Delaunay Deformable Models: Topology-Adaptive Meshes Based on the Restricted Delaunay Triangulation Jean-Philippe Pons, Jean-Daniel Boissonnat To cite this version: Jean-Philippe Pons, Jean-Daniel Boissonnat. Delaunay Deformable Models: Topology-Adaptive Meshes Based on the Restricted Delaunay Triangulation. IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2007, Minneapolis, France. pp.200. hal-00488042 HAL Id: hal-00488042 https://hal.archives-ouvertes.fr/hal-00488042 Submitted on 1 Jun 2010 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. Delaunay Deformable Models: Topology-Adaptive Meshes Based on the Restricted Delaunay Triangulation Jean-Philippe Pons Jean-Daniel Boissonnat WILLOW, INRIA / ENS / Ecole´ des Ponts GEOMETRICA, INRIA Paris, France Sophia-Antipolis, France [email protected] [email protected] Abstract mitted that both viewpoints have strengths and weaknesses and that only a hybrid approach can overcome the limita- In this paper, we propose a robust and efficient La- tions of both. In this paper, we propose a method which grangian approach, which we call Delaunay Deformable has the particularity of being purely Lagrangian with all the Models, for modeling moving surfaces undergoing large de- associated advantages, while achieving topological adaptiv- formations and topology changes. Our work uses the con- ity with a comparable robustness and efficiency to Eulerian cept of restricted Delaunay triangulation, borrowed from methods. computational geometry. In our approach, the interface is represented by a triangular mesh embedded in the Delau- 1.1. Eulerian methods nay tetrahedralization of interface points. The mesh is it- eratively updated by computing the restricted Delaunay tri- The Eulerian formulation casts deformation as a time angulation of the deformed objects. Our method has many variation of quantities defined over a fixed grid. The inter- advantages over popular Eulerian techniques such as the faces have to be represented implicitly, since the grid does level set method and over hybrid Eulerian-Lagrangian tech- not conform to them. In computational physics, this is also niques such as the particle level set method: localization known as the front capturing method. Two notable front accuracy, adaptive resolution, ability to track properties as- capturing techniques are the level set method, introduced sociated to the interface, seamless handling of triple junc- by Osher and Sethian [34], and the volume-of-fluid (VOF) tions. Our work brings a rigorous and efficient alternative method, pioneered by Hirt and Nichols [23]. to existing topology-adaptive mesh techniques such as T- Hereafter, we will mainly focus on the level set method snakes. because it is an established technique in computer vision. Basically, this method consists in representing the interface as the zero level set of a higher-dimensional scalar function. 1. Introduction The movement of the interface can be cast as an evolution of the embedding level set function by an Eulerian PDE (par- Deformable models, also known in the literature tial differential equation). We refer the reader to some good as snakes, active contours/surfaces, deformable con- reviews [33, 40] for all the details about the theory, the re- tours/surfaces, constitute a widely used computerized tech- cent developments, the implementation and the applications nique to address various shape reconstruction problems in of the level set method. image processing. They have been initially proposed for the On the one hand, this approach has several advantages purpose of image segmentation by Kass, Witkin and Ter- over an explicit Lagrangian representation of the interface: zopoulos in [24], but they have proven successful in many no parameterization is needed, topology changes are han- other contexts, in computer vision and in medical imaging, dled automatically, intrinsic geometric properties such as including region tracking and shape from X. More gener- normal or curvature can be computed easily from the level ally, modeling dynamic interfaces between several materi- set function. Last but not least, the theory of viscosity solu- als undergoing large deformations is a ubiquitous task in tions provides robust numerical schemes and strong math- science and engineering. Let us mention computer aided ematical results to deal with the evolution PDE. These ad- design, physics simulation and computer graphics. vantages explain the popularity of the level set method not The existing techniques roughly fall into two categories: only in computer vision but also for multi-phase fluid flow Eulerian and Lagrangian formulations. It is commonly ad- simulation [42] in CFD (computational fluid dynamics), as well as for computer animation of fluids with free surfaces defective parts of the interface must be detected, then re- [18, 19, 20, 21, 28]. moved through intricate delooping procedures. On the other hand, several serious shortcomings limit the Another major shortcoming of the mesh-based La- applicability of the level set method: grangian approach is that a fully automatic, robust and effi- First, the higher dimensional embedding makes the level cient handling of topology changes remains an open issue, set method much more expensive computationally than ex- despite several heuristic solutions proposed in computer vi- plicit representations. Much effort has been done to alle- sion [8, 12, 13, 17, 25, 26, 30, 31]. viate this drawback, leading to the narrow band methodol- McInerney and Terzopoulos [30, 31] propose topology ogy [1] and to the PDE-based fast local level set method adaptive deformable curves and meshes, called T-snakes [36]. More recently, octree decompositions have been pro- and T-surfaces. During the evolution, the model is periodi- posed [10, 19, 27, 28] to circumvent the typically fixed uni- cally resampled by computing its intersections with a regu- form sampling of the level set method, in order to reach lar simplicial decomposition of space. A labeling of the ver- high resolution (typically an effective resolution of 5123) tices of the simplicial grid as inside or outside of the model while keeping the computational and memory cost sustain- is maintained. This procedure loses the desirable adaptivity able. However, these methods somewhat lose the simplicity of the Lagrangian formulation, by imposing a fixed uniform of the original level set method, as an efficient implemen- spatial resolution. Also, not all motions are admissible: this tation of such tree-based methods turns out to be a tricky approach only works when the model inflates or deflates task. everywhere, which considerably restricts the range of ap- Second, as discussed and numerically demonstrated by plications. Enright et al. in [18], the level set method is strongly af- Several authors have proposed alternatives to the T- fected by mass loss, smearing of high curvature regions snakes approach: Lachaud and coworkers [25, 26], Bredno and inability to resolve very thin parts. These limitations et al. [8], Duan and Qin [17], Delingette and Montagnat have motivated the development of some hybrid Eulerian- [12, 13]. Basically, these approaches consist in detect- Lagrangian methods, such as the particle level set method ing self-intersections in the evolving mesh and in merg- outlined by Foster and Fedkiw [21] and later improved ing the colliding regions using a set of heuristic remeshing by Enright and coworkers [18, 19, 20]. While the latter rules. Unfortunately, the detection of intersection is com- method yields state-of-the-art results, an objection could be putationally expensive, even when optimizing pairwise dis- the large number of parameters controlling the particle re- tance computations with an octree structure. In practice, it seeding strategy included in this approach. requires the most part of total computation time. Also, these Third, purely Eulerian formulation is not very appropri- approaches lack a systematic and provably correct remesh- ate for tracking interface properties such as color or texture ing strategy. Consequently, they are very likely to break in coordinates, as may be needed in computer graphics appli- some complex or degenerate practical cases. cations. Some approaches based on a coupled system of Eu- This major shortcoming of mesh-based methods have lerian PDEs were recently proposed to overcome this lim- gained popularity to the particle-based approach [9, 14, 15, itation within the level set framework [2, 37, 46], but this 22, 32, 35, 38, 43, 44] for representing dynamic interfaces capability comes at a significant additional computational undergoing complex topology changes. However, it is gen- cost. erally admitted that with the meshless approach, the local- ization of the interface and the computation of interface 1.2. Previous Lagrangian methods properties such as normal and curvature gets cumbersome. The Lagrangian formulation adopts a more “natural” 1.3. Novelty of our method point of view. It explicitly tracks the interfaces between the different materials with some