Computation of injective (or inversion-free) maps is a key task in geometry processing, physical simulation, and shape optimization. Despite being a longstanding problem, it remains challenging due to its highly nonconvex and combinatoric nature. We propose computation of
- NSF-PAR ID:
- 10185279
- Date Published:
- Journal Name:
- European Conference on Computer Vision (ECCV2020)
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
more » « less
variational quasi-harmonic maps to obtain smooth inversion-free maps. Our work is built on a key observation about inversion-free maps: A planar map is a diffeomorphism if and only if it is quasi-harmonic and satisfies a special Cauchy boundary condition. We hence equate the inversion-free mapping problem to an optimal control problem derived from our theoretical result, in which we search in the space of parameters that define an elliptic PDE. We show that this problem can be solved by minimizing within a family of functionals. Similarly, our discretized functionals admit exactly injective maps as the minimizers, empirically producing inversion-free discrete maps of triangle meshes. We design efficient numerical procedures for our problem that prioritize robust convergence paths. Experiments show that on challenging examples our methods can achieve up to orders of magnitude improvement over state-of-the-art, in terms of speed or quality. Moreover, we demonstrate how to optimize a generic energy in our framework while restricting to quasi-harmonic maps. -
more » « less
Abstract We consider the tasks of representing, analysing and manipulating maps between shapes. We model maps as densities over the product manifold of the input shapes; these densities can be treated as scalar functions and therefore are manipulable using the language of signal processing on manifolds. Being a manifold itself, the product space endows the set of maps with a geometry of its own, which we exploit to define map operations in the spectral domain; we also derive relationships with other existing representations (soft maps and functional maps). To apply these ideas in practice, we discretize product manifolds and their Laplace–Beltrami operators, and we introduce localized spectral analysis of the product manifold as a novel tool for map processing. Our framework applies to maps defined between and across 2D and 3D shapes without requiring special adjustment, and it can be implemented efficiently with simple operations on sparse matrices.
-
Vehicles are highly likely to lose control unexpectedly when encountering unforeseen hazardous road friction conditions. With automation and connectivity increasingly available to assist drivers, vehicle performance can significantly benefit from a road friction preview map, particularly to identify where and how friction ahead of a vehicle may be suddenly decreasing. Although many techniques enable the vehicle to measure the local friction as driving upon a surface, these encounters limit the ability of a vehicle to slow down before a low-friction surface is already encountered. Using the connectivity of connected and autonomous vehicles (CAVs), a global road friction map can be created by aggregating information from vehicles. A challenge in the creation of these global friction maps is the very large quantity of data involved, and that the measurements populating the map are generated by vehicle trajectories that do not uniformly cover the grid. This paper presents a road friction map generation strategy that aggregates the measured road-tire friction coefficients along the individual trajectories of CAVs into a road surface grid. And through clustering the friction grids further, an insight of this work is that the friction map can be represented compactly by rectangular boxes defined by a pair of corner coordinates in space and a friction value within the box. To demonstrate the method, a simulation is presented that integrates traffic simulations, vehicle dynamics and on-vehicle friction estimators, and a highway road surface where friction is changing in space, particularly over a bridge segment. The experimental results indicate that the road friction distribution can be measured effectively by collecting and aggregating the friction data from CAVs. By defining a cloud-based data sharing method for the networks of CAVs, this road friction mapping strategy provides great potential for improving CAVs' control performance and stability via database-mediated feedback systems.more » « less
-
Abstract Line intensity mapping (LIM) is a rapidly emerging technique for constraining cosmology and galaxy formation using multi-frequency, low angular resolution maps.Many LIM applications crucially rely on cross-correlations of two line intensity maps, or of intensity maps with galaxy surveys or galaxy/CMB lensing.We present a consistent halo model to predict all these cross-correlations and enable joint analyses, in 3D redshift-space and for 2D projected maps.We extend the conditional luminosity function formalism to the multi-line case, to consistently account for correlated scatter between multiple galaxy line luminosities.This allows us to model the scale-dependent decorrelation between two line intensity maps,a key input for foreground rejection and for approaches that estimate auto-spectra from cross-spectra.This also enables LIM cross-correlations to reveal astrophysical properties of the interstellar medium inacessible with LIM auto-spectra.We expose the different sources of luminosity scatter or “line noise” in LIM, and clarify their effects on the 1-halo and galaxy shot noise terms.In particular, we show that the effective number density of halos can in some cases exceed that of galaxies, counterintuitively.Using observational and simulation input, we implement this halo model for the Hα, [Oiii], Lyman-α, CO and [Cii] lines.We encourage observers and simulators to measure galaxy luminosity correlation coefficients for pairs of lines whenever possible.Our code is publicly available at https://github.com/EmmanuelSchaan/HaloGen/tree/LIM .In a companion paper, we use this halo model formalism and codeto highlight the degeneracies between cosmology and astrophysics in LIM, and to compare the LIM observables to galaxy detection for a number of surveys.more » « less
-
more » « less
Abstract We revisit the inverse source problem in a two dimensional absorbing and scattering medium and present a direct reconstruction method, which does not require iterative solvability of the forward problem, using measurements of the radiating flux at the boundary. The attenuation and scattering coefficients are known and the unknown source is isotropic. The approach is based on the Cauchy problem for a Beltrami-like equation for the sequence valued maps, and extends the original ideas of Bukhgeim from the non-scattering to the scattering media. We demonstrate the feasibility of the method in a numerical experiment in which the scattering is modeled by the two dimensional Henyey–Greenstein kernel with parameters meaningful in optical tomography.
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1007/978-3-030-58577-8_22
