Abstract-Blue noise point sampling is one of the core algorithms in computer graphics. In this paper we present a new and versatile variational framework for generating point distributions with high-quality blue noise characteristics while precisely adapting to given density functions. Different from previous approaches based on discrete settings of capacity-constrained Voronoi tessellation, we cast the blue noise sampling generation as a variational problem with continuous settings. Based on an accurate evaluation of the gradient of an energy function, an efficient optimization is developed which delivers significantly faster performance than the previous optimization-based methods. Our framework can easily be extended to generating blue noise point samples on manifold surfaces and for multi-class sampling. The optimization formulation also allows us to naturally deal with dynamic domains, such as deformable surfaces, and to yield blue noise samplings with temporal coherence. We present experimental results to validate the efficacy of our variational framework. Finally, we show a variety of applications of the proposed methods, including non-photorealistic image stippling, color stippling, and blue noise sampling on deformable surfaces.

Abstract-Collision detection and avoidance are important in robotics. Compared with commonly used circular disks, elliptic disks provide a more compact shape representation for robots or other vehicles confined to move in the plane. Furthermore, elliptic disks allow a simpler analytic representation than rectangular boxes, which makes it easier to perform continuous collision detection (CCD). We shall present a fast and accurate method for CCD between two moving elliptic disks, which avoids any need to sample the time domain of the motion, thus avoiding the possibility of missing collisions between time samples. Based on some new algebraic conditions on the separation of two ellipses, we reduce collision detection for two moving ellipses to the problem of detecting real roots of a univariate equation, which is the discriminant of the characteristic polynomial of the two ellipses. Several techniques are investigated for robust and accurate processing of this univariate equation for two classes of commonly used motions: planar cycloidal motions and planar rational motions. Experimental results demonstrate the efficiency, accuracy, and robustness of our method.

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