Restricted Delaunay Research Articles

OA009] Motion inclusive variations in bladder dose surface maps during the course of high-precision radiotherapy for prostate cancer

Purpose Modern radiotherapy (RT) protocols for prostate cancer often involve the use of narrow margins and image-based monitoring of rectum/bladder fill status. These protocols have allowed safe prostate dose escalation, maintaining acceptable levels for organs at risk (bladder and rectum). However, daily cone-beam CT (CBCT) image-guided RT has demonstrated considerable bladder volume variation throughout treatment, which suggests dose delivered may vary. By using CBCT-based parametrized 2D dose surface maps (DSM) of the inferior bladder, this study aims to evaluate bladder volume impact on bladder DSMs during high-precision RT for prostate cancer. Methods Seven prostate cancer patients treated using daily CBCT-based image-guided VMAT/IMRT were included in this study (81.0 Gy prescription dose). Planning CT and RT delivery adhered to a full bladder/empty rectum protocol, where daily CBCTs were used for patient realignment and to assess bladder/rectum filling status. Fourteen CBCTs per patient were rigidly registered to the planning CT using recorded treatment shifts, and the bladder was manually contoured on each CBCT. Contours were validated by the responsible radiation oncologist. For the planning CT and each CBCT, bladder wall dose was digitally projected onto 1024x1024 DSMs using orthographic ray-traced surface dose sampling. Bladder wall volumetric meshes were generated from contours using restricted Delaunay triangulation. Surface dose was sampled at ray-wall intersections, sub-pixel sampling was used to improve image quality, and rays were oriented parallel to the prostate-bladder center-of-mass connecting line. DSM dose distributions were compared between planned and delivered. Correlation with bladder volume variations was evaluated. Results Bladder volumes varied considerably during RT (15–42%), with slightly larger volumes at planning compared to treatment (p = 0.16). Differences at the central part of the DSM ranged between 1% and 7%. Overall, delivered doses were lower at the central part of the DSM compared to planned (range: −5.4 to 0.9 Gy). Plan-vs-delivery mean dose differences were slightly correlated with bladder volume differences (Rs = 0.55, p = 0.3). Conclusions No significant variations were observed in delivered doses at the interior part of the bladder although considerable bladder volume changes occurred during the RT course. The generally smaller treatment bladder volume possibly explains lower dose delivered to the inferior bladder sector.

Conforming Restricted Delaunay Mesh Generation for Piecewise Smooth Complexes

A Frontal-Delaunay refinement algorithm for mesh generation in piecewise smooth domains is described. Built using a restricted Delaunay framework, this new algorithm combines a number of novel features, including: (i) an unweighted, conforming restricted Delaunay representation for domains specified as a (non-manifold) collection of piecewise smooth surface patches and curve segments, (ii) a protection strategy for domains containing curve segments that subtend sharply acute angles, and (iii) a new class of off-centre refinement rules designed to achieve high-quality point-placement along embedded curve features. Experimental comparisons show that the new Frontal-Delaunay algorithm outperforms a classical (statically weighted) restricted Delaunay- refinement technique for a number of three-dimensional benchmark problems.

CGALmesh

CGALmesh is the mesh generation software package of the Computational Geometry Algorithm Library (CGAL). It generates isotropic simplicial meshes—surface triangular meshes or volume tetrahedral meshes—from input surfaces, 3D domains, and 3D multidomains, with or without sharp features. The underlying meshing algorithm relies on restricted Delaunay triangulations to approximate domains and surfaces and on Delaunay refinement to ensure both approximation accuracy and mesh quality. CGALmesh provides guarantees on approximation quality and on the size and shape of the mesh elements. It provides four optional mesh optimization algorithms to further improve the mesh quality. A distinctive property of CGALmesh is its high flexibility with respect to the input domain representation. Such a flexibility is achieved through a careful software design, gathering into a single abstract concept, denoted by the oracle, all required interface features between the meshing engine and the input domain. We already provide oracles for domains defined by polyhedral and implicit surfaces.

Poisson-driven seamless completion of triangular meshes

The production of high-quality 3D mesh models has seen important technological advancements in recent years and is increasingly becoming a crucial asset for several application domains. However, the intrinsic nature of the problem (acquisition constraints and real objects complexity) makes hole filling and mesh completion still critical tasks for the effectual finalization of the models. In this work, we propose a new solution for filling holes and gaps and for reconstructing missing parts of mesh models that is capable to guarantee the full preservation of the input mesh. On the implicit function obtained from a Poisson Reconstruction computed on a set of oriented points derived from the input mesh, we develop an interpolation method which allows a fast and continuous surface-oracle computation which is exploited to efficiently guide a restricted Delaunay triangulation. This can be used to confine the tessellation inside the holed zones, while protecting the input mesh and the boundary curves so that simple tailoring routines can merge the obtained surface-reconstruction driven patches with the input mesh without stitching artifacts. The proposed technique is versatile and evidences its effectiveness in dealing with multiple complex holes, gaps and surface blending, and it compares favorably with respect to different, and usually more specialized, hole filling and global mesh repair techniques. As a parallel benefit, we also show how the same proposed solution can be exploited as an effective and high-quality meshing or remeshing tool.

Anisotropic Delaunay Meshes of Surfaces

Anisotropic simplicial meshes are triangulations with elements elongated along prescribed directions. Anisotropic meshes have been shown well suited for interpolation of functions or solving PDEs. They can also significantly enhance the accuracy of a surface representation. Given a surface S endowed with a metric tensor field, we propose a new approach to generate an anisotropic mesh that approximates S with elements shaped according to the metric field. The algorithm relies on the well-established concepts of restricted Delaunay triangulation and Delaunay refinement and comes with theoretical guarantees. The star of each vertex in the output mesh is Delaunay for the metric attached to this vertex. Each facet has a good aspect ratio with respect to the metric specified at any of its vertices. The algorithm is easy to implement. It can mesh various types of surfaces like implicit surfaces, polyhedra, or isosurfaces in 3D images. It can handle complicated geometries and topologies, and very anisotropic metric fields.

Differentiation of Calcium Oxalate Monohydrate and Calcium Oxalate Dihydrate Stones Using Quantitative Morphological Information from Micro-Computerized and Clinical Computerized Tomography

We differentiated calcium oxalate monohydrate and calcium oxalate dihydrate kidney stones using micro and clinical computerized tomography images. A total of 22 calcium oxalate monohydrate and 15 calcium oxalate dihydrate human kidney stones were scanned using a commercial micro-computerized tomography scanner with a pixel size of 7 to 23 μm. Under an institutional review board approved protocol, image data on 10 calcium oxalate monohydrate and 9 calcium oxalate dihydrate stones greater than 5 mm were retrieved from a total of 80 patients who underwent clinical dual energy computerized tomography for clinical indications and had stones available for infrared spectroscopic compositional analysis. Micro and clinical computerized tomography images were processed using in-house software, which quantified stone surface morphology with curvature based calculations. A shape index was generated as a quantitative shape metric to differentiate calcium oxalate monohydrate from calcium oxalate dihydrate stones. Statistical tests were used to test the performance of the shape index. On micro-computerized tomography images the shape index of calcium oxalate monohydrate and calcium oxalate dihydrate stones significantly differed (ROC curve AUC 0.92, p <0.0001). At the optimal cutoff sensitivity was 0.93 and specificity was 0.91. On clinical computerized tomography images a significant morphological difference was also detected (p = 0.007). AUC, sensitivity and specificity were 0.90, 1 and 0.73, respectively. On micro and clinical computerized tomography images a morphological difference was detectable in calcium oxalate monohydrate and calcium oxalate dihydrate stones larger than 5 mm. The shape index is a highly promising method that can distinguish calcium oxalate monohydrate and calcium oxalate dihydrate stones with reasonable accuracy.

Surface Reconstruction from Multi-View Stereo of Large-Scale Outdoor Scenes

This article describes an original method to reconstruct a 3D scene from a sequence of images. Our approach uses both the dense 3D point cloud extracted by multi-view stereovision and the calibrated images. It combines depth-maps construction in the image planes with surface reconstruction through restricted Delaunay triangulation. The method may handle very large scale outdoor scenes. Its accuracy has been tested on numerous outdoor scenes including the dense multi-view benchmark proposed by Strecha et al. Our results show that the proposed method compares favorably with the current state of the art.

Towards Sub-cellular Modeling with Delaunay Triangulation

In this article a novel model framework to simulate cells and their internal structure is described. The model is agent-based and suitable to simulate single cells with a detailed internal structure as well as multi-cellular compounds. Cells are simulated as a set of many interacting particles, with neighborhood relations defined via a Delaunay triangulation. The interacting sub-particles of a cell can assume specific roles – i.e., membrane sub-particle, internal sub-particle, organelles, etc –, distinguished by specific interaction potentials and, eventually, also by the use of modified interaction criteria. For example, membrane sub-particles may interact only on a two-dimensional surface embedded on three-dimensional space, described via a restricted Delaunay triangulation. The model can be used not only to study cell shape and movement, but also has the potential to investigate the coupling between internal space-resolved movement of molecules and determined cell behaviors.

Manifold Reconstruction in Arbitrary Dimensions Using Witness Complexes

It is a well-established fact that the witness complex is closely related to the restricted Delaunay triangulation in low dimensions. Specifically, it has been proved that the witness complex coincides with the restricted Delaunay triangulation on curves, and is still a subset of it on surfaces, under mild sampling conditions. In this paper, we prove that these results do not extend to higher-dimensional manifolds, even under strong sampling conditions such as uniform point density. On the positive side, we show how the sets of witnesses and landmarks can be enriched, so that the nice relations that exist between restricted Delaunay triangulation and witness complex hold on higher-dimensional manifolds as well. We derive from our structural results an algorithm that reconstructs manifolds of any arbitrary dimension or co-dimension at different scales. The algorithm combines a farthest-point refinement scheme with a vertex pumping strategy. It is very simple conceptually, and it does not require the input point sample to be sparse. Its running time is bounded by c(d)n 2, where n is the size of the input point cloud, and c(d) is a constant depending solely (yet exponentially) on the dimension d of the ambient space. Although this running time makes our reconstruction algorithm rather theoretical, recent work has shown that a variant of our approach can be made tractable in arbitrary dimensions, by building upon the results of this paper.

Reconstruction Using Witness Complexes

We present a novel reconstruction algorithm that, given an input point set sampled from an object S, builds a one-parameter family of complexes that approximate S at different scales. At a high level, our method is very similar in spirit to Chew's surface meshing algorithm, with one notable difference though: the restricted Delaunay triangulation is replaced by the witness complex, which makes our algorithm applicable in any metric space. To prove its correctness on curves and surfaces, we highlight the relationship between the witness complex and the restricted Delaunay triangulation in 2d and in 3d. Specifically, we prove that both complexes are equal in 2d and closely related in 3d, under some mild sampling assumptions.

Delaunay meshing of isosurfaces

We present an isosurface meshing algorithm, DelIso, based on the Delaunay refinement paradigm. This paradigm has been successfully applied to mesh a variety of domains with guarantees for topology, geometry, mesh gradedness, and triangle shape. A restricted Delaunay triangulation, dual of the intersection between the surface and the three-dimensional Voronoi diagram, is often the main ingredient in Delaunay refinement. Computing and storing three-dimensional Voronoi/Delaunay diagrams become bottlenecks for Delaunay refinement techniques since isosurface computations generally have large input datasets and output meshes. A highlight of our algorithm is that we find a simple way to recover the restricted Delaunay triangulation of the surface without computing the full 3D structure. We employ techniques for efficient ray tracing of isosurfaces to generate surface sample points, and demonstrate the effectiveness of our implementation using a variety of volume datasets.

Fast and Efficient Restricted Delaunay Triangulation in Random Geometric Graphs

Let _G_ = 𝓖(_n, r_) be a random geometric graph resulting from placing _n_ nodes uniformly at random in the unit square (or the unit disk) and connecting every two nodes if and only if their Euclidean distance is at most _r_. Let <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="uinm_a_10129164_o_ilf0001.gif"></inline-graphic> be the known critical radius for connectivity when _n_ → ∞. The _restricted Delaunay graph_ RDG(_G_) is a subgraph of _G_ with the following properties: it is a planar graph and a spanner of _G_, and in particular it contains all the short edges of the Delaunay triangulation of _G_. While in general graphs, the construction of RDG(_G_) requires Θ(_n_) messages, we show that when _r_ = _O_(_r__con) and _G_ = 𝓖(_n, r_), then with high probability, RDG(_G_) can be constructed locally in one round of communication with <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="uinm_a_10129164_o_ilf0002.gif"></inline-graphic> messages, and with only one-hop neighborhood information. This size of _r_ proves that the existence of long Delaunay edges (an order larger than _r__con) in the unit square (disk) does not significantly affect the efficiency with which good routing graphs can be maintained.

Geometric spanners for routing in mobile networks

We propose a new routing graph, the restricted Delaunay graph (RDG), for mobile ad hoc networks. Combined with a node clustering algorithm, the RDG can be used as an underlying graph for geographic routing protocols. This graph has the following attractive properties: 1) it is planar; 2) between any two graph nodes there exists a path whose length, whether measured in terms of topological or Euclidean distance, is only a constant times the minimum length possible; and 3) the graph can be maintained efficiently in a distributed manner when the nodes move around. Furthermore, each node only needs constant time to make routing decisions. We show by simulation that the RDG outperforms previously proposed routing graphs in the context of the Greedy perimeter stateless routing (GPSR) protocol. Finally, we investigate theoretical bounds on the quality of paths discovered using GPSR.

Geometric spanners for routing in mobile networks

We propose a new routing graph, the restricted Delaunay graph (RDG), for mobile ad hoc networks. Combined with a node clustering algorithm, the RDG can be used as an underlying graph for geographic routing protocols. This graph has the following attractive properties: 1) it is planar; 2) between any two graph nodes there exists a path whose length, whether measured in terms of topological or Euclidean distance, is only a constant times the minimum length possible; and 3) the graph can be maintained efficiently in a distributed manner when the nodes move around. Furthermore, each node only needs constant time to make routing decisions. We show by simulation that the RDG outperforms previously proposed routing graphs in the context of the Greedy perimeter stateless routing (GPSR) protocol. Finally, we investigate theoretical bounds on the quality of paths discovered using GPSR.

Approximation of the Normal Vector Field and the Area of a Smooth Surface

This paper deals with the comparison of the normal vector field of a smooth surface S with the normal vector field of another surface differentiable almost everywhere. The main result gives an upper bound on angles between the normals of S and the normals of a triangulation T close to S. This upper bound is expressed in terms of the geometry of T, the curvature of S and the Hausdorff distance between both surfaces. This kind of result is really useful: in particular, results of the approximation of the normal vector field of a smooth surface S can induce results of the approximation of the area; indeed, in a very general case (T is only supposed to be locally the graph of a lipschitz function), if we know the angle between the normals of both surfaces, then we can explicitly express the area of S in terms of geometrical invariants of T, the curvature of S and of the Hausdorff distance between both surfaces. We also apply our results in surface reconstruction: we obtain convergence results when T is the restricted Delaunay triangulation of an e-sample of S; using Chew’s algorithm, we also build sequences of triangulations inscribed in S whose curvature measures tend to the curvatures measures of S.

An improved incremental algorithm for constructing restricted Delaunay triangulations

This work presents an algorithm that given a generalized planar graph obtains its Constrained Delaunay triangulation (CDT). The proposed method, which follows the general approach of de Floriani and Puppo ( Computer Vision, Graphics and Image Processing, 1992, 54, pp. 290–300, [1]), works incrementally based on two improved procedures. The first improvement gives a procedure that inserts a new point in a CDT; the second one is an algorithm that enforces a new constraining edge in a CDT. In particular, an algorithm that generates the CDT of a given polygon (possibly with holes) is also obtained. These algorithms have been implemented and included in several applications, showing their robustness and efficiency, even when the original graph has many vertices or edges.

Triangulation of trimmed surfaces in parametric space

The paper presents a new approach for triangulating trimmed surfaces. The basic idea is to perform the triangulation completely in parametric space. The trimmed regions of the surfaces are first mapped into parametric space and approximated by 2D polygonal regions, which are then pretriangulated by a restricted Delaunay triangulation algorithm. The generated triangles are subdivided further until each edge of the triangles is smaller than the allowed length that results from the surface definition and the specified tolerance. All the triangles are finally mapped back into Euclidean space so that the coordinate triples for the triangle vertices can be calculated. This approach makes the triangulation more reliable and faster, and it is also easy to avoid cracks between patches and surfaces. Thus, the algorithm is particularly suitable for the generation of a valid triangulation model for many engineering applications, for example stereolothography.