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Feature Preserving Delaunay Mesh Generation from 3D Multi-Material Images Dobrina Boltcheva, Mariette Yvinec, Jean-Daniel Boissonnat

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Feature Preserving Delaunay Mesh Generation from 3D Multi-Material Images Dobrina Boltcheva, Mariette Yvinec, Jean-Daniel Boissonnat Feature preserving Delaunay mesh generation from 3D multi-material images Dobrina Boltcheva, Mariette Yvinec, Jean-Daniel Boissonnat To cite this version: Dobrina Boltcheva, Mariette Yvinec, Jean-Daniel Boissonnat. Feature preserving Delaunay mesh generation from 3D multi-material images. Computer Graphics Forum, Wiley, 2009, pp.1455-1464. 10.1111/j.1467-8659.2009.01522.x. inria-00413248 HAL Id: inria-00413248 https://hal.inria.fr/inria-00413248 Submitted on 3 Sep 2009 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Eurographics Symposium on Geometry Processing 2009 Volume 28 (2009), Number 5 Marc Alexa and Michael Kazhdan (Guest Editors) Feature preserving Delaunay mesh generation from 3D multi-material images Dobrina Boltcheva, Mariette Yvinec, Jean-Daniel Boissonnat GEOMETRICA – INRIA Sophia Antipolis, France fi[email protected] Abstract Generating realistic geometric models from 3D segmented images is an important task in many biomedical appli- cations. Segmented 3D images impose particular challenges for meshing algorithms because they contain multi- material junctions forming features such as surface patches, edges and corners. The resulting meshes should preserve these features to ensure the visual quality and the mechanical soundness of the models. We present a feature preserving Delaunay refinement algorithm which can be used to generate high-quality tetrahedral meshes from segmented images. The idea is to explicitly sample corners and edges from the input image and to constrain the Delaunay refinement algorithm to preserve these features in addition to the surface patches. Our experimental results on segmented medical images have shown that, within a few seconds, the algorithm outputs a tetrahedral mesh in which each material is represented as a consistent submesh without gaps and overlaps. The optimization property of the Delaunay triangulation makes these meshes suitable for the purpose of realistic visualization or finite element simulations. Categories and Subject Descriptors (according to ACM CCS): Computational Geometry and Object Modeling [I.3.5]: Curve, surface, solid, and object representations—Modeling packages 1. Introduction the interface between 2 different regions of the image. Sec- ondly, the 1-junctions are edges defined as the intersection of Advanced medical applications, like numerical simulations 3 or more image regions. Thirdly, the 0-junctions are corner of physical or physiological processes, frequently require vertices appearing at the meeting point of 4 or more regions. meshed models of human organs rather than grey-level MRI or CT-scan images. Extracting geometric representa- tion from a medical image usually supposes that the image All these multi-material junctions form features in the in- has been segmented into several regions of interest made of put and are expected to be preserved and accurately repre- different materials. During this segmentation, every voxel is sented in the output mesh for many applications. Besides, if assigned a label so that connected voxels with the same label the image regions are meshed separately, it is not obvious to represent an anatomical structure. This paper focuses on the guarantee that the resulting meshes have conforming bound- next step and presents a method that, given a segmented 3D aries without gaps or overlaps along their common junctions. image, meshes simultaneously a set of regions with different In this paper, we present a feature preserving Delaunay re- labels. The resulting meshes are either surface meshes ap- finement algorithm which addresses this issue. The method proximating the boundaries of anatomical structures or vol- simultaneously meshes all image regions and builds a unique ume meshes of these structures. high-quality tetrahedral mesh where each anatomical struc- Building meshed models from multi-material images is ture is represented by a submesh. In addition, the method is very challenging. Indeed, in dimension 3, the different ma- constrained to preserve the 0-junctions between regions and terials are often organized in intricate geometric configura- to reconstruct accurately the 1-junctions as a set of consec- tions and they present three kinds of multi-material junc- utive Delaunay edges and the 2-junctions as surface patches tions. First, the 2-junctions are surface patches defined as composed of Delaunay facets. c 2009 The Author(s) Journal compilation c 2009 The Eurographics Association and Blackwell Publishing Ltd. Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA. D. Boltcheva & M. Yvinec & JD. Boissonnat / Feature preserving Delaunay mesh generation from 3D multi-material images 1.1. Related work [NF97, HK98]. More recently, an octree-based method has There are only a few surface mesh generation methods been proposed [ZHB07] that outputs tetrahedral or hexahe- which deal with multi-material 3D images. The most popu- dral meshes with consistent multi-material junctions. How- lar are extensions of the Marching Cubes algorithm handling ever, like the other grid-based methods, this algorithm does multi-material junction configurations. In an early work, not offer a mesh quality control and elements of poor qual- Hege et al. [HSSZ97] proposed the first generalization of ity are always generated along boundaries. Quality improve- this algorithm for segmented data. Their look-up table sup- ment techniques are usually used as a post-processing step ports up to three different materials meeting at one grid cell. to deliver acceptable meshes. More recently, Wu and Sullivan [WS03] proposed another A powerful strategy based on Delaunay refinement, has extension dealing with up to eight different labels per cell. been proposed for meshing smooth surfaces [BO05] and vol- A well-known issue with MC methods is that they pro- umes bounded by smooth and piecewise smooth surfaces duce very dense meshes presenting staircase artefacts and [ORY05, RY07]. The refinement process is controlled by that they are composed of numerous distorted triangles. highly customizable quality and size criteria on triangular ∗ Meshes generated from a typical segmented medical image facets and on tetrahedra. Pons et al. [PSB 07] have adapted contain an average of 106 triangles. Thus, the initially gen- this mesh generation strategy to segmented images and have erated meshes are usually smoothed (using Laplacian based shown that the sizing criteria can be material-dependent filters) and decimated to obtain a mesh with reasonable size leading to high-quality volume meshes of the different ma- and quality. However, concurrently controlling the accuracy terials where 2-junctions are accurately represented by con- of this mesh filtering and preserving its topological proper- nected sets of Delaunay facets. However, in this work, 0- ties (such as the absence of self-intersections) turns out to and 1-junctions are not explicitly handled and they are not be a difficult and time consuming task. In addition, the usual accurately represented in the output mesh. One way to cir- methods do not guarantee that the resulting meshes are con- cumvent this drawback is to extract the 0- and 1-junctions forming at the 0- and 1-junctions where ambiguous topolog- from the input 3D image and to constrain the Delaunay re- ical configurations exist. finement to preserve them. However, it is well recognized that the presence of constrained edges where meeting sur- Less popular, but much simpler are Dual Contouring face patches form small angles, jeopardize the termination methods which operate on the dual grid obtained by shift- of the Delaunay refinement in 3D [She00]. Recent meth- ing the hexahedral image grid by half a cell width in each ods [CDR07] deal with this problem using the idea of pro- direction. These methods have been initially proposed for bi- tecting small angle regions with balls so that during the re- nary images [Gib98,Nie04] and output quadrilateral meshes finement no point is inserted into these protecting balls. corresponding to the intervoxel boundaries. As for the MC methods, the generated meshes need smoothing and deci- Very recently, Mayer et al. [MWK∗08] proposed a sam- mation in order to give acceptable tessellations in terms of pling strategy for segmented 3D images based on a dynamic quality and size. Dual contouring methods do not rely on particle system which explicitly samples all multi-material look-up tables and their extension to multi-material images junctions. The resulting set of points is then meshed using a is straightforward. Delaunay-based algorithm which outputs high-quality sur- face meshes where, in particular, corner vertices are pre- In a recent work, Bertram et al. [BH05] proposed an al- served and junction edges are accurately represented. How- gorithm for dual contouring of segmented medical images. ever, their sampling approach relies on heavy preprocessing The obtained non-manifold mesh is made of all quadrilat- computations and takes between 3 to 12 hours for small med-
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