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Automatic mesh-healing
technique for model
repair and finite element
model generation
BASED ON PAPER BY C.S. CHONG, A. SENTHIL KUMAR, H.P. LEE
PRESENTED BY NURIT MOSCOVICI
Good Polygonal Models


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What is a good model?

Equilateral triangles

Valence close to 6

Equal edge lengths

Feature preservation
Used for?

3D printing

Finite Element Analysis

…
Link: DGP lecture 9
Problems in Polygonal Models
Holes
Gaps
T-Joints
Overlaps
3
Slivers
This Paper

Black-box algorithm: automatic* model repair
and mesh generation

Work on a triangle-meshed version of the model

Original geometric model is not repaired

Fix mesh so it is watertight

Reconstruct mesh to get good quality
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This Paper
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Previous Work

Directly healing geometry
Edge merging [Steinbrenner et al.2001]
 Stitching using geometric hashing [Barequet 1997]
 Spatial partitioning [Murali et al. 1997]


User-controlled tools
[Petersen et al. 2001]
 [Morvan et al. 1996]


Focus on specific elements
Zipping small gaps [Sheng and Meier 1995]
 Healing slivers [Chew 1997]
 Healing slivers using Delaunay [Edelsbrunner 2000]
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Important terms
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Triangle Mesh

Elements

Nodes

Edges
Inner Edge
Inner Node
Free Node
Free Edge/
Boundary Edge
Element
General description of algorithm
Initial
imperfect
model
Create initial
triangular mesh
Search for free
edges
Mesh healing
Identify and heal
small gaps and
overlaps
Identify and heal
t-joints
Surface
reconstruction
Identify and heal
large gaps/holes
Healed
mesh
model
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Creating the Initial Mesh
Initial
imperfect
model
Create initial
triangular mesh
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Search for free
edges
Mesh healing
Identify and heal
small gaps and
overlaps
Identify and heal
t-joints
Surface
reconstruction
Identify and heal
large gaps/holes
Healed
mesh
model
Creating the Initial Mesh

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Create mesh by triangulating given model

Paving algorithm

Create mesh nodes along model boundaries


Element edge length defined by user
Create triangular elements that conform to mesh nodes
http://mathworld.wolfram.com/Triangulation.html
http://www.cs.cmu.edu/~quake/triangle.delaunay.html
Search for Free Edges
Initial
imperfect
model
Create initial
triangular mesh
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Search for free
edges
Mesh healing
Identify and heal
small gaps and
overlaps
Identify and heal
t-joints
Surface
reconstruction
Identify and heal
large gaps/holes
Healed
mesh
model
Healing Small Gaps and Overlaps
Initial
imperfect
model
Create initial
triangular mesh
Search for free
edges
Mesh healing
Identify and heal
small gaps and
overlaps
Identify and heal
t-joints
Surface
reconstruction
Identify and heal
large gaps/holes
Healed
mesh
model
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Small Gaps and Overlaps

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In this paper gaps are defined as an accidental
space between two surfaces that should be
connected

Usually occur because of numerical and translational errors
Gaps
Overlaps
Small Gaps and Overlaps
Detection

General idea: check proximity between two
candidate surface edges

Traditionally detected by checking proximity
between two element edges
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
Can be difficult if edges are different sizes
In this article: check proximity between nodes
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Nodal Merging Algorithm

Node Pair: a pair of free nodes within a user
specified tolerance distance
Node-pair exists in
model?
yes
Determine location of
merged node
Collapse both nodes
to determined
location
no
Continue to T-Joint
healing
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Determine New Node Location
Calculate line intersection between
tangential planes at the two nodes.

Distance between nodes and line
larger than tolerance distance:


Location is at midpoint
Else:

Project both points onto this line

Location is at midpoint on line
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Healing T-Joints
Initial
imperfect
model
Create initial
triangular mesh
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Search for free
edges
Mesh healing
Identify and heal
small gaps and
overlaps
Identify and heal
t-joints
Surface
reconstruction
Identify and heal
large gaps/holes
Healed
mesh
model
T-Joints

An area in the mesh where the node of one or more
elements intersects the edge of another.


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Modeling error or large truncation error during translation
In this article: also an area where a free node is within a
small distance from a free edge
T-Joints
T-Joint detection

Still checking proximity between two candidate
surface edges

Find free edge with a free node within tolerance
distance.

Edge-node pair
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T-Joint Stitching Algorithm
Node-pair exists in
model?
Yes
Return to nodal
merging algorithm
No
Go to hole-healing
algorithm
No
Edge-pair exists in
model?
Yes
T-Joint found: Nodal
Insertion and element
splitting
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Stitching

Project free node onto free edge in
edge-node pair

Create new node at this location
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Split element belonging to free
edge at new node location

Merge free node with new node
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Stitching example
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Healing Holes
Initial
imperfect
model
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Create initial
triangular mesh
Search for free
edges
Mesh healing
Identify and heal
small gaps and
overlaps
Identify and heal
t-joints
Surface
reconstruction
Identify and heal
large gaps/holes
Healed
mesh
model
Types of Holes

Simple hole: a single loop of free
edges


Can be any shape
Ring hole: a hole consisting of at
least two peripheral loops
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Holes with “islands”
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Hole Filling Algorithm
For every closed loop of free edges (simple hole),
fill hole with triangle elements:
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Calculate the minimum angle between two
adjacent free edges in loop
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3 options:
   75
 75    135
   135
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Option 1:

  75
Form triangle using these two free elements.
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Option 2: 75    135

Create two equilateral
triangles using free edges
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Calculate normal of new
triangles to decide on plane
neightbor
elem ents
n N ode 
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
ni
n1 
i 1
neightbor
elem ents
n2 
 n  nA 
2
 n  nB 
2
n1
n2
Option 2: 75    135

Find average location
between two new nodes
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Merge to a single node at
this location.

Check if any node is within
tolerance distance of the
new node and merge if
found.
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Option 3:   1 3 5
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Create two equilateral triangles using free edges.


Calculate normal of new triangles to decide on plane
No nodal merging
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Ring holes
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Interactively identify ring holes
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Connect islands to the main loop with a bridge line
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
Bridge line converted to triangle elements during meshing
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Leaving us with only simple holes
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If this isn’t done the islands will eventually be discarded
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Consistent model, worse fidelity.
A word about values
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Why 75 and 135?

Good meshes are as close as possible to equilateral
triangles (60 degrees)
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The author decided 75 degrees is tolerable
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Why? They don’t say
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If angle is between 75 and 135 degrees you can split into triangles with
an angle between 37.5 and 75
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If angle is too small it will be handled in next stage
Reconstructing Surface Elements
Initial
imperfect
model
Create initial
triangular mesh
Search for free
edges
Mesh healing
Identify and heal
small gaps and
overlaps
Identify and heal
t-joints
Surface
reconstruction
Identify and heal
large gaps/holes
Healed
mesh
model
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Slivers
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Often generated during t-joint and hole healing
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Long and slender
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Ratio of surface area/perimeter is small
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Bad for numerical solution.
Slivers
Reconstruction options
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Nodal Merging
Edge Flipping
Feature Edges
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Feature Edge: edge that is likely part of shape
definition and should be kept.
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Identified by Smoothness Angle: angle along the
edge of a skewed triangle
Smoothness Angle
Link: DGP lecture 9
Automatic mesh repair thesis
Agnes Larsson

Shape Factor
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Value indicating skewedness of a triangle
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Way to determine if triangle should be fixed
Based on equilateral factor
2
Equilateral factor:
 
2

E     j    m ax  E   
3
3
j 1 
Shape factor:
f 
3
2
3
 E
2
2
Where  j are the angles
of a given triangle
Reconstruction algorithm
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User input: shortest edge length
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Element with edge shorter than the tolerance will be fixed
User input: tolerance factor
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Element with shape factor lower than the tolerance will be fixed
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In the paper, 0.7
Smoothness angle: default 135o
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If angle < 135o consider as feature edge
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If angle > 135o consider as planar
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Reconstruction algorithm
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When to invoke nodal merging:
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Low shape factor and edge shorter than edge length
One of the edges is a feature edge
No feature edge
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Reconstruction algorithm
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When to invoke edge flipping:
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Low shape factor and no short edges
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Take smoothness angle into account
Edge A is not a
feature edge
Edge A is a
feature edge
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Results
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Handles many different input formats
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Divergence from original model
bounded by mesh element size
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Results
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Future Work

Find non-heuristic tolerance values
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Should be connected to size of edges
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Do more during remeshing to ensure optimal
mesh
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Find a way to deal with open loops, missing
surfaces, complex holes, etc.
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Not mentioned: large overlaps, intersections,
bad normals
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Later Work
Repairing and meshing imperfect shapes with
Delaunay refinement [Busaryev, Dey, Levine 2009]
 Handles small errors
 Output: good quality Delaunay mesh
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Weighted Delaunay triangulation
Merge gaps using original boundary data, instead
of merging nodes from new mesh
Sealing Faceted Surfaces to Achieve Watertight CAD
Models[Smith, Tautges, Wilson 2010]
 Use original CAD face data to fill holes more
accurately
 Can heal non-manifold models
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Later Work: Surveys
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Polygon Mesh Repairing: An Application Perspective [Attene et al., 2013]

Surface based reconstructions are suitable when there are few errors, and are generally
not very robust. Volumetric solutions are more robust.
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No common strategy for handling self intersections. Existing solutions are tailored to
specific problems.
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No automatic way of dealing with ring holes.
Fixing Geometric Errors on Polygonal Models: A Survey [Ju, 2009]
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Surface based reconstructions better for preserving small details of the model
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Less robust, and can be very time consuming when large holes/gaps must be filled.
Questions?
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