Fast generation of 3D deformable moving surfaces

Dynamic surface modeling is an important subject of geometric modeling due to their extensive applications in engineering design, entertainment and medical visualization. Many deformable objects in the real world are dynamic objects as their shapes change over time. Traditional geometric modeling methods are mainly concerned with static problems, therefore unsuitable for the representation of dynamic objects. Apart from the definition of a dynamic modeling problem, another key issue is how to solve the problem. Because of the complexity of the representations, currently the finite element method or finite difference method is usually used. Their major shortcoming is the excessive computational cost, hence not ideal for applications requiring real-time performance. We propose a representation of dynamic surface modeling with a set of fourth order dynamic partial differential equations (PDEs). To solve these dynamic PDEs accurately and efficiently, we also develop an effective resolution method. This method is further extended to achieve local deformation and produce n-sided patches. It is demonstrated that this new method is almost as fast and accurate as the analytical closed form resolution method and much more efficient and accurate than the numerical methods

Compact convex sets of the plane and probability theory

The Gauss-Minkowski correspondence in $\mathbb{R}^2$ states the existence of a homeomorphism between the probability measures $\mu$ on $[0,2\pi]$ such that $\int_0^{2\pi} e^{ix}d\mu(x)=0$ and the compact convex sets (CCS) of the plane with perimeter~1. In this article, we bring out explicit formulas relating the border of a CCS to its probability measure. As a consequence, we show that some natural operations on CCS -- for example, the Minkowski sum -- have natural translations in terms of probability measure operations, and reciprocally, the convolution of measures translates into a new notion of convolution of CCS. Additionally, we give a proof that a polygonal curve associated with a sample of $n$ random variables (satisfying $\int_0^{2\pi} e^{ix}d\mu(x)=0$) converges to a CCS associated with $\mu$ at speed $\sqrt{n}$, a result much similar to the convergence of the empirical process in statistics. Finally, we employ this correspondence to present models of smooth random CCS and simulations

The White Dwarf -- White Dwarf galactic background in the LISA data

LISA (Laser Interferometer Space Antenna) is a proposed space mission, which will use coherent laser beams exchanged between three remote spacecraft to detect and study low-frequency cosmic gravitational radiation. In the low-part of its frequency band, the LISA strain sensitivity will be dominated by the incoherent superposition of hundreds of millions of gravitational wave signals radiated by inspiraling white-dwarf binaries present in our own galaxy. In order to estimate the magnitude of the LISA response to this background, we have simulated a synthesized population that recently appeared in the literature. We find the amplitude of the galactic white-dwarf binary background in the LISA data to be modulated in time, reaching a minimum equal to about twice that of the LISA noise for a period of about two months around the time when the Sun-LISA direction is roughly oriented towards the Autumn equinox. Since the galactic white-dwarfs background will be observed by LISA not as a stationary but rather as a cyclostationary random process with a period of one year, we summarize the theory of cyclostationary random processes, present the corresponding generalized spectral method needed to characterize such process, and make a comparison between our analytic results and those obtained by applying our method to the simulated data. We find that, by measuring the generalized spectral components of the white-dwarf background, LISA will be able to infer properties of the distribution of the white-dwarfs binary systems present in our Galaxy.Comment: 36 pages, 15 figure

PDE Face: A Novel 3D Face Model

YesWe introduce a novel approach to face models, which exploits the use of Partial Differential Equations (PDE) to generate the 3D face. This addresses some common problems of existing face models. The PDE face benefits from seamless merging of surface patches by using only a relatively small number of parameters based on boundary curves. The PDE face also provides users with a great degree of freedom to individualise the 3D face by adjusting a set of facial boundary curves. Furthermore, we introduce a uv-mesh texture mapping method. By associating the texels of the texture map with the vertices of the uv mesh in the PDE face, the new texture mapping method eliminates the 3D-to-2D association routine in texture mapping. Any specific PDE face can be textured without the need for the facial expression in the texture map to match exactly that of the 3D face model

Lattice topology and spontaneous parametric down-conversion in quadratic nonlinear waveguide arrays

We analyze spontaneous parametric down-conversion in various experimentally feasible 1D quadratic nonlinear waveguide arrays, with emphasis on the relationship between the lattice's topological invariants and the biphoton correlations. Nontrivial topology results in a nontrivial "winding" of the array's Bloch waves, which introduces additional selection rules for the generation of biphotons. These selection rules are in addition to, and independent of existing control using the pump beam's spatial profile and phase matching conditions. In finite lattices, nontrivial topology produces single photon edge modes, resulting in "hybrid" biphoton edge modes, with one photon localized at the edge and the other propagating into the bulk. When the single photon band gap is sufficiently large, these hybrid biphoton modes reside in a band gap of the bulk biphoton Bloch wave spectrum. Numerical simulations support our analytical results.Comment: 11 pages, 12 figure