Static Scene Illumination Estimation from Videos with Applications

We present a system that automatically recovers scene geometry and illumination from a video, providing a basis for various applications. Previous image based illumination estimation methods require either user interaction or external information in the form of a database. We adopt structure-from-motion and multi-view stereo for initial scene reconstruction, and then estimate an environment map represented by spherical harmonics (as these perform better than other bases). We also demonstrate several video editing applications that exploit the recovered geometry and illumination, including object insertion (e.g., for augmented reality), shadow detection, and video relighting.     

Non-unified Compact Residual-Distribution Methods for Scalar Advection–Diffusion Problems

This paper solves the advection–diffusion equation by treating both advection and diffusion residuals in a separate (non-unified) manner. An alternative residual distribution (RD) method combined with the Galerkin method is proposed to solve the advection–diffusion problem. This Flux-Difference RD method maintains a compact-stencil and the whole process of solving advection–diffusion does not require additional equations to be solved. A general mathematical analysis reveals that the new RD method is linearity preserving on arbitrary grids for the steady-state advection–diffusion equation. The numerical results show that the flux difference RD method preserves second-order accuracy on various unstructured grids including highly randomized anisotropic grids on both the linear and nonlinear scalar advection–diffusion cases.     

A $$C^0$$ Linear Finite Element Method for Biharmonic Problems

In this paper, a \(C^0\) linear finite element method for biharmonic equations is constructed and analyzed. In our construction, the popular post-processing gradient recovery operators are used to calculate approximately the second order partial derivatives of a \(C^0\) linear finite element function which do not exist in traditional meaning. The proposed scheme is straightforward and simple. More importantly, it is shown that the numerical solution of the proposed method converges to the exact one with optimal orders both under \(L^2\) and discrete \(H^2\) norms, while the recovered numerical gradient converges to the exact one with a superconvergence order. Some novel properties of gradient recovery operators are discovered in the analysis of our method. In several numerical experiments, our theoretical findings are verified and a comparison of the proposed method with the nonconforming Morley element and \(C^0\) interior penalty method is given.     

Interior Penalty Discontinuous Galerkin Methods for Second Order Linear Non-divergence Form Elliptic PDEs

This paper develops interior penalty discontinuous Galerkin (IP-DG) methods to approximate \(W^{2,p}\) strong solutions of second order linear elliptic partial differential equations (PDEs) in non-divergence form with continuous coefficients. The proposed IP-DG methods are closely related to the IP-DG methods for advection-diffusion equations, and they are easy to implement on existing standard IP-DG software platforms. It is proved that the proposed IP-DG methods have unique solutions and converge with optimal rate to the \(W^{2,p}\) strong solution in a discrete \(W^{2,p}\)-norm. The crux of the analysis is to establish a DG discrete counterpart of the Calderon–Zygmund estimate and to adapt a freezing coefficient technique used for the PDE analysis at the discrete level. To obtain such a crucial estimate, we need to establish broken \(W^{1,p}\)-norm error estimates for IP-DG approximations of constant coefficient elliptic PDEs, which is also of independent interest. Numerical experiments are provided to gauge the performance of the proposed IP-DG methods and to validate the theoretical convergence results.     

POD/DEIM Reduced-Order Modeling of Time-Fractional Partial Differential Equations with Applications in Parameter Identification

In this paper, a reduced-order model (ROM) based on the proper orthogonal decomposition and the discrete empirical interpolation method is proposed for efficiently simulating time-fractional partial differential equations (TFPDEs). Both linear and nonlinear equations are considered. We demonstrate the effectiveness of the ROM by several numerical examples, in which the ROM achieves the same accuracy of the full-order model (FOM) over a long-term simulation while greatly reducing the computational cost. The proposed ROM is then regarded as a surrogate of FOM and is applied to an inverse problem for identifying the order of the time-fractional derivative of the TFPDE model. Based on the Levenberg–Marquardt regularization iterative method with the Armijo rule, we develop a ROM-based algorithm for solving the inverse problem. For cases in which the observation data is either uncontaminated or contaminated by random noise, the proposed approach is able to achieve accurate parameter estimation efficiently.     

An Improved High Order Finite Difference Method for Non-conforming Grid Interfaces for the Wave Equation

This paper presents an extension of a recently developed high order finite difference method for the wave equation on a grid with non-conforming interfaces. The stability proof of the existing methods relies on the interpolation operators being norm-contracting, which is satisfied by the second and fourth order operators, but not by the sixth order operator. We construct new penalty terms to impose interface conditions such that the stability proof does not require the norm-contracting condition. As a consequence, the sixth order accurate scheme is also provably stable. Numerical experiments demonstrate the improved stability and accuracy property.     

Numerical Preservation of Velocity Induced Invariant Regions for Reaction–Diffusion Systems on Evolving Surfaces

We propose and analyse a finite element method with mass lumping (LESFEM) for the numerical approximation of reaction–diffusion systems (RDSs) on surfaces in \({\mathbb {R}}^3\) that evolve under a given velocity field. A fully-discrete method based on the implicit–explicit (IMEX) Euler time-discretisation is formulated and dilation rates which act as indicators of the surface evolution are introduced. Under the assumption that the mesh preserves the Delaunay regularity under evolution, we prove a sufficient condition, that depends on the dilation rates, for the existence of invariant regions (i) at the spatially discrete level with no restriction on the mesh size and (ii) at the fully-discrete level under a timestep restriction that depends on the kinetics, only. In the specific case of the linear heat equation, we prove a semi- and a fully-discrete maximum principle. For the well-known activator-depleted and Thomas reaction–diffusion models we prove the existence of a family of rectangles in the phase space that are invariant only under specific growth laws. Two numerical examples are provided to computationally demonstrate (i) the discrete maximum principle and optimal convergence for the heat equation on a linearly growing sphere and (ii) the existence of an invariant region for the LESFEM–IMEX Euler discretisation of a RDS on a logistically growing surface.     

A survey of decision making methods based on two classes of hybrid soft set models

To the best of our knowledge, the tool of soft set theory is a new efficacious technique to dispose uncertainties and it focuses on the parameterization, while fuzzy set theory emphasizes the truth degree and rough set theory as another tool to handle uncertainties, it places emphasis on granular. However, the real-world problems that under considerations are usual very complicated. Consequently, it is very difficult to solve them by a single mathematical tool. It is worth noting that decision making (briefly, DM) in an imprecise environment has been showing more and more role in real-world applications. Researches on the idiographic applications of the above three uncertain theories as well as their hybrid models in DM have attracted many researchers’ widespread interest. DM methods are not yet proposed based on fusions of the above three uncertain theories. In view of the reason, by compromising the above three uncertain theories, we elaborate some reviews to DM methods based on two classes of hybrid soft models: SRF-sets and SFR-sets. We test all algorithms for DM and computation time on data sets produced by soft sets and FS-sets. The numerical experimentation programs are written for given pseudo codes in MATLAB. At the same time, the comparisons of all algorithms are given. Finally, we expatiate on an overview of techniques based on the involved hybrid soft set models.     

A view-invariant gait recognition algorithm based on a joint-direct linear discriminant analysis

This paper proposes a view-invariant gait recognition algorithm, which builds a unique view invariant model taking advantage of the dimensionality reduction provided by the Direct Linear Discriminant Analysis (DLDA). Proposed scheme is able to reduce the under-sampling problem (USP) that appears usually when the number of training samples is much smaller than the dimension of the feature space. Proposed approach uses the Gait Energy Images (GEIs) and DLDA to create a view invariant model that is able to determine with high accuracy the identity of the person under analysis independently of incoming angles. Evaluation results show that the proposed scheme provides a recognition performance quite independent of the view angles and higher accuracy compared with other previously proposed gait recognition methods, in terms of computational complexity and recognition accuracy.