Minimal Surface Problem Research Articles

An algorithm for prescribed mean curvature using isogeometric methods

We present a Newton type algorithm to find parametric surfaces of prescribed mean curvature with a fixed given boundary. In particular, it applies to the problem of minimal surfaces. The algorithm relies on some global regularity of the spaces where it is posed, which is naturally fitted for discretization with isogeometric type of spaces. We introduce a discretization of the continuous algorithm and present a simple implementation using the recently released isogeometric software library igatools. Finally, we show several numerical experiments which highlight the convergence properties of the scheme.

On a Problem in the Calculus of Variations

The paper has a scientific-methodical nature. The classical soap film shape (minimal surface) problem is considered, the film being stretched between two parallel coaxial rings. An analytic approach based on the relations to the Sturm–Liouville problem is proposed. An interpretation of the energy terms of the classical Goldschmidt condition is discussed. The appearance of a soliton potential when analyzing the second variation is noticed. Bibliography: 3 titles.

A relaxation method for the energy and morphology of grain boundaries and interfaces

The energy density of crystal interfaces exhibits a characteristic “cusp” structure that renders it non-convex. Furthermore, crystal interfaces are often observed to be faceted, i.e., to be composed of flat facets in alternating directions. In this work, we forge a connection between these two observations by positing that the faceted morphology of crystal interfaces results from energy minimization. Specifically, we posit that the lack of convexity of the interfacial energy density drives the development of finely faceted microstructures and accounts for their geometry and morphology. We formulate the problem as a generalized minimal surface problem couched in a geometric measure-theoretical framework. We then show that the effective, or relaxed, interfacial energy density, with all possible interfacial morphologies accounted for, corresponds to the convexification of the bare or unrelaxed interfacial energy density, and that the requisite convexification can be attained by means of a faceting construction. We validate the approach by means of comparisons with experiment and atomistic simulations including symmetric and asymmetric tilt boundaries in face-centered cubic (FCC) and body-centered cubic (BCC) crystals. By comparison with simulated and experimental data, we show that this simple model of interfacial energy combined with a general microstructure construction based on convexification is able to replicate complex interfacial morphologies, including thermally induced morphological transitions.

Existence of solutions for the anti-plane stress for a new class of “strain-limiting” elastic bodies

The main purpose of this study is to establish the existence of a weak solution to the anti-plane stress problem on V-notch domains for a class of recently proposed new models that could describe elastic materials in which the stress can increase unboundedly while the strain yet remains small. We shall also investigate the qualitative properties of the solution that is established. Although the equations governing the deformation that are being considered share certain similarities with the minimal surface problem, the boundary conditions and the presence of an additional model parameter that appears in the equation and its specific range makes the problem, as well as the result, different from those associated with the minimal surface problem.

Image segmentation with arbitrary noise models by solving minimal surface problems

Segmentation is one of the fundamental tasks in computer vision applications. For natural images there exist a vast amount of sophisticated segmentation methods in the literature. However, these standard methods tend to fail in the presence of non-Gaussian image noise, e.g., in biomedical imaging or astronomy. In this paper we propose an adequate variational segmentation model for segmentation of images perturbed by arbitrary noise models without adding a-priori assumptions about the unknown physical noise model. For this, we discuss the prominent minimal surface problem and two different numerical minimization schemes to solve it. The first approach efficiently computes the set of all possible minimal surface solutions for the given data via convex optimization and reduces the segmentation problem to the estimation of a proper threshold. The second approach is based on level set methods and is especially suitable for the separation of inhomogeneous image regions. The advantage of this approach is both its simpleness and robustness: the noise in the data does not have to be modeled explicitly since the image intensities are separated using histogram-based thresholding techniques. The proposed model can be interpreted as a generalization of many traditional segmentation methods which implicitly assume a perturbation by additive Gaussian noise. The superiority of this approach over standard methods such as the popular Chan–Vese formulation is demonstrated on synthetic images as well as real application data from biomedical imaging.

The Björling problem for minimal surfaces in a Lorentzian three-dimensional Lie group

In this paper, we will show the existence and uniqueness of the solution of the Bjorling problem for minimal surfaces in a three-dimensional Lorentzian Lie group.

Wilson loops and Riemann theta functions II

In this paper we extend and simplify previous results regarding the computation of Euclidean Wilson loops in the context of the AdS/CFT correspondence, or, equivalently, the problem of finding minimal area surfaces in hyperbolic space (Euclidean AdS3). If the Wilson loop is given by a boundary curve X(s) we define, using the integrable properties of the system, a family of curves X(lambda,s) depending on a complex parameter lambda known as the spectral parameter. This family has remarkable properties. As a function of lambda, X(lambda,s) has cuts and therefore is appropriately defined on a hyperelliptic Riemann surface, namely it determines the spectral curve of the problem. Moreover, X(lambda,s) has an essential singularity at the origin lambda=0. The coefficients of the expansion of X(lambda,s) around lambda=0, when appropriately integrated along the curve give the area of the corresponding minimal area surface. Furthermore we show that the same construction allows the computation of certain surfaces with one or more boundaries corresponding to Wilson loop correlators. We extend the area formula for that case and give some concrete examples. As the main example we consider a surface ending on two concentric circles and show how the boundary circles can be deformed by introducing extra cuts in the spectral curve.

Fast Geodesic Active Fields for Image Registration Based on Splitting and Augmented Lagrangian Approaches

In this paper, we present an efficient numerical scheme for the recently introduced geodesic active fields (GAF) framework for geometric image registration. This framework considers the registration task as a weighted minimal surface problem. Hence, the data-term and the regularization-term are combined through multiplication in a single, parametrization invariant and geometric cost functional. The multiplicative coupling provides an intrinsic, spatially varying and data-dependent tuning of the regularization strength, and the parametrization invariance allows working with images of nonflat geometry, generally defined on any smoothly parametrizable manifold. The resulting energy-minimizing flow, however, has poor numerical properties. Here, we provide an efficient numerical scheme that uses a splitting approach; data and regularity terms are optimized over two distinct deformation fields that are constrained to be equal via an augmented Lagrangian approach. Our approach is more flexible than standard Gaussian regularization, since one can interpolate freely between isotropic Gaussian and anisotropic TV-like smoothing. In this paper, we compare the geodesic active fields method with the popular Demons method and three more recent state-of-the-art algorithms: NL-optical flow, MRF image registration, and landmark-enhanced large displacement optical flow. Thus, we can show the advantages of the proposed FastGAF method. It compares favorably against Demons, both in terms of registration speed and quality. Over the range of example applications, it also consistently produces results not far from more dedicated state-of-the-art methods, illustrating the flexibility of the proposed framework.

Relaxations for Minimizing Metric Distortion and Elastic Energies for 3D Shape Matching

We present two methods for non-rigid shape matching. Both methods formulate shape matching as an energy minimization problem, where the energy measures distortion of the metric defined on the shapes in one case, or directly describes the physical deformation relating the two shapes in the other case. The first method considers a parametrized relaxation of the widely adopted quadratic assignment problem (QAP) formulation for minimum distortion correspondence between deformable shapes. In order to control the accuracy/sparsity trade-o a weighting parameter is introduced to combine two existing relaxations, namely spectral and game-theoretic. This leads to an approach for deformable shape matching with controllable sparsity. The second method focuses on computing a geometrically consistent and spatially dense matching between two 3D shapes. Rather than mapping points to points it matches infinitesimal surface patches while preserving the geometric structures. In this spirit, matchings are considered as dieomorphisms between the objects’ surfaces which are by definition geometrically consistent. Based on the observation that such dieomorphisms can be represented as closed and continuous surfaces in the product space of the two shapes, this leads to a minimal surface problem in this product space. The proposed discrete formulation describes the search space with linear constraints. Computationally, the approach results in a binary linear program whose relaxed version can be solved eciently in a globally optimal manner.

Shape optimisation in the design of thin-walled shells as components of aerospace structures

AbstractOne of the most resent efforts in aircraft design is the replacement of aluminium structures by carbon fibre reinforced polymer composites. Due to lower material and manufacturing costs, doubly curved shapes covering big areas are preferred over simpler surfaces which integrate stiffening profiles. In this context, CAD parameterisation of surfaces allows design solutions by means of classical shape optimisation. Related geometrical parameters are manipulated towards optimal design, generating innovative geometries and detailing.The presented structure is optimised by reducing the overall weight. The final optimum is guided using stability and strength restrictions in order to assure the safety of the component. Geometrical considerations are also included due to operational reasons. A hierarchical design procedure is developed which results in a work flow from preliminary ‘parameter-free’ form finding motivated by solving the minimal surface problem. The geometrical model for optimisation is recovered by generating B-Spline surface patches to preserve continuity requirements over large regions. The number of geometrical coefficients are defined by the accuracy in surface generation and the required freedom in surface control. The hierarchical approach reduces the possibilities of ending with an unsatisfactory optimum when several local minima characterise the non-linear problem, as it is usually the case in shape optimal design. A geometrical non-linear analysis is used to verify the performance of the optimum.

Remarks on holographic Wilson loops and the Schwinger effect

We extend Douglas' solution of the problem of finding minimal surfaces to anti-de Sitter space. The case of a circle as a boundary contour is elaborated. We discuss applications to $\mathcal{N}=4$ super Yang-Mills: a circular Wilson loop and the Schwinger process, where we calculate the $1/\sqrt{\ensuremath{\lambda}}$ correction to the critical value of constant electric field.

Doubly Connected Minimal Surfaces and Extremal Harmonic Mappings

The concept of a conformal deformation has two natural extensions: quasiconformal and harmonic mappings. Both classes do not preserve the conformal type of the domain, however they cannot change it in an arbitrary way. Doubly connected domains are where one first observes nontrivial conformal invariants. Herbert Groetzsch and Johannes C. C. Nitsche addressed this issue for quasiconformal and harmonic mappings, respectively. Combining these concepts we obtain sharp estimates for quasiconformal harmonic mappings between doubly connected domains. We then apply our results to the Cauchy problem for minimal surfaces, also known as the Bjorling problem. Specifically, we obtain a sharp estimate of the modulus of a doubly connected minimal surface that evolves from its inner boundary with a given initial slope.

Geodesic Active Fields—A Geometric Framework for Image Registration

In this paper we present a novel geometric framework called geodesic active fields for general image registration. In image registration, one looks for the underlying deformation field that best maps one image onto another. This is a classic ill-posed inverse problem, which is usually solved by adding a regularization term. Here, we propose a multiplicative coupling between the registration term and the regularization term, which turns out to be equivalent to embed the deformation field in a weighted minimal surface problem. Then, the deformation field is driven by a minimization flow toward a harmonic map corresponding to the solution of the registration problem. This proposed approach for registration shares close similarities with the well-known geodesic active contours model in image segmentation, where the segmentation term (the edge detector function) is coupled with the regularization term (the length functional) via multiplication as well. As a matter of fact, our proposed geometric model is actually the exact mathematical generalization to vector fields of the weighted length problem for curves and surfaces introduced by Caselles-Kimmel-Sapiro. The energy of the deformation field is measured with the Polyakov energy weighted by a suitable image distance, borrowed from standard registration models. We investigate three different weighting functions, the squared error and the approximated absolute error for monomodal images, and the local joint entropy for multimodal images. As compared to specialized state-of-the-art methods tailored for specific applications, our geometric framework involves important contributions. Firstly, our general formulation for registration works on any parametrizable, smooth and differentiable surface, including nonflat and multiscale images. In the latter case, multiscale images are registered at all scales simultaneously, and the relations between space and scale are intrinsically being accounted for. Second, this method is, to the best of our knowledge, the first reparametrization invariant registration method introduced in the literature. Thirdly, the multiplicative coupling between the registration term, i.e. local image discrepancy, and the regularization term naturally results in a data-dependent tuning of the regularization strength. Finally, by choosing the metric on the deformation field one can freely interpolate between classic Gaussian and more interesting anisotropic, TV-like regularization.

A KIND OF SOAP FILM PROBLEM

We numerically discuss a kind of minimal surface problem with fixed boundary by the software Surface Evolver. Compared with the analytic result of catenoid, it indicates that there is an obvious range for the existence of this kind of minimal surface.

Minimal Surfaces Extend Shortest Path Segmentation Methods to 3D

Shortest paths have been used to segment object boundaries with both continuous and discrete image models. Although these techniques are well defined in 2D, the character of the path as an object boundary is not preserved in 3D. An object boundary in three dimensions is a 2D surface. However, many different extensions of the shortest path techniques to 3D have been previously proposed in which the 3D object is segmented via a collection of shortest paths rather than a minimal surface, leading to a solution which bears an uncertain relationship to the true minimal surface. Specifically, there is no guarantee that a minimal path between points on two closed contours will lie on the minimal surface joining these contours. We observe that an elegant solution to the computation of a minimal surface on a cellular complex (e.g., a 3D lattice) was given by Sullivan [47]. Sullivan showed that the discrete minimal surface connecting one or more closed contours may be found efficiently by solving a Minimum-cost Circulation Network Flow (MCNF) problem. In this work, we detail why a minimal surface properly extends a shortest path (in the context of a boundary) to three dimensions, present Sullivan's solution to this minimal surface problem via an MCNF calculation, and demonstrate the use of these minimal surfaces on the segmentation of image data.

Multilevel Algorithms for Large-Scale Interior Point Methods

We develop and compare multilevel algorithms for solving constrained nonlinear variational problems via interior point methods. Several equivalent formulations of the linear systems arising at each iteration of the interior point method are compared from the point of view of conditioning and iterative solution. Furthermore, we show how a multilevel continuation strategy can be used to obtain good initial guesses (“hot starts”) for each nonlinear iteration. Some minimal surface and volume-constrained image registration problems are used to illustrate the various approaches.

A GPU Laplacian solver for diffusion curves and Poisson image editing

We present a new Laplacian solver for minimal surfaces---surfaces having a mean curvature of zero everywhere except at some fixed (Dirichlet) boundary conditions. Our solution has two main contributions: First, we provide a robust rasterization technique to transform continuous boundary values (diffusion curves) to a discrete domain. Second, we define a variable stencil size diffusion solver that solves the minimal surface problem. We prove that the solver converges to the right solution, and demonstrate that it is at least as fast as commonly proposed multigrid solvers, but much simpler to implement. It also works for arbitrary image resolutions, as well as 8 bit data. We show examples of robust diffusion curve rendering where our curve rasterization and diffusion solver eliminate the strobing artifacts present in previous methods. We also show results for real-time seamless cloning and stitching of large image panoramas.

Minimal Surfaces in the Roto-Translation Group with Applications to a Neuro-Biological Image Completion Model

We investigate solutions to the minimal surface problem with Dirichlet boundary conditions in the roto-translation group equipped with a sub-Riemannian metric. By work of G. Citti and A. Sarti, such solutions are completions of occluded visual data when using a model of the first layer of the visual cortex. Using a characterization of smooth non-characteristic minimal surfaces as ruled surfaces, we give a method to compute a minimal spanning surface given fixed boundary data presuming such a surface exists. Moreover, we describe a number of obstructions to existence and uniqueness but also show that under suitable conditions, smooth minimal spanning surfaces with good properties exist. Not only does this provide an explicit realization of the disocclusion process for the neurobiological model, but it also has application to constructing disocclusion algorithms in digital image processing.

Proximal point method and elliptic regularization

A generalized proximal point method for solving variational inequalities with maximal monotone operators is developed. It admits a successive approximation of the feasible set and of a symmetric component of the operator as well as an inexact solving of the regularized problems. The conditions on the approximation are coordinated with the properties of finite element methods for solving problems in mathematical physics. The choice of the regularizing functional exploits a possible “reserve of monotonicity” of the operator in the variational inequality. For the minimal surface problem and related variational inequalities as well as for the convection-diffusion problem the studied method extends the principle of elliptic regularization. A special convergence analysis shows a more qualitative convergence of the method applied to these problems than it follows from the general theory of proximal point methods. Also applications to some variational inequalities from the elasticity theory are investigated.

Solution of a Class of Minimal Surface Problem with Obstacle

Plateau’s problem is to determine the surface with minimal area that lies above an obstacle with given boundaryconditions. In this paper, a special example of this class of the problem is given and solved with the linearfinite element method. First, we triangulate the domain of definition, and transform the linear finite elementapproximation into a constrained nonlinear optimization problem. Then we introduce a simple and efficientmethod, named sequential quadratic programming, for solving the constrained nonlinear optimization problem.The sequential quadratic programming is implemented by the fmincon function in the optimization toolbox ofMATLAB. Also, we discuss the relations between the number of grids and the computing time as well as theprecision of the result.