Seamless Parametrization with Arbitrarily Prescribed Cones

10/04/2018
by   Marcel Campen, et al.
0

Seamless global parametrization of surfaces is a key operation in geometry processing, e.g. for high-quality quad mesh generation. A common approach is to prescribe the parametric domain structure, in particular the locations of parametrization singularities (cones), and solve a non-convex optimization problem minimizing a distortion measure, with local injectivity imposed through either constraints or barrier terms. In both cases, an initial valid parametrization is essential to serve as feasible starting point for obtaining an optimized solution. While convexified versions of the constraints eliminate this initialization requirement, they narrow the range of solutions, causing some problem instances that actually do have a solution to become infeasible. We demonstrate that for arbitrary given sets of topologically admissible parametric cones with prescribed curvature, a global seamless parametrization always exists (with the exception of one well-known case). Importantly, our proof is constructive and directly leads to a general algorithm for computing such parametrizations. Most distinctively, this algorithm is bootstrapped with a convex optimization problem (solving for a conformal map), in tandem with a simple linear equation system (determining a seamless modification of this map). This initial map can then serve as valid starting point and be optimized with respect to application specific distortion measures using existing injectivity preserving methods.

READ FULL TEXT

page 1

page 11

page 12

research
11/20/2017

Solution of network localization problem with noisy distances and its convergence

The network localization problem with convex and non-convex distance con...
research
06/10/2022

Inverting Incomplete Fourier Transforms by a Sparse Regularization Model and Applications in Seismic Wavefield Modeling

We propose a sparse regularization model for inversion of incomplete Fou...
research
12/19/2019

Learning Convex Optimization Control Policies

Many control policies used in various applications determine the input o...
research
02/13/2016

Convex Optimization for Linear Query Processing under Approximate Differential Privacy

Differential privacy enables organizations to collect accurate aggregate...
research
11/16/2020

Cinematic-L1 Video Stabilization with a Log-Homography Model

We present a method for stabilizing handheld video that simulates the ca...
research
05/18/2021

Approximate solutions of convex semi-infinite optimization problems in finitely many iterations

We develop two adaptive discretization algorithms for convex semi-infini...
research
11/18/2022

An optimization-based registration approach to geometry reduction

We develop and assess an optimization-based approach to parametric geome...

Please sign up or login with your details

Forgot password? Click here to reset