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update after Mael's review
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<h3><a href="https://geometryfactory.com/who-we-are/">Jane Tournois</a></h3>
<h4><a href="http://www.geometryfactory.com" target="_blank">GeometryFactory</a></h4>
<br>


<p>Introduced in CGAL 3.5, the package
<p>The CGAL component
<a href="https://doc.cgal.org/latest/Manual/packages.html#PkgMesh3" text="3D Mesh Generation">3D Mesh Generation</a>
enables the generation of isotropic simplicial meshes discretizing 3D domains.
Since then, multiple improvements in mesh quality, performance, and versatility have been made available in CGAL.
Lately, we have focused more on the quality of the output mesh, the ability of the
mesh generation algorithm to represent the input surface in a more accurate way, and for a wider range of input data types.
</p>
is the primary component to generate isotropic simplicial meshes discretizing 3D domains with CGAL.
First introduced in CGAL-3.5 (2009!), this component has been constantly updated through the years
to improve its performance, its versatility, and the quality of the meshes it produces.</p>

<br>
<p>Recent releases of CGAL have introduced improvements to the 3D Tetrahedral Mesh Generation package:
detection and protection of triple lines from labeled images,
generation and use of weighted labeled images,
and a new approximation error criterion for polyline features.</p>
<p>Recent releases of CGAL have once again introduced new enhancements that enable more accurate
representations of input surfaces, for a wider range of input data types. It is now possible to:
<ul>
<li>detect and recover sharp features in labeled images,</li>
<li>generate meshes of <i>weighted</i> labeled images,</li>
<li>control the maximum error on the approximation error of polyline features.</li>
</ul></p>


<br>
<h3>Triple Lines from Labeled Images</h3>
<h3>Recovery of Sharp Features in Labeled Images</h3>

<p>CGAL's tetrahedral mesh generation algorithm is able to compute tetrahedral meshes from 3D segmented images.
For all types of inputs, detection and construction of one-dimensional features must be done a priori for the
feature protection algorithm to be able to represent them properly in the generated mesh.
In 3D segmented images, such features correspond to the intersections of the bounding box of the image with the surfaces
<p>CGAL's tetrahedral mesh generation algorithm is able to generate tetrahedral meshes from 3D segmented images.
Regardless of input type, one-dimensional feature detection and construction must be performed before
the generation of the mesh.
In 3D segmented images, these features correspond to the intersections of the bounding box of the image with the surfaces
defined by the image, as well as polylines that lie at the intersection of three or more subdomains (including the outside).</p>

<p>The article
<a href="https://opus4.kobv.de/opus4-zib/frontdoor/index/index/docId/274"><i>Hege, H. C., Seebass, M., Stalling, D., & Zöckler, M. (1997),
A generalized marching cubes algorithm based on non-binary classifications</i></a>, provides a list of possible voxel configurations.
This description has been generalized for tetrahedral mesh generation needs, and
implemented in CGAL.</p>
A generalized marching cubes algorithm based on non-binary classifications</i></a> provides a classification
of possible voxel configurations.
This description has been generalized for tetrahedral mesh generation needs, and implemented in CGAL.</p>

<p>After feature lines detection, similarly to what is done for polyhedral surfaces, the polyline features
are added to the feature graph of the domain for feature protection.</p>
<p>Following feature lines detection, polyline features are added to the feature graph of the domain
for feature protection, similarly to what is done for polyhedral surfaces.</p>

<p>Two detection functors are provided :
<p>Two new detection functors are introduced to support the protection of triple lines:
<ul>
<li><code>CGAL::Detect_features_in_image()</code> detects and constructs all
<li><code>CGAL::Detect_features_in_image()</code>: detect and construct all
polylines lying at the intersection of three or more subdomains, and</li>
<li><code>CGAL::Detect_features_on_image_bbox()</code> detects and constructs the polylines
<li><code>CGAL::Detect_features_on_image_bbox()</code>: detect and construct the polylines
that lie at the intersection of two or more subdomains and the bounding box of the input labeled image.</li>
</ul>
</p>
Expand All @@ -57,21 +57,24 @@ are added to the feature graph of the domain for feature protection.</p>
<a href="../../../../images/mesh3_triple_lines_protection.png"><img src="../../../../images/mesh3_triple_lines_protection.png" style="max-width:80%"/></a><br>
<br><small>Mesh generation from labeled image without (left)
and with triple lines detection and protection (right).
The polylines lying at the intersection of three or more subdomains are
a lot more precise with a lot fewer vertices.</small>
Protection of the feature lines at the intersection of the three subdomains (pink, blue, outside)
enables a much more faithful mesh, with fewer vertices.</small>
</div>


<br>
<h3>Weighted Labeled Images</h3>

<p>When a segmented image is given as input for tetrahedral mesh generation,
the generated mesh surface sometimes sticks too closely to the voxels surface, causing an aliasing effect. A solution to generate
a smooth and accurate output surface was described by Stalling et al. in
the generated mesh surface sometimes sticks too closely to the voxels surface, causing an aliasing effect.
A solution to generate a smooth and accurate output surface was described by Stalling et al. in
<a href="https://opus4.kobv.de/opus4-zib/frontdoor/index/index/docId/382"><i>Stalling, D., Zöckler, M., Sander, O., & Hege, H. C. (1998),
Weighted labels for 3D image segmentation</i></a>.
It consists in generating a second input image, made of integer coefficients called weights, and use those weights to define
smoother domain boundaries. The 3D image of weights can be generated by <code>CGAL::Mesh_3::generate_label_weights()</code>.</p>
The approach consists in generating a second input image made of integer coefficients called <i>weights</i>,
and to use those weights to define smoother domain boundaries.

This addition has been added to CGAL, and the 3D image of weights can be generated using the new function
<code>CGAL::Mesh_3::generate_label_weights()</code>.</p>

<div style="text-align:center;">
<a href="../../../../images/mesh3_weighted_images.png"><img src="../../../../images/mesh3_weighted_images.png" style="max-width:100%"/></a><br>
Expand All @@ -82,33 +85,33 @@ smoother domain boundaries. The 3D image of weights can be generated by <code>CG
<br>
<p>The following example shows how to use weighted labeled images: <a href="https://doc.cgal.org/6.0.1/Mesh_3/Mesh_3_2mesh_3D_weighted_image_8cpp-example.html">mesh_3D_weighted_image.cpp</a>.</p>

<p>Utilizing tetrahedral mesh generation from labeled images, incorporating both feature detection and weighted labels,
presents challenges. A new method that combines feature detection and weighted labels has been
designed and implemented in CGAL, achieving smooth and feature preserving output meshes.
<p>A new method combining triple line protection and weighted labels has been
designed and implemented in CGAL, achieving smooth and feature preserving meshes.
This method leverages the best of these two new functionalities.</p>


<br>
<h3>Approximation Criterion on Feature Edges</h3>
<h3>Approximation Criterion for Feature Edges</h3>

<p>A Delaunay refinement algorithm is guided by meshing criteria that trigger Steiner vertices insertions,
until they are satisfied.
The list of CGAL meshing criteria has been enriched with a new criterion that enables the user to
define an upper bound on the distance between the input feature graph and the output feature edges.</p>
<p>Our Delaunay refinement algorithm is guided by meshing criteria (size, shape, etc.).
When these criteria are not honored, new vertices are inserted until the criteria are satisfied,
a process guaranteed to terminate.
The list of CGAL meshing criteria has been enriched with a new criterion that enables users to
define an upper bound on the distance between the input feature graph and the output feature edges,
providing finer control over the mesh features.</p>

<p>The example <a href="https://doc.cgal.org/6.0.1/Mesh_3/Mesh_3_2mesh_polyhedral_domain_with_edge_distance_8cpp-example.html">
mesh_polyhedral_domain_with_edge_distance.cpp</a> shows how to use this new criterion, called <code>edge_distance</code>.</p>

<div style="text-align:center;">
<a href="../../../../images/mesh3_edge_distance_stool.png"><img src="../../../../images/mesh3_edge_distance_stool.png" style="max-width:65%"/></a><br>
<br><small>Mesh generation from polyhedral surface with varying <code>edge_distance</code> meshing criterion.
The quality improvement is noticeable at the bottom section of the stool legs.</small>
<br><small>Mesh generation from polyhedral surface with a varying <code>edge_distance</code> meshing criterion.
The quality improvement is noticeable at the extremities of the stool legs.</small>
</div>


<h3>Status</h3>

<p>All these new improvements have been integrated in CGAL-6.0.1, which has just been officially released.</p>
<p>All these new improvements are part of CGAL-6.0.1, which has just been officially released.</p>

<i class="bi bi-book"></i>
<a href="https://www.cgal.org/2024/10/22/cgal601/">CGAL-6.0.1 announcement</a>
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