Molecular Shape Analysis based uponthe Morse-Smale Complexand the Connolly Function

Docking is the process by which two or several molecules form a complex. Docking involves the geometry of the molecular surfaces, as well as chemical and energetical considerations. In the mid-eighties, Connolly proposed a docking algorithm matching surface {\em knobs} with surface {\em depressions}. Knobs and depressions refer to the extrema of the {\em Connolly} function, which is defined as follows. Given a surface \calM bounding a three-dimensional domain $X$, and a sphere $S$ centered at a point $p$ of \calM, the Connolly function is equal to the solid angle of the portion of $S$ containing within $X$. We recast the notions of knob and depression of the Connolly function in the framework of Morse theory for functions defined over two-dimensional manifolds. First, we study the critical points of the Connolly function for smooth surfaces. Second, we provide an efficient algorithm for computing the Connolly function over a triangulated surface. Third, we introduce a Morse-Smale decomposition based on Forman's discrete Morse theory, and provide an $O(n\log n)$ algorithm to construct it. This decomposition induces a partition of the surface into regions of homogeneous flow, and provides an elegant way to relate local quantities to global ones ---from critical points to Euler's characteristic of the surface. Fourth, we apply this Morse-Smale decomposition to the discrete gradient vector field induced by Connolly's function, and present experimental results for several mesh models

BodyNet: Volumetric Inference of 3D Human Body Shapes

Human shape estimation is an important task for video editing, animation and fashion industry. Predicting 3D human body shape from natural images, however, is highly challenging due to factors such as variation in human bodies, clothing and viewpoint. Prior methods addressing this problem typically attempt to fit parametric body models with certain priors on pose and shape. In this work we argue for an alternative representation and propose BodyNet, a neural network for direct inference of volumetric body shape from a single image. BodyNet is an end-to-end trainable network that benefits from (i) a volumetric 3D loss, (ii) a multi-view re-projection loss, and (iii) intermediate supervision of 2D pose, 2D body part segmentation, and 3D pose. Each of them results in performance improvement as demonstrated by our experiments. To evaluate the method, we fit the SMPL model to our network output and show state-of-the-art results on the SURREAL and Unite the People datasets, outperforming recent approaches. Besides achieving state-of-the-art performance, our method also enables volumetric body-part segmentation.Comment: Appears in: European Conference on Computer Vision 2018 (ECCV 2018). 27 page

Theory of the anomalous Hall effect from the Kubo formula and the Dirac equation

A model to treat the anomalous Hall effect is developed. Based on the Kubo formalism and on the Dirac equation, this model allows the simultaneous calculation of the skew-scattering and side-jump contributions to the anomalous Hall conductivity. The continuity and the consistency with the weak-relativistic limit described by the Pauli Hamiltonian is shown. For both approaches, Dirac and Pauli, the Feynman diagrams, which lead to the skew-scattering and the side-jump contributions, are underlined. In order to illustrate this method, we apply it to a particular case: a ferromagnetic bulk compound in the limit of weak-scattering and free-electrons approximation. Explicit expressions for the anomalous Hall conductivity for both skew-scattering and side-jump mechanisms are obtained. Within this model, the recently predicted ''spin Hall effect'' appears naturally

Regularized Implicit Surface Reconstruction from Points and Normals ∗

We consider the problem of surface reconstruction of a geometric object from a finite set of sample points with normals. Our contribution is to present a new scheme for implicit surface reconstruction. Similarly to the multilevel partition of unity (MPU) method we hierarchically divide the domain obtaining local approximation for the object on each part, and then patch all together obtaining a global description of the object. Our new scheme uses ridge regression and weighted gradient one fitting techniques to get better stability on local approximations. The method behaves reasonably on sparse set of points and data with holes as those which comes from 3D scanning of real objects