For capillary driven flow the interface curvature is essential in the modelling of surface tension via the imposition of the Young--Laplace jump condition. We show that traditional geometric volume of fluid (VOF) methods, that are based on a piecewise linear approximation of the interface, do not lead to an interface curvature which is convergent under mesh refinement in time-dependent problems. Instead, we propose to use a piecewise parabolic approximation of the interface, resulting in a class of piecewise parabolic interface calculation (PPIC) methods. In particular, we introduce the parabolic LVIRA and MOF methods, PLVIRA and PMOF, respectively. We show that a Lagrangian remapping method is sufficiently accurate for the advection of such a parabolic interface. It is numerically demonstrated that the newly proposed PPIC methods result in an increase of reconstruction accuracy by one order, convergence of the interface curvature in time-dependent advection problems and Weber number independent convergence of a droplet translation problem, where the advection method is coupled to a two-phase Navier--Stokes solver. The PLVIRA method is applied to the simulation of a 2D rising bubble, which shows good agreement to a reference solution.
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We propose BareSkinNet, a novel method that simultaneously removes makeup and lighting influences from the face image. Our method leverages a 3D morphable model and does not require a reference clean face image or a specified light condition. By combining the process of 3D face reconstruction, we can easily obtain 3D geometry and coarse 3D textures. Using this information, we can infer normalized 3D face texture maps (diffuse, normal, roughness, and specular) by an image-translation network. Consequently, reconstructed 3D face textures without undesirable information will significantly benefit subsequent processes, such as re-lighting or re-makeup. In experiments, we show that BareSkinNet outperforms state-of-the-art makeup removal methods. In addition, our method is remarkably helpful in removing makeup to generate consistent high-fidelity texture maps, which makes it extendable to many realistic face generation applications. It can also automatically build graphic assets of face makeup images before and after with corresponding 3D data. This will assist artists in accelerating their work, such as 3D makeup avatar creation.
Linear programming (LP) is an extremely useful tool which has been successfully applied to solve various problems in a wide range of areas, including operations research, engineering, economics, or even more abstract mathematical areas such as combinatorics. It is also used in many machine learning applications, such as $\ell_1$-regularized SVMs, basis pursuit, nonnegative matrix factorization, etc. Interior Point Methods (IPMs) are one of the most popular methods to solve LPs both in theory and in practice. Their underlying complexity is dominated by the cost of solving a system of linear equations at each iteration. In this paper, we consider both feasible and infeasible IPMs for the special case where the number of variables is much larger than the number of constraints. Using tools from Randomized Linear Algebra, we present a preconditioning technique that, when combined with the iterative solvers such as Conjugate Gradient or Chebyshev Iteration, provably guarantees that IPM algorithms (suitably modified to account for the error incurred by the approximate solver), converge to a feasible, approximately optimal solution, without increasing their iteration complexity. Our empirical evaluations verify our theoretical results on both real-world and synthetic data.
In this paper, we tackle the problem of active robotic 3D reconstruction of an object. In particular, we study how a mobile robot with an arm-held camera can select a favorable number of views to recover an object's 3D shape efficiently. Contrary to the existing solution to this problem, we leverage the popular neural radiance fields-based object representation, which has recently shown impressive results for various computer vision tasks. However, it is not straightforward to directly reason about an object's explicit 3D geometric details using such a representation, making the next-best-view selection problem for dense 3D reconstruction challenging. This paper introduces a ray-based volumetric uncertainty estimator, which computes the entropy of the weight distribution of the color samples along each ray of the object's implicit neural representation. We show that it is possible to infer the uncertainty of the underlying 3D geometry given a novel view with the proposed estimator. We then present a next-best-view selection policy guided by the ray-based volumetric uncertainty in neural radiance fields-based representations. Encouraging experimental results on synthetic and real-world data suggest that the approach presented in this paper can enable a new research direction of using an implicit 3D object representation for the next-best-view problem in robot vision applications, distinguishing our approach from the existing approaches that rely on explicit 3D geometric modeling.
In this paper, we consider a coupled chemotaxis-fluid system that models self-organized collective behavior of oxytactic bacteria in a sessile drop. This model describes the biological chemotaxis phenomenon in the fluid environment and couples a convective chemotaxis system for the oxygen-consuming and oxytactic bacteria with the incompressible Navier-Stokes equations subject to a gravitational force, which is proportional to the relative surplus of the cell density compared to the water density. We develop a new positivity preserving and high-resolution method for the studied chemotaxis-fluid system. Our method is based on the diffuse-domain approach, which we use to derive a new chemotaxis-fluid diffuse-domain (cf-DD) model for simulating bioconvection in complex geometries. The drop domain is imbedded into a larger rectangular domain, and the original boundary is replaced by a diffuse interface with finite thickness. The original chemotaxis-fluid system is reformulated on the larger domain with additional source terms that approximate the boundary conditions on the physical interface. We show that the cf-DD model converges to the chemotaxis-fluid model asymptotically as the width of the diffuse interface shrinks to zero. We numerically solve the resulting cf-DD system by a second-order hybrid finite-volume finite-difference method and demonstrate the performance of the proposed approach on a number of numerical experiments that showcase several interesting chemotactic phenomena in sessile drops of different shapes, where the bacterial patterns depend on the droplet geometries.
The method of common lines is a well-established reconstruction technique in cryogenic electron microscopy (cryo-EM), which can be used to extract the relative orientations of an object in tomographic projection images from different directions. In this paper, we deal with an analogous problem in optical diffraction tomography. Based on the Fourier diffraction theorem, we show that rigid motions, i.e., a map composed of rotations and translations, can be determined by detecting common circles in the Fourier-transformed data. We introduce two methods based on the idea of identifying common circles to reconstruct the object motion: While the first one is motivated by the common line approach for projection images and detects the relative orientation by the shape of the common circles in the two images, the second one assumes a smooth motion over time and calculates the angular velocity of the rotational motion from an infinitesimal version of the common circle method. Interestingly, using the stereographic projection, both methods can be reformulated as common line methods, but these lines are, in contrast to those used in cryo-EM, not confined to pass through the origin and allow for a full reconstruction of the relative orientation. Numerical proof-of-the-concept examples demonstrate the performance of our reconstruction methods.
A two dimensional eigenvalue problem (2DEVP) of a Hermitian matrix pair $(A, C)$ is introduced in this paper. The 2DEVP can be viewed as a linear algebraic formulation of the well-known eigenvalue optimization problem of the parameter matrix $H(\mu) = A - \mu C$. We present fundamental properties of the 2DEVP such as the existence, the necessary and sufficient condition for the finite number of 2D-eigenvalues and variational characterizations. We use eigenvalue optimization problems from the minmax of two Rayleigh quotients and the computation of distance to instability to show their connections with the 2DEVP and new insights of these problems derived from the properties of the 2DEVP.
We propose to combine the Carleman estimate and the Newton method to solve an inverse source problem for nonlinear parabolic equations from lateral boundary data. The stability of this inverse source problem is conditionally logarithmic. Hence, numerical results due to the conventional least squares optimization might not be reliable. In order to enhance the stability, we approximate this problem by truncating the high frequency terms of the Fourier series that represents the solution to the governing equation. By this, we derive a system of nonlinear elliptic PDEs whose solution consists of Fourier coefficients of the solution to the parabolic governing equation. We solve this system by the Carleman-Newton method. The Carleman-Newton method is a newly developed algorithm to solve nonlinear PDEs. The strength of the Carleman-Newton method includes (1) no good initial guess is required and (2) the computational cost is not expensive. These features are rigorously proved. Having the solutions to this system in hand, we can directly compute the solution to the proposed inverse problem. Some numerical examples are displayed.
For the first time, a nonlinear interface problem on an unbounded domain with nonmonotone set-valued transmission conditions is analyzed. The investigated problem involves a nonlinear monotone partial differential equation in the interior domain and the Laplacian in the exterior domain. Such a scalar interface problem models nonmonotone frictional contact of elastic infinite media. The variational formulation of the interface problem leads to a hemivariational inequality, which lives on the unbounded domain, and so cannot be treated numerically in a direct way. By boundary integral methods the problem is transformed and a novel hemivariational inequality (HVI) is obtained that lives on the interior domain and on the coupling boundary, only. Thus for discretization the coupling of finite elements and boundary elements is the method of choice. In addition smoothing techniques of nondifferentiable optimization are adapted and the nonsmooth part in the HVI is regularized. Thus we reduce the original variational problem to a finite dimensional problem that can be solved by standard optimization tools. We establish not only convergence results for the total approximation procedure, but also an asymptotic error estimate for the regularized HVI.
We investigate hypothesis testing in nonparametric additive models estimated using simplified smooth backfitting (Huang and Yu, Journal of Computational and Graphical Statistics, \textbf{28(2)}, 386--400, 2019). Simplified smooth backfitting achieves oracle properties under regularity conditions and provides closed-form expressions of the estimators that are useful for deriving asymptotic properties. We develop a generalized likelihood ratio (GLR) and a loss function (LF) based testing framework for inference. Under the null hypothesis, both the GLR and LF tests have asymptotically rescaled chi-squared distributions, and both exhibit the Wilks phenomenon, which means the scaling constants and degrees of freedom are independent of nuisance parameters. These tests are asymptotically optimal in terms of rates of convergence for nonparametric hypothesis testing. Additionally, the bandwidths that are well-suited for model estimation may be useful for testing. We show that in additive models, the LF test is asymptotically more powerful than the GLR test. We use simulations to demonstrate the Wilks phenomenon and the power of these proposed GLR and LF tests, and a real example to illustrate their usefulness.
Semantic reconstruction of indoor scenes refers to both scene understanding and object reconstruction. Existing works either address one part of this problem or focus on independent objects. In this paper, we bridge the gap between understanding and reconstruction, and propose an end-to-end solution to jointly reconstruct room layout, object bounding boxes and meshes from a single image. Instead of separately resolving scene understanding and object reconstruction, our method builds upon a holistic scene context and proposes a coarse-to-fine hierarchy with three components: 1. room layout with camera pose; 2. 3D object bounding boxes; 3. object meshes. We argue that understanding the context of each component can assist the task of parsing the others, which enables joint understanding and reconstruction. The experiments on the SUN RGB-D and Pix3D datasets demonstrate that our method consistently outperforms existing methods in indoor layout estimation, 3D object detection and mesh reconstruction.
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