ).

( )

( )

v

F v

G v

G v

Sketch of the Algorithm

• Step 1: tangent vector estimation

• Step 2: planar contours and tangent ribbons

• Step 3: surrounding surface

• Step 4: trimming

• Step 5: hole filling

Step 1: tangent vector estimation

Step 2: Planar Contours and Tangent Ribbons

Step 3: Surrounding Surface

Step 4: Trimming

• Trimming curve Y(t)=S•X(t)

• Cross-tangent of trimming curve

(1) ( , ) ( , )( ) ( ) ( )S S

u v u vt a t b t

u v

S S

Y

Step 5: Hole Filling

• Selecting center point K

xy coordinate

centroid of Mk

z coordinate Kz

K

K

Mk

Mk-1Mk+1

1

z i

i i

K zr

z z

Step 5: Hole Filling

• Guide curves

• Gordon-Coons surface

K

Mk

Mk-1Mk+1

Gk

Gk+1Gk-1

The “many-to-many” Problem

The Separating Strip

Symmetric Data Set

Symmetric Construction

Non-Symmetric Case

Integral Construction

Open Set Case

The Bulbous Hull Example

Failure of the Algorithm

Improvement by Voronoi Diagram

Conclusions

• Ensuring G1 continuity

• Preserving data symmetry

• Automatic solution is available

• Topological instability

• Failure for multiple-connected residuals

Thanks!

Q&A