Power Voronoi Diagram Research Articles

Voronoi diagram based simulation and experimental validation of austenite in C22 and C35 steels

This paper examines the ability of power Voronoi diagram assisted simulation in microstructure modeling during heat treatment. A model is developed for predicting fraction of austenite evolved during continuous heating of steel to austenite range, by integrating geometrical features of power Voronoi diagram and classical nucleation theory. From the simulation results, it is possible to predict the transformed fraction. The simulation results are validated using experiments conducted on two varieties of steels. The maximum error obtained is 2.08%. Thus, power Voronoi assisted simulation can be considered as an effective tool in modeling microstructure evolution during austenitization.

Power Voronoi assisted modelling and simulation of austenite formation during continuous heating in C35 steel

In the development of materials with desired properties, modelling of transformation kinetics is an important aspect. Ability of power Voronoi assisted simulation in predicting transformation kinetics is examined in this work. Initially, a mathematical model is derived incorporating the geometrical properties of power Voronoi diagram and classical nucleation theory. This is the governing equation for the power Voronoi assisted simulation. Transformed fractions at different temperatures are calculated using this equation. Simulation of microstructure evolution is carried out using power Voronoi diagram by giving nucleation rate and interface velocity as input. From the resultant geometry, it is possible to calculate the transformed fraction. The results are validated and verified using the available experimental results in literature. On comparison, it is evident that, transformed fraction can be predicted with less than 4% error using power Voronoi assisted simulation model with the derived governing equation. Thus, power Voronoi diagram can be used to simulate microstructure evolution during austenitisation with reasonably good accuracy in phase fraction.

Power Voronoi assisted modelling and simulation of austenite formation during continuous heating in C35 steel

In the development of materials with desired properties, modelling of transformation kinetics is an important aspect. Ability of power Voronoi assisted simulation in predicting transformation kinetics is examined in this work. Initially, a mathematical model is derived incorporating the geometrical properties of power Voronoi diagram and classical nucleation theory. This is the governing equation for the power Voronoi assisted simulation. Transformed fractions at different temperatures are calculated using this equation. Simulation of microstructure evolution is carried out using power Voronoi diagram by giving nucleation rate and interface velocity as input. From the resultant geometry, it is possible to calculate the transformed fraction. The results are validated and verified using the available experimental results in literature. On comparison, it is evident that, transformed fraction can be predicted with less than 4% error using power Voronoi assisted simulation model with the derived governing equation. Thus, power Voronoi diagram can be used to simulate microstructure evolution during austenitisation with reasonably good accuracy in phase fraction.

Hyperbolic Wasserstein Distance for Shape Indexing.

Shape space is an active research topic in computer vision and medical imaging fields. The distance defined in a shape space may provide a simple and refined index to represent a unique shape. This work studies the Wasserstein space and proposes a novel framework to compute the Wasserstein distance between general topological surfaces by integrating hyperbolic Ricci flow, hyperbolic harmonic map, and hyperbolic power Voronoi diagram algorithms. The resulting hyperbolic Wasserstein distance can intrinsically measure the similarity between general topological surfaces. Our proposed algorithms are theoretically rigorous and practically efficient. It has the potential to be a powerful tool for 3D shape indexing research. We tested our algorithm with human face classification and Alzheimer's disease (AD) progression tracking studies. Experimental results demonstrated that our work may provide a succinct and effective shape index.

Stable and predictable Voronoi treemaps for software quality monitoring

Context: Voronoi treemaps can be used to effectively visualize software quality attributes of a given software system. Algorithms for computing Voronoi treemaps are non-deterministic making them unsuited for monitoring the development of such attributes over time.Objective: We adapt an existing sweep line algorithm to efficiently compute Voronoi treemaps and we introduce a novel algorithm that adds stability and predictability.Method: We introduce stable and predictable Voronoi treemaps based on additively weighted power Voronoi diagrams. We employ scaled Hilbert curves to place Voronoi sites in the plane, retaining the order in which sites are placed along the curve for easy comparison with revisions of the same software system.Results: Our algorithm achieves a predictable first good approximation of the final location of the sites. We show that our algorithm not only provides more stability, but also that because of better placement it needs fewer iterations to compute its result. As part of our implementation we introduce a visualization to show the difference between two versions of a software system. We also present a small case study in which we use a web based application that implements our work to investigate the usefulness of stability and predictability of visualizations.Conclusion: It is possible to achieve stable and predictable visualizations of software system attributes, while, as a pleasant side effect, decreasing the number of iterations necessary to arrive at the visualization.

Shape Analysis with Hyperbolic Wasserstein Distance.

Shape space is an active research field in computer vision study. The shape distance defined in a shape space may provide a simple and refined index to represent a unique shape. Wasserstein distance defines a Riemannian metric for the Wasserstein space. It intrinsically measures the similarities between shapes and is robust to image noise. Thus it has the potential for the 3D shape indexing and classification research. While the algorithms for computing Wasserstein distance have been extensively studied, most of them only work for genus-0 surfaces. This paper proposes a novel framework to compute Wasserstein distance between general topological surfaces with hyperbolic metric. The computational algorithms are based on Ricci flow, hyperbolic harmonic map, and hyperbolic power Voronoi diagram and the method is general and robust. We apply our method to study human facial expression, longitudinal brain cortical morphometry with normal aging, and cortical shape classification in Alzheimer's disease (AD). Experimental results demonstrate that our method may be used as an effective shape index, which outperforms some other standard shape measures in our AD versus healthy control classification study.

Shape Classification Using Wasserstein Distance for Brain Morphometry Analysis.

Brain morphometry study plays a fundamental role in medical imaging analysis and diagnosis. This work proposes a novel framework for brain cortical surface classification using Wasserstein distance, based on uniformization theory and Riemannian optimal mass transport theory. By Poincare uniformization theorem, all shapes can be conformally deformed to one of the three canonical spaces: the unit sphere, the Euclidean plane or the hyperbolic plane. The uniformization map will distort the surface area elements. The area-distortion factor gives a probability measure on the canonical uniformization space. All the probability measures on a Riemannian manifold form the Wasserstein space. Given any 2 probability measures, there is a unique optimal mass transport map between them, the transportation cost defines the Wasserstein distance between them. Wasserstein distance gives a Riemannian metric for the Wasserstein space. It intrinsically measures the dissimilarities between shapes and thus has the potential for shape classification. To the best of our knowledge, this is the first. work to introduce the optimal mass transport map to general Riemannian manifolds. The method is based on geodesic power Voronoi diagram. Comparing to the conventional methods, our approach solely depends on Riemannian metrics and is invariant under rigid motions and scalings, thus it intrinsically measures shape distance. Experimental results on classifying brain cortical surfaces with different intelligence quotients demonstrated the efficiency and efficacy of our method.

Optimal mass transport for geometric modeling based on variational principles in convex geometry

In geometric modeling, surface parameterization plays an important role for converting triangle meshes to spline surfaces. Parameterization will introduce distortions. Conventional parameterization methods emphasize on angle-preservation, which may induce huge area distortions and cause large spline fitting errors and trigger numerical instabilities.To overcome this difficulty, this work proposes a novel area-preserving parameterization method, which is based on an optimal mass transport theory and convex geometry. Optimal mass transport mapping is measure-preserving and minimizes the transportation cost. According to Brenier's theorem, for quadratic distance transportation costs, the optimal mass transport map is the gradient of a convex function. The graph of the convex function is a convex polyhedron with prescribed normal and areas. The existence and the uniqueness of such a polyhedron have been proved by the Minkowski-Alexandrov theorem in convex geometry. This work gives an explicit method to construct such a polyhedron based on the variational principle, and formulates the solution to the optimal transport map as the unique optimum of a convex energy. In practice, the energy optimization can be carried out using Newton's method, and each iteration constructs a power Voronoi diagram dynamically. We tested the proposal algorithms on 3D surfaces scanned from real life. Experimental results demonstrate the efficiency and efficacy of the proposed variational approach for the optimal transport map.

Optimal partitioning for multi-vehicle systems using quadratic performance criteria

We consider the problem of characterizing a generalized Voronoi diagram that is relevant to a special class of area assignment problems for multi-vehicle systems. It is assumed that the motion of each vehicle is described by a second order mechanical system with time-varying linear or affine dynamics. The proposed generalized Voronoi diagram encodes information regarding the proximity relations between the vehicles and arbitrary target points in the plane. These proximity relations are induced by an anisotropic (generalized) distance function that incorporates the vehicle dynamics. In particular, the generalized distance is taken to be the minimum control effort required for the transition of a vehicle to an arbitrary target point with a small terminal speed at a fixed final time. The space we wish to partition corresponds to the union of all the terminal positions that can be attained by each vehicle using finite control effort. Consequently, the partition space has lower dimension than the state space of each vehicle. We show that, in the general case, the solution to the proposed partitioning problem can be associated with a power Voronoi diagram generated by a set of spheres in a five-dimensional Euclidean space for the computation of which efficient techniques exist in the relevant literature.

Splat representation of parametric surfaces

Point and splat-based representations have become a suitable technique both for modeling and rendering complex 3D shapes. Converting other kinds of models as parametric surfaces to splat-based representations will allow to mix surface and splat-based models and to take advantage of the existing point-based rendering methods. In this work, we present an approach to convert a parametric surface into a splat-based representation. It works in parametric space, performs an adaptive sampling based on the surface curvature and a given error tolerance and uses power Voronoi diagrams. The goal is to approximate the surface with an optimized set of elliptical splats.

画像の処理と符号化 重み付きボロノイ分割の２スキャンアルゴリズム

We developed a two-scan algorithm for discrete Voronoi tessellations of digital images. The computation time of our method is independent of the number of Voronoi sites. In addition to the previous additively, multiplicatively, compoundly weighted Voronoi diagrams and the additively weighted power Voronoi diagram, we proposed a new weighted Voronoi diagram, namely the compoundly weighted power Voronoi diagram. These five weighted Voronoi diagrams were efficiently computed with our two-scan algorithm.