Interactive surface reconstruction on triangle meshes with subdivision surfaces

Matthias Bein

Fraunhofer-Institut für Graphische Datenverarbeitung IGDFraunhoferstraße 564283 Darmstadt

Tel.: +49 6151 155 – 465

Email: mbein@igd.fraunhofer.dehttp://www.igd.fraunhofer.de

07

_10

_26

_Pre

sen

tati

on

.pp

t

2

Exposure of the problem

Input: triangle mesh (scanned data)

Aim: connected control mesh for subdivision surfaces

Constraints: time, complexity, control, accuracy and more

07

_10

_26

_Pre

sen

tati

on

.pp

t

3

Exposure of the problem

Difficulties

Holes

Varying point density

Errors

Bad triangulation

07

_10

_26

_Pre

sen

tati

on

.pp

t

4

Related Work

Academic

Prof. Klein (primitive fitting, parametrization)

Prof. Kobbelt (quad dominant remeshing)

07

_10

_26

_Pre

sen

tati

on

.pp

t

6

Related Work

Commercial

Geo Magic

07

_10

_26

_Pre

sen

tati

on

.pp

t

7

Motivation

Technical models can be reconstructed pretty good

Freeform models too?

Fully automatic reconstruction is hardly possible (and not even wanted)

Human‘s shape recognition unreached

Designer‘s intention can not be predicted

Control over the reconstructed model has to be assured

Challenge: What is the minimum user interaction for surface reconstruction?

07

_10

_26

_Pre

sen

tati

on

.pp

t

8

Idea

User picks vertices

Patch borders are extracted automatically

Points in borders are approximated automatically

Hole filling is implicit

Pick1

Pick2

Pick3

Pick4

07

_10

_26

_Pre

sen

tati

on

.pp

t

9

Components

Principle curvature analysis

Feature recognition

Patch border alignment

Visualisation and tools

Support the user to understand the shape

Patch approximation

Patch borders need to track curvature lines for alignment

Approximate the surface inside the patch

Holes

Bad triangulation

Regular grid wanted

07

_10

_26

_Pre

sen

tati

on

.pp

t

10

Principle Curvature Analysis

Principle Curvature Analysis

Discrete

Taubin, modifications and others

Inadequate for noisy scanned meshes

Analytic

Primitive fitting

Polynom fitting

Moving Least Squares

...

07

_10

_26

_Pre

sen

tati

on

.pp

t

11

Our Principle Curvature Analysis

Polynom fitting in a local coordinate system

Bivariate polynom of total grade 2 or 3

Radius neighbourhood search triangulation used for neighbourhood information only

Least squares approximation

Analytic derivation of main curvature direction and values

25 seconds for 100.000 vertices (2GHz, 2GB ram)Local coordinate system with radius

Polynom surface and normal

Vertices

Main curvature 1Main curvature 2

07

_10

_26

_Pre

sen

tati

on

.pp

t

12

Visualisation and tools

Visualisation

Main curvature direction

Scaled normals

07

_10

_26

_Pre

sen

tati

on

.pp

t

13

Visualisation and tools

Tools

Region growing

Pick a vertex

BFS growing with constraints

Main curvature lines

Pick a vertex

Track main curvature lines

Modified shortest path

Pick two points

Search path following main curvature lines

07

_10

_26

_Pre

sen

tati

on

.pp

t

14

Modified shortest path

Experiments with Dijkstra algorithm

Robust

Symmetric

Works with the intended user interaction

Weight every edge length

reduce its length if the edge is „good“ small angle to a main curvature direction Small angle to the current path

Surface around the edge is orientable

Prefere the main curvature direction with the lower curvature value (along an edge, not across it)

1-neighbourhood (triangulation) is not sufficient

Radius neighbourhood search

Path calculation within a second

07

_10

_26

_Pre

sen

tati

on

.pp

t

15

Patch Approximation

Sequential in u and v direction

Input: four borders and number of segments in u and v

Analyse the patch (aspect ratio)

Calculate cutting planes and cutting curves inside the patch in u or v

Robust against holes and bad triangulations

Approximate the border curves and cutting curves in one direction => first set of controle points

Interpolate first set of control points in the other direction => final set of controle points

Reduction of the patch approximation to several curve approximations

07

_10

_26

_Pre

sen

tati

on

.pp

t

16

B-Spline Curve Approximation

Input: set of points dk with parameters tk

B-Spline definition: c(t) = Σ Ni(t)pi

Linear equation system: dk = Σ Ni(tk)pi

D = N * P

Overestimated (# points > # control points)

Multiply with transposed N

NT * D = NT * N * P

Q = M * P

Solve this linear equation system to gain control points P

M is symmetric and positive definite => LU decomposition

Least squares approximation. Error = Σ || dk - c(tk) ||2

Catmull&Clark subdivision derived from uniform B-Splines

=> Subdivision control net with this approximation

07

_10

_26

_Pre

sen

tati

on

.pp

t

17

Results

Whole seat

206k triangles

105k vertices

Backrest

108k triangles

57k vertices

Reconstruction

19 patches

~300 quads

7 minutes

07

_10

_26

_Pre

sen

tati

on

.pp

t

18

Results

Vase

50k triangles

25k vertices

Reconstruction

11 patches

~150 quads

07

_10

_26

_Pre

sen

tati

on

.pp

t

19

Results

Vase

50k triangles

25k vertices

Reconstruction

11 patches

~150 quads

07

_10

_26

_Pre

sen

tati

on

.pp

t

20

Results

Vase

50k triangles

25k vertices

Reconstruction

11 patches

~150 quads

07

_10

_26

_Pre

sen

tati

on

.pp

t

21

Future Work

Connecting patches and iterative reconstruction (in progress)

Error visualisation (in progress)

Refining patch approximation

Parametrize all points inside a patch

Approximate patch by solving one linear equation system

Attach semantics to features

Extrapolate parametric GML model

07

_10

_26

_Pre

sen

tati

on

.pp

t

22

Acknowledgement

European Project Focus K3D

European Project 3D-COFORM

Volkswagen AG

AIM@SHAPE Digital Shape Workbench

07

_10

_26

_Pre

sen

tati

on

.pp

t

23

Questions and Discussion

Thank you for listening

Feel free to ask any questions.

Suggestions for improvements welcome...