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"content": "Voronoi diagrams are powerful for understanding spatial properties. However, few reports have been made for moving generators despite their important applications. We present a topology-oriented event-increment (TOI-E) algorithm for constructing a Voronoi diagram of moving circular disks in the plane over the time horizon <inline-formula><tex-math notation=\"LaTeX\">$[0, t^{\\infty })$</tex-math><alternatives><mml:math><mml:mrow><mml:mo>[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:msup><mml:mi>t</mml:mi><mml:mi>∞</mml:mi></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\"kim-ieq1-2959321.gif\"/></alternatives></inline-formula>. The proposed TOI-E algorithm computes the event history of the Voronoi diagram over the entire time horizon in <inline-formula><tex-math notation=\"LaTeX\">$O(k_F \\log n + k_C n \\log n)$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mi>F</mml:mi></mml:msub><mml:mo form=\"prefix\">log</mml:mo><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mi>C</mml:mi></mml:msub><mml:mi>n</mml:mi><mml:mo form=\"prefix\">log</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\"kim-ieq2-2959321.gif\"/></alternatives></inline-formula> time with <inline-formula><tex-math notation=\"LaTeX\">$O(n \\log n)$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo form=\"prefix\">log</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\"kim-ieq3-2959321.gif\"/></alternatives></inline-formula> preprocessing time and <inline-formula><tex-math notation=\"LaTeX\">$O(n + k_F + k_C)$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mi>F</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mi>C</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\"kim-ieq4-2959321.gif\"/></alternatives></inline-formula> memory for <inline-formula><tex-math notation=\"LaTeX\">$n$</tex-math><alternatives><mml:math><mml:mi>n</mml:mi></mml:math><inline-graphic xlink:href=\"kim-ieq5-2959321.gif\"/></alternatives></inline-formula> disk generators, <inline-formula><tex-math notation=\"LaTeX\">$k_F$</tex-math><alternatives><mml:math><mml:msub><mml:mi>k</mml:mi><mml:mi>F</mml:mi></mml:msub></mml:math><inline-graphic xlink:href=\"kim-ieq6-2959321.gif\"/></alternatives></inline-formula> edge flips, and <inline-formula><tex-math notation=\"LaTeX\">$k_C$</tex-math><alternatives><mml:math><mml:msub><mml:mi>k</mml:mi><mml:mi>C</mml:mi></mml:msub></mml:math><inline-graphic xlink:href=\"kim-ieq7-2959321.gif\"/></alternatives></inline-formula> disk collisions during the time horizon. Given an event history, the Voronoi diagram of an arbitrary moment <inline-formula><tex-math notation=\"LaTeX\">$t^{\\ast}<t^{\\infty }$</tex-math><alternatives><mml:math><mml:mrow><mml:msup><mml:mi>t</mml:mi><mml:mo>*</mml:mo></mml:msup><mml:mo><</mml:mo><mml:msup><mml:mi>t</mml:mi><mml:mi>∞</mml:mi></mml:msup></mml:mrow></mml:math><inline-graphic xlink:href=\"kim-ieq8-2959321.gif\"/></alternatives></inline-formula> can be constructed in <inline-formula><tex-math notation=\"LaTeX\">$O(k^{\\ast} + n)$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:msup><mml:mi>k</mml:mi><mml:mo>*</mml:mo></mml:msup><mml:mo>+</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\"kim-ieq9-2959321.gif\"/></alternatives></inline-formula> time where <inline-formula><tex-math notation=\"LaTeX\">$k^{\\ast}$</tex-math><alternatives><mml:math><mml:msup><mml:mi>k</mml:mi><mml:mo>*</mml:mo></mml:msup></mml:math><inline-graphic xlink:href=\"kim-ieq10-2959321.gif\"/></alternatives></inline-formula> represents the number of events in <inline-formula><tex-math notation=\"LaTeX\">$[0, t^{\\ast})$</tex-math><alternatives><mml:math><mml:mrow><mml:mo>[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:msup><mml:mi>t</mml:mi><mml:mo>*</mml:mo></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href=\"kim-ieq11-2959321.gif\"/></alternatives></inline-formula>. An example of the collision avoidance problem among moving disks is given by predicting future conjunctions among the disks using the proposed algorithm. Dynamic Voronoi diagrams will be very useful as a platform for the planning and management of the traffics of unmanned vehicles such as cars on street, vessels on surface, drones and airplanes in air, and satellites in geospace.",
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"abstract": "In geometry processing, symmetry is a universal type of high-level structural information of 3D models and benefits many geometry processing tasks including shape segmentation, alignment, matching, and completion. Thus it is an important problem to analyze various symmetry forms of 3D shapes. Planar reflective symmetry is the most fundamental one. Traditional methods based on spatial sampling can be time-consuming and may not be able to identify all the symmetry planes. In this article, we present a novel learning framework to automatically discover global planar reflective symmetry of a 3D shape. Our framework trains an unsupervised 3D convolutional neural network to extract global model features and then outputs possible global symmetry parameters, where input shapes are represented using voxels. We introduce a dedicated symmetry distance loss along with a regularization loss to avoid generating duplicated symmetry planes. Our network can also identify generalized cylinders by predicting their rotation axes. We further provide a method to remove invalid and duplicated planes and axes. We demonstrate that our method is able to produce reliable and accurate results. Our neural network based method is hundreds of times faster than the state-of-the-art methods, which are based on sampling. Our method is also robust even with noisy or incomplete input surfaces.",
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"abstract": "We present a neural optimization model trained with reinforcement learning to solve the coordinate ordering problem for sets of star glyphs. Given a set of star glyphs associated to multiple class labels, we propose to use shape context descriptors to measure the perceptual distance between pairs of glyphs, and use the derived silhouette coefficient to measure the perception of class separability within the entire set. To find the optimal coordinate order for the given set, we train a neural network using reinforcement learning to reward orderings with high silhouette coefficients. The network consists of an encoder and a decoder with an attention mechanism. The encoder employs a recurrent neural network (RNN) to encode input shape and class information, while the decoder together with the attention mechanism employs another RNN to output a sequence with the new coordinate order. In addition, we introduce a neural network to efficiently estimate the similarity between shape context descriptors, which allows to speed up the computation of silhouette coefficients and thus the training of the axis ordering network. Two user studies demonstrate that the orders provided by our method are preferred by users for perceiving class separation. We tested our model on different settings to show its robustness and generalization abilities and demonstrate that it allows to order input sets with unseen data size, data dimension, or number of classes. We also demonstrate that our model can be adapted to coordinate ordering of other types of plots such as RadViz by replacing the proposed shape-aware silhouette coefficient with the corresponding quality metric to guide network training.",
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"abstract": "Turbulent vortices in smoke flows are crucial for a visually interesting appearance. Unfortunately, it is challenging to efficiently simulate these appealing effects in the framework of vortex filament methods. The vortex filaments in grids scheme allows to efficiently generate turbulent smoke with macroscopic vortical structures, but suffers from the projection-related dissipation, and thus the small-scale vortical structures under grid resolution are hard to capture. In addition, this scheme cannot be applied in wall-bounded turbulent smoke simulation, which requires efficiently handling smoke-obstacle interaction and creating vorticity at the obstacle boundary. To tackle above issues, we propose an effective filament-mesh particle-particle (FMPP) method for fast wall-bounded turbulent smoke simulation with ample details. The Filament-Mesh component approximates the smooth long-range interactions by splatting vortex filaments on grid, solving the Poisson problem with a fast solver, and then interpolating back to smoke particles. The Particle-Particle component introduces smoothed particle hydrodynamics (SPH) turbulence model for particles in the same grid, where interactions between particles cannot be properly captured under grid resolution. Then, we sample the surface of obstacles with boundary particles, allowing the interaction between smoke and obstacle being treated as pressure forces in SPH. Besides, the vortex formation region is defined at the back of obstacles, providing smoke particles flowing by the separation particles with a vorticity force to simulate the subsequent vortex shedding phenomenon. The proposed approach can synthesize the lost small-scale vortical structures and also achieve the smoke-obstacle interaction with vortex shedding at obstacle boundaries in a lightweight manner. The experimental results demonstrate that our FMPP method can achieve more appealing visual effects than vortex filaments in grids scheme by efficiently simulating more vivid thin turbulent features.",
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