Adequate Inner Bound for Geometric Modeling with Compact Field Function

International audienceRecent advances in implicit surface modeling now provide highly controllable blending effects. These effects rely on the field functions of $\mathbb{R}^3 \rightarrow \mathbb{R}$ in which the implicit surfaces are defined. In these fields, there is an outside part in which blending is defined and an inside part. The implicit surface is the interface between these two parts. As recent operators often focus on blending, most efforts have been made on the outer part of field functions and little attention has been paid on the inner part. Yet, the inner fields are important as soon as difference and intersection operators are used. This makes its quality as crucial as the quality of the outside. In this paper, we analyze these shortcomings, and deduce new constraints on field functions such that differences and intersections can be seamlessly applied without introducing discontinuities or field distortions. In particular, we show how to adapt state of the art gradient-based union and blending operators to our new constraints. Our approach enables a precise control of the shape of both the inner or outer field boundaries. We also introduce a new set of asymmetric operators tailored for the modeling of fine details while preserving the integrity of the resulting fields

An inexact Newton-Krylov algorithm for constrained diffeomorphic image registration

We propose numerical algorithms for solving large deformation diffeomorphic image registration problems. We formulate the nonrigid image registration problem as a problem of optimal control. This leads to an infinite-dimensional partial differential equation (PDE) constrained optimization problem. The PDE constraint consists, in its simplest form, of a hyperbolic transport equation for the evolution of the image intensity. The control variable is the velocity field. Tikhonov regularization on the control ensures well-posedness. We consider standard smoothness regularization based on $H^1$- or $H^2$-seminorms. We augment this regularization scheme with a constraint on the divergence of the velocity field rendering the deformation incompressible and thus ensuring that the determinant of the deformation gradient is equal to one, up to the numerical error. We use a Fourier pseudospectral discretization in space and a Chebyshev pseudospectral discretization in time. We use a preconditioned, globalized, matrix-free, inexact Newton-Krylov method for numerical optimization. A parameter continuation is designed to estimate an optimal regularization parameter. Regularity is ensured by controlling the geometric properties of the deformation field. Overall, we arrive at a black-box solver. We study spectral properties of the Hessian, grid convergence, numerical accuracy, computational efficiency, and deformation regularity of our scheme. We compare the designed Newton-Krylov methods with a globalized preconditioned gradient descent. We study the influence of a varying number of unknowns in time. The reported results demonstrate excellent numerical accuracy, guaranteed local deformation regularity, and computational efficiency with an optional control on local mass conservation. The Newton-Krylov methods clearly outperform the Picard method if high accuracy of the inversion is required.Comment: 32 pages; 10 figures; 9 table

Colliding Interfaces in Old and New Diffuse-interface Approximations of Willmore-flow

This paper is concerned with diffuse-interface approximations of the Willmore flow. We first present numerical results of standard diffuse-interface models for colliding one dimensional interfaces. In such a scenario evolutions towards interfaces with corners can occur that do not necessarily describe the adequate sharp-interface dynamics. We therefore propose and investigate alternative diffuse-interface approximations that lead to a different and more regular behavior if interfaces collide. These dynamics are derived from approximate energies that converge to the $L^1$-lower-semicontinuous envelope of the Willmore energy, which is in general not true for the more standard Willmore approximation

Computing the Casimir energy using the point-matching method

We use a point-matching approach to numerically compute the Casimir interaction energy for a two perfect-conductor waveguide of arbitrary section. We present the method and describe the procedure used to obtain the numerical results. At first, our technique is tested for geometries with known solutions, such as concentric and eccentric cylinders. Then, we apply the point-matching technique to compute the Casimir interaction energy for new geometries such as concentric corrugated cylinders and cylinders inside conductors with focal lines.Comment: 11 pages, 18 figure