[PPT] - Model Repair Leif Kobbelt RWTH Aachen University 1 PowerPoint Presentation

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Model Repair

Leif Kobbelt RWTH Aachen University

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Leif Kobbelt RWTH Aachen University

Model Repair

model repair is the removal of

artifacts from a geometric model such that it becomes suitable for further processing.

produce a nice, manifold triangle mesh

– with boundary or – without boundary (watertight)

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Leif Kobbelt RWTH Aachen University

Model Repair

types of input
surface-oriented algorithms

– Filling holes in meshes [Liepa 2003]

volumetric algorithms

– Simplification and repair of polygonal models using volumetric techniques [Nooruddin and Turk 2003] – Automatic restoration of polygon models [Bischoff, Pavic, Kobbelt 2005]

conclusion & outlook

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Leif Kobbelt RWTH Aachen University

Range Images

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registered range images are a set of patches that

describe different parts of an object.

Registration

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Leif Kobbelt RWTH Aachen University

Range Images

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large areas of overlap are ...

– ... necessary for registration but – ... bad for consistency

how to merge the patches into

a single mesh?

– inconsistent geometry – incompatible connectivities

large scale

verlaps

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Leif Kobbelt RWTH Aachen University

Range Images

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successfully merged range

images are manifold meshes with holes and islands (i.e. boundaries)

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Leif Kobbelt RWTH Aachen University

Range Images

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holes and islands are due to obstructions in the

line of sight of the scanner

identify correspondences

between holes and islands

fill holes

– smoothly – geometry transfer/synthesis

avoid intersections

holes and isles

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Leif Kobbelt RWTH Aachen University

Contoured Meshes

contoured meshes have been

extracted from a volumetric representation (e.g. by marching cubes)

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Leif Kobbelt RWTH Aachen University

Contoured Meshes

contoured meshes are usually manifold and

closed, but may contain topological noise

– disconnected components – spurious handles – cavities

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Leif Kobbelt RWTH Aachen University

Triangulated NURBS

set of patches that contain small scale gaps and
verlaps

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Leif Kobbelt RWTH Aachen University

Triangulated NURBS

set of patches that contain small scale gaps and
verlaps

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Leif Kobbelt RWTH Aachen University

Triangulated NURBS

gaps and overlaps are due to triangulating a

common (trimmed) patch boundary differently from both sides

issues

– consistent orientation – structure preservation

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small scale gaps and overlaps

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Leif Kobbelt RWTH Aachen University

Triangulated NURBS

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typical workflow, e.g., in CAD/CAM:

Editing Repair

Simulation

manual

ften:manu

al(!) automatic NURBS triangle mesh

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Leif Kobbelt RWTH Aachen University

Triangle Soups

a triangle soup is a set of triangles without

connectivity information

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Leif Kobbelt RWTH Aachen University

Triangle Soups

a triangle soup is a set of triangles without

connectivity information

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Leif Kobbelt RWTH Aachen University

Triangle Soups

good for visualization but bad for downstream

applications that require manifold meshes

in addition to the artifacts we already encountered:

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intersections singular vertex complex edges incompatible

rientations

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Leif Kobbelt RWTH Aachen University

Not Covered In This Lecture ...

geometrical noise

➙ smoothing (Mark)

badly meshed manifolds

➙ remeshing (Pierre)

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Leif Kobbelt RWTH Aachen University

Model Repair

types of input
surface-oriented algorithms

– Filling holes in meshes [Liepa 2003]

volumetric algorithms

– Simplification and repair of polygonal models using volumetric techniques [Nooruddin and Turk 2003] – Automatic restoration of polygon models [Bischoff, Pavic, Kobbelt 2005]

conclusion & outlook

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Leif Kobbelt RWTH Aachen University

Surface-Oriented Algorithms

surface oriented approaches

explicitly identify and resolve artifacts

methods

– snapping – splitting – stitching – ...

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Leif Kobbelt RWTH Aachen University

Surface-Oriented Algorithms

advantages

– fast – conceptually easy – memory friendly – structure preserving, minimal modification

f the input

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Leif Kobbelt RWTH Aachen University

Surface-Oriented Algorithms

problems

– not robust

numerical issues
inherent non-robustness

– no quality guarantees on the output

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Leif Kobbelt RWTH Aachen University

Example Algorithm

algorithm for filling holes

Peter Liepa Filling Holes in Meshes In Proc. Symposium on Geometry Processing 2003

three stages
1. compute a coarse triangulation T to fill a hole
2. refine the triangulation, T → Tʼ, to match the vertex

densities of the surrounding area

3. smooth the triangulation Tʼ to match the geometry
f the surrounding

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Leif Kobbelt RWTH Aachen University

Filling Holes in Meshes - 1

compute a coarse triangulation T

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Leif Kobbelt RWTH Aachen University

Filling Holes in Meshes - 1

compute a coarse triangulation T
f minimal weight w(T)

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n vertices, n−2 triangles

1 2 11 10 9 8 7 6 5 4 3

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Leif Kobbelt RWTH Aachen University

Filling Holes in Meshes - 1

weight w(T) is a mixture of

– area(T) = area(∆ ) – maximum dihedral angle in T

thus, we favour triangulations of low area and low

normal variation

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∆ ∈ T

∑

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Leif Kobbelt RWTH Aachen University

Filling Holes in Meshes - 1

let w[a,c] be the minimal weight that can be

achieved in triangulating the polygon a,a+1,...,c

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1 2 11 10 9 8 7 6 5 4 3

w[2,9] = ?

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Leif Kobbelt RWTH Aachen University

Filling Holes in Meshes - 1

let w[a,c] be the minimal weight that can be

achieved in triangulating the polygon a,a+1,...,c

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1 2 11 10 9 8 7 6 5 4 3

w[2,9] = min( w(∆(2,3,9)) + w[3,9], w[2,4] + w(∆(2,4,9)) + w[4,9], w[2,5] + w(∆(2,5,9)) + w[5,9], w[2,6] + w(∆(2,6,9)) + w[6,9], w[2,7] + w(∆(2,7,9)) + w[7,9], w[2,8] + w(∆(2,8,9)) )

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Leif Kobbelt RWTH Aachen University

Filling Holes in Meshes - 1

let w[a,c] be the minimal weight that can be

achieved in triangulating the polygon a,a+1,...,c

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1 2 11 10 9 8 7 6 5 4 3

w[2,9] = min( w(∆(2,3,9)) + w[3,9], w[2,4] + w(∆(2,4,9)) + w[4,9], w[2,5] + w(∆(2,5,9)) + w[5,9], w[2,6] + w(∆(2,6,9)) + w[6,9], w[2,7] + w(∆(2,7,9)) + w[7,9], w[2,8] + w(∆(2,8,9)) )

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Leif Kobbelt RWTH Aachen University

Filling Holes in Meshes - 1

let w[a,c] be the minimal weight that can be

achieved in triangulating the polygon a,a+1,...,c

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1 2 11 10 9 8 7 6 5 4 3

w[2,9] = min( w(∆(2,3,9)) + w[3,9], w[2,4] + w(∆(2,4,9)) + w[4,9], w[2,5] + w(∆(2,5,9)) + w[5,9], w[2,6] + w(∆(2,6,9)) + w[6,9], w[2,7] + w(∆(2,7,9)) + w[7,9], w[2,8] + w(∆(2,8,9)) )

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Leif Kobbelt RWTH Aachen University

Filling Holes in Meshes - 1

let w[a,c] be the minimal weight that can be

achieved in triangulating the polygon a,a+1,...,c

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1 2 11 10 9 8 7 6 5 4 3

w[2,9] = min( w(∆(2,3,9)) + w[3,9], w[2,4] + w(∆(2,4,9)) + w[4,9], w[2,5] + w(∆(2,5,9)) + w[5,9], w[2,6] + w(∆(2,6,9)) + w[6,9], w[2,7] + w(∆(2,7,9)) + w[7,9], w[2,8] + w(∆(2,8,9)) )

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Leif Kobbelt RWTH Aachen University

Filling Holes in Meshes - 1

let w[a,c] be the minimal weight that can be

achieved in triangulating the polygon a,a+1,...,c

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1 2 11 10 9 8 7 6 5 4 3

w[2,9] = min( w(∆(2,3,9)) + w[3,9], w[2,4] + w(∆(2,4,9)) + w[4,9], w[2,5] + w(∆(2,5,9)) + w[5,9], w[2,6] + w(∆(2,6,9)) + w[6,9], w[2,7] + w(∆(2,7,9)) + w[7,9], w[2,8] + w(∆(2,8,9)) )

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Leif Kobbelt RWTH Aachen University

Filling Holes in Meshes - 1

let w[a,c] be the minimal weight that can be

achieved in triangulating the polygon a,a+1,...,c

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1 2 11 10 9 8 7 6 5 4 3

w[2,9] = min( w(∆(2,3,9)) + w[3,9], w[2,4] + w(∆(2,4,9)) + w[4,9], w[2,5] + w(∆(2,5,9)) + w[5,9], w[2,6] + w(∆(2,6,9)) + w[6,9], w[2,7] + w(∆(2,7,9)) + w[7,9], w[2,8] + w(∆(2,8,9)) )

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Leif Kobbelt RWTH Aachen University

Filling Holes in Meshes - 1

let w[a,c] be the minimal weight that can be

achieved in triangulating the polygon a,a+1,...,c

recursion formula
dynamic programming leads to an O(n3) algorithm

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w[a,c] = min w[a,b] + w(∆(a,b,c)) + w[b,c]

a<b<c

w[x,x+1] = 0

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Leif Kobbelt RWTH Aachen University

Filling Holes in Meshes - 2+3

refine the triangulation such that its vertex

density matches that of the surrounding area ➡ Pierreʼs talk about remeshing

smooth the filling such that its geometry matches

that of the surrounding area ➡ Markʼs talk about mesh smoothing

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Leif Kobbelt RWTH Aachen University

Filling Holes in Meshes - 2+3

refinement and smoothing

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Leif Kobbelt RWTH Aachen University 34 Input model Minimal triangulation Refined triangulation Output model Output model

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Leif Kobbelt RWTH Aachen University

Filling Holes in Meshes

what problems do we encounter?

– islands are not incorporated – self-intersections cannot be excluded – quality depends on boundary distortion

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Leif Kobbelt RWTH Aachen University

Model Repair

types of input
surface-oriented algorithms

– Filling holes in meshes [Liepa 2003]

volumetric algorithms

– Simplification and repair of polygonal models using volumetric techniques [Nooruddin and Turk 2003] – Automatic restoration of polygon models [Bischoff, Pavic, Kobbelt 2005]

conclusion & outlook

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Leif Kobbelt RWTH Aachen University

Volumetric Algorithms

1. convert the input model into an intermediate

volumetric representation ➙ loss of information

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voxel grid adaptive octree BSP tree

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Leif Kobbelt RWTH Aachen University

Volumetric Algorithms

1. convert the input model into an intermediate

volumetric representation ➙ loss of information

2. discrete volumetric representation ➙ robust

and reliable processing

– morphological operators (dilation, erosion) – smoothing – flood-fill to determine interior/exterior – ...

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Leif Kobbelt RWTH Aachen University

Volumetric Algorithms

1. convert the input model into an intermediate

volumetric representation ➙ loss of information

2. discrete volumetric representation ➙ robust

and reliable processing

– morphological operators (dilation, erosion) – smoothing – flood-fill to determine interior/exterior

3. extract the surface of a solid object from the

volume ➙ manifold and watertight

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Leif Kobbelt RWTH Aachen University

Volumetric Algorithms

1. convert the input model into an intermediate

volumetric representation ➙ loss of information

2. discrete volumetric representation ➙ robust

and reliable processing

– morphological operators (dilation, erosion) – smoothing – flood-fill to determine interior/exterior

3. extract the surface of a solid object from the

volume ➙ manifold and watertight

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Leif Kobbelt RWTH Aachen University

Volumetric Algorithms

advantages

– fully automatic – few (intuitive) user parameters – robust – guaranteed manifold output

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Leif Kobbelt RWTH Aachen University

Volumetric Algorithms

problems

– slow and memory intensive ➙ adaptive data structures – aliasing and loss of features ➙ feature sensitive reconstruction (EMC, DC) – loss of mesh structure ➙ bad luck (... hybrid approaches) – large output ➙ mesh decimation (Markʼs talk)

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Leif Kobbelt RWTH Aachen University

Example 1

example algorithm 1
F. S. Nooruddin and G. Turk
Simplification and Repair of Polygonal Models Using Volumetric Techniques
IEEE Transactions on Visualization and Computer Graphics 2003
issues

– classification of sample points as being inside or outside of the object

(parity count, ray stabbing)

– sampling the volume – extracting the mesh

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Leif Kobbelt RWTH Aachen University

Nooruddin and Turkʼs Method

point classification: Layered depth images (LDI)

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x y

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Leif Kobbelt RWTH Aachen University

Nooruddin and Turkʼs Method

point classification: Layered depth images (LDI)

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x y

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Leif Kobbelt RWTH Aachen University

Nooruddin and Turkʼs Method

point classification: Layered depth images (LDI)

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x y

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Leif Kobbelt RWTH Aachen University

Nooruddin and Turkʼs Method

point classification: Layered depth images (LDI)

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x y

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Leif Kobbelt RWTH Aachen University

point classification: Layered depth images (LDI)
1. record n layered depth images
2. project the query point x into each depth image
3. if any of the images classifies x as exterior, then

x is globally classified as exterior else as interior

Nooruddin and Turkʼs Method

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Leif Kobbelt RWTH Aachen University

supersampling
filtering

– Gaussian – morphological filters (dilation, erosion)

model simplification
reduction of topological

noise

marching cubes

Nooruddin and Turkʼs Method

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Leif Kobbelt RWTH Aachen University

Nooruddin and Turkʼs Method

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supersampling
filtering

– Gaussian – morphological filters (dilation, erosion)

model simplification
reduction of topological

noise

marching cubes

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Leif Kobbelt RWTH Aachen University

supersampling
filtering

– Gaussian – morphological filters (dilation, erosion)

model simplification
reduction of topological

noise

marching cubes

Nooruddin and Turkʼs Method

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Leif Kobbelt RWTH Aachen University 52 100×100×100 Input model Input model 200×200×200 300×300×300 50×50×50

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Leif Kobbelt RWTH Aachen University 53 Supersampling Raw Smoothing Supersampling + smoothing Marching Cubes

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Leif Kobbelt RWTH Aachen University

voxelization

– characteristic function / signed distance function – cannot handle all kinds of inconsistencies

repair

– uniform treatment of voxel – cannot exploit local shape information

extraction

– thresholding – sampling artifacts

Nooruddin and Turkʼs Method

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Leif Kobbelt RWTH Aachen University

Example 2

example algorithm 2
S. Bischoff, D. Pavic, L. Kobbelt
Automatic Restoration of Polygon Models
Transactions on Graphics 2005

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Leif Kobbelt RWTH Aachen University

Overview

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volumetric representation volumetric representation manifold mesh extraction gap filling, removal of interior geometry voxelization

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Leif Kobbelt RWTH Aachen University

Overview

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Leif Kobbelt RWTH Aachen University

Overview

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Leif Kobbelt RWTH Aachen University

Overview

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Leif Kobbelt RWTH Aachen University

Overview

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Leif Kobbelt RWTH Aachen University

Overview

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Leif Kobbelt RWTH Aachen University

Overview

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Leif Kobbelt RWTH Aachen University

Overview

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Leif Kobbelt RWTH Aachen University

Conversion

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adaptive octree: subdivide a cell, if it contains

multiple planes or a boundary

voxel topology = polygon topology

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Closing Gaps

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close gaps by dilating the boundary voxels

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Leif Kobbelt RWTH Aachen University

Closing Gaps

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close gaps by dilating the boundary voxels

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Leif Kobbelt RWTH Aachen University

Closing Gaps

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close gaps by dilating the boundary voxels

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Determine Exterior

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determine the exterior by flood filling & dilation

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Leif Kobbelt RWTH Aachen University

Determine Exterior

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determine the exterior by flood filling & dilation

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Leif Kobbelt RWTH Aachen University

Determine Exterior

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determine the exterior by flood filling & dilation

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Leif Kobbelt RWTH Aachen University

Determine Exterior

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determine the exterior by flood filling & dilation

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Extract the Surface

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extract the surface by a variant of Dual Contouring

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Leif Kobbelt RWTH Aachen University

Extract the Surface

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extract the surface by a variant of Dual Contouring

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Leif Kobbelt RWTH Aachen University

Extract the Surface

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extract the surface by a variant of Dual Contouring

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Leif Kobbelt RWTH Aachen University

Results

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Results

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Leif Kobbelt RWTH Aachen University

Results

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riginal

1124 triangles reconstruction 279892 triangles (at 1000³) decimated 7018 triangles

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Leif Kobbelt RWTH Aachen University

Results

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without gap filling with gap filling

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Leif Kobbelt RWTH Aachen University

Results

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Leif Kobbelt RWTH Aachen University

Model Repair

types of input
surface-oriented algorithms

– Filling holes in meshes [Liepa 2003]

volumetric algorithms

– Simplification and repair of polygonal models using volumetric techniques [Nooruddin and Turk 2003] – Automatic restoration of polygon models [Bischoff, Pavic, Kobbelt 2005]

conclusion & outlook

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Leif Kobbelt RWTH Aachen University

Conclusion

mesh repair to remove artifacts that arise in

various types of input models

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Leif Kobbelt RWTH Aachen University

Conclusion

surface-oriented algorithms ...

– fast, structure preserving – often not robust, need user interaction and cannot give quality guarantees on the output

volumetric algorithms ...

– use an intermediate volumetric representation and thus produce guaranteed watertight meshes – suffer from (topological) sampling problems

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Leif Kobbelt RWTH Aachen University

History of Mesh Repair

– Bøhn, Wozny: Automatic CAD Model Repair: Shell-Closure.

1992

– Mäkelä, Dolenc: Some Efficient Procedures for Correcting Triangulated Models.

1993

– Turk, Levoy: Zippered Polygon Meshes from Range Images.

1994

– Barequet, Sharir: Filling Gaps in the Boundary of a Polyhedron.

1995

– Curless, Levoy: A Volumetric Method for Building Complex Models from Range Images. 1996 – Barequet, Kumar: Repairing CAD Models.

1997

– Murali, Funkhouser. Consistent Solid and Boundary Representations.

1997

– Guéziec, Taubin, Lazarus, Horn: Cutting and Stitching: [...]

2001

– Guskov, Wood: Topological Noise Removal.

2001

– Borodin, Novotni, Klein: Progressive Gap Closing for Mesh Repairing.

2002

– Davis, Marschner, Garr, Levoy: Filling Holes in Complex Surfaces Using Volumetric Diffusion. 2002 – Liepa: Filling Holes in Meshes.

2003

– Greß, Klein: Efficient Representation and Extraction of 2-Manifold Isosurfaces Using kd-Trees. 2003 – Nooruddin, Turk: Simplification and Repair of Polygonal Models Using Volumetric Techniques. 2003 – Borodin, Zachmann Klein: Consistent Normal Orientation for Polygonal Meshes.

2004

– Ju: Robust Repair of Polygonal Models.

2004

– Bischoff, Pavic, Kobbelt: Automatic Restoration of Polygon Models.

2005

– Podolak, Rusinkiewicz: Atomic Volumes for Mesh Completion.

2005

– Shen, O'Brien, Shewchuk: Interpolating and Approximating Implicit Surfaces from Polygon Soup. 2005

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Surface-oriented
Volumetric

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Outlook

hybrid algorithms that are ...

– ... robust and – ... structure preserving

Bischoff, Kobbelt: Structure Preserving CAD Model Repair. Eurographics 2005

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