Biharmonic Functions Research Articles

Retargeting of facial model for unordered dense point cloud

Facial retargeting is a widely used technique in the game and film industries that replicates the expressions of a source facial model onto a target model. Existing methods for facial retargeting rely on either hand-crafted uniform triangle meshes or sparse points obtained from motion capture(mocap). In this paper, we propose an end-to-end facial retargeting algorithm that copies facial expressions from unordered dense point clouds onto the target model. First, a corresponding building method based on bi-harmonic function is introduced to ensure that the template model and a cluster of point clouds share the same triangle topology. Second, a deformation transferring method is presented to transfer the calculated deformation onto the target model. Several experiments are conducted on the SIAT-3DFE dataset to demonstrate the accuracy and efficiency of our method.

The Trefftz methods for 3D biharmonic equation using directors and in-plane biharmonic functions

The Trefftz methods for 3D biharmonic equation using directors and in-plane biharmonic functions

A class of solutions of the n-dimensional generalized Helmholtz equation which describes generalized Weingarten hypersurfaces

In this paper, we introduce the \(n\)-dimensional generalized Helmholtz equation and present explicit solutions to this equation in terms of biharmonic functions, in particular, we get solutions that depend on holomorphic functions. Also, we present explicit radial solutions for this equation and we provide explicit solutions to the \(n\)-dimensional Helmholtz equation. In addition, as an application we introduced two classes of generalized Weingarten hypersurfaces, namely, the RSHGW-hypersurfaces and the RSGW-hypersurfaces, associated with solutions of the \(n\)-dimensional generalized Helmholtz equation and classify the RSHGW-hypersurfaces of rotation. For \(n=2\), we obtain a Weierstrass type representation for these surfaces which depend of three holomorphic functions and we classify the RSHGW-surfaces and the RSGW-surfaces of rotation.

On biharmonic functions on vector bundles

On biharmonic functions on vector bundles

A four-node rectangular plate finite element using Airy functions with transverse shear

PurposeAmong different types of engineering structures, plates play a significant role. Their analysis necessitates numerical modeling with finite elements, such as triangular, quadrangular or sector plate elements, owing to the intricate geometrical shapes and applied loads. The scope of this study is the development of a new rectangular finite element for thin plate bending based on the strain approach using Airy's function. It is called a rectangular plate finite element using Airy function (RPFEUAF) and has four nodes. Each node had three degrees of freedom: one transverse displacement (w) and two normal rotations (x, y).Design/methodology/approachEquilibrium conditions are used to generate the interpolation functions for the fields of strain, displacements and stresses. The evolution of the Airy function solutions yielded the selection of these polynomial bi-harmonic functions. The variational principle and the analytical integration approach are used to evaluate the basic stiffness matrix.FindingsThe numerical findings for thin plates quickly approach the Kirchhoff solution. The results obtained are compared to the analytical solution based on Kirchhoff theory.Originality/valueThe efficiency of the strain based approach using Airy's function is confirmed, and the robustness of the presented element RPFEUAF is demonstrated. Because of this, the current element is more reliable, better suited for computations and especially intriguing for modeling this kind of structure.

A Unified Measurement Method for Shape Sensing of Plate Structure

This study proposes an enhanced inverse finite-element method (iFEM) for reconstructing the deformation of homogeneity structures, including thin to thick plate models. The methodology develops the explicit relation between the structural mid-plane deformation field by biharmonic function and discrete surface strain measurements consistent with scaled boundary element theory. The present method fills the gap in existing iFEM for the deployment of strain sensors and expands the engineering application of iFEM based on single-surface (top/bottom) strain information. In addition, the algorithm accommodates for the effect of transverse shear strain measurements on reconstructing accuracy, and the calculated process of shear strain measurement based on discrete surface strain is formulated by the equilibrium equations of elasticity theory and strain–stress constitutive relation. Overall, by performing the shape sensing of an array antenna plate model subjected to different loading cases, the accuracy and applicability of the present method are numerically and experimentally validated for various thickness–width ratio plate structures.

Political Improved Invasive Weed Optimization-Driven Hybrid Exemplar Technique for Video Inpainting Process

Video inpainting aspires to fill the Spatio-temporal holes in videos with probable and coherent content. This process recovers the missing content of corrupted video effectively, which is useful in many fields, including removal of watermarking and video restoration. The difficulties of creating video contents with exquisite detail while maintaining spatiotemporal coherence in the missing areas is the main difficulty in the video inpainting process. Modern studies ignore semantic structural coherence maintenance between frames in favor of using flow information to synthesize temporally smooth pixels. In this paper, Political Improved Invasive Weed Optimization (PIIWO)-based optimal exemplar is designed for the productive video inpainting process. Accordingly, the developed PIIWO algorithm is newly designed by combining Political Optimizer (PO) and Improved Invasive Weed Optimization (IIWO). Here, the inpainting results obtained from context-aware Ant Lion Gray Wolf Optimization (ALGWO)-based Markov Random Field (MRF) modeling, Whale Monarch Butterfly Optimization (Whale MBO)-based Deep Convolutional Neural Network (DCNN), K-Nearest Neighbors (KNN) with Bhattacharya distance, Bi-harmonic function modules and developed PIIWO-based exemplar model are fused using Bayes probabilistic fusion for producing the final result. Three metrics, peak signal-to-noise ratio (PSNR), second derivative like the measure of enhancement (SDME) and structural similarity (SSIM) of 40.19[Formula: see text]dB, 78.07[Formula: see text]dB and 0.9857, respectively, are used to assess the performance of the developed video inpainting technique.

Generalized Spherical type hypersurfaces

In this paper we present a classification of Laguerre minimal surfaces with planar curvature lines. We define Spherical type hypersurfaces, show that these hypersurfaces are determined by a harmonic function, we also introduce the class of Generalized Spherical type hypersurfaces (GS-Hypersurfaces) and show that these hypersurfaces are determined by a biharmonic function. As an application, we classify rotation GS-Hypersurfaces and present examples of families of GS-Hypersurfaces parameterized by planar curvature lines.

A comparison of deep-learning-based inpainting techniques for experimental X-ray scattering.

The implementation is proposed of image inpainting techniques for the reconstruction of gaps in experimental X-ray scattering data. The proposed methods use deep learning neural network architectures, such as convolutional autoencoders, tunable U-Nets, partial convolution neural networks and mixed-scale dense networks, to reconstruct the missing information in experimental scattering images. In particular, the recovered pixel intensities are evaluated against their corresponding ground-truth values using the mean absolute error and the correlation coefficient metrics. The results demonstrate that the proposed methods achieve better performance than traditional inpainting algorithms such as biharmonic functions. Overall, tunable U-Net and mixed-scale dense network architectures achieved the best reconstruction performance among all the tested algorithms, with correlation coefficient scores greater than 0.9980.

On the origin of chaotic attractors with two zero Lyapunov exponents in a system of five biharmonically coupled phase oscillators.

We study chaotic dynamics in a system of four differential equations describing the interaction of five identical phase oscillators coupled via biharmonic function. We show that this system exhibits strange spiral attractors (Shilnikov attractors) with two zero (indistinguishable from zero in numerics) Lyapunov exponents in a wide region of the parameter space. We explain this phenomenon by means of bifurcation analysis of a three-dimensional Poincaré map for the system under consideration. We show that chaotic dynamics develop here near a codimension three bifurcation, when a periodic orbit (fixed point of the Poincaré map) has the triplet of multipliers ( 1 , 1 , 1 ). As it is known, the flow normal form for such bifurcation is the well-known three-dimensional Arneodó-Coullet-Spiegel-Tresser (ACST) system, which exhibits spiral attractors. According to this, we conclude that the additional zero Lyapunov exponent for orbits in the observed attractors appears due to the fact that the corresponding three-dimensional Poincaré map is very close to the time-shift map of the ACST-system.

About One Variational Problem, Leading to а Biharmonic Equation, and about the Approximate Solution of the Main Boundary Value Problem for this Equation

. Many important questions in the theory of elasticity lead to a variational problem associated with a biharmonic equation and to the corresponding boundary value problems for such an equation. The paper considers the main boundary value problem for the biharmonic equation in the unit circle. This problem leads, for example, to the study of plate deflections in the case of kinematic boundary conditions, when the displacements and their derivatives depend on the circular coordinate. The exact solution of the considered boundary value problem is known. The desired biharmonic function can be represented explicitly in the unit circle by means of the Poisson integral. An approximate solution of this problem is sometimes foundusing difference schemes. To do this, a grid with cells of small diameter is thrown onto the circle, and at each grid node all partial derivatives of the problem are replaced by their finite-difference relations. As a result, a system of linear algebraic equations arises for unknown approximate values of the biharmonic function, from which they are uniquely found. The disadvantage of this method is that the above system is not always easy to solve. In addition, we get the solution not at any point of the circle, but only at the nodes of the grid. For real calculations and numerical analysis of solutions to applied problems, the authors have constructed its unified analytical approximate representation on the basis of the known exact solution of the boundary value problem while using logarithms. The approximate formula has a simple form and can be easily implemented numerically. Uniform error estimates make it possible to perform calculations with a given accuracy. All coefficients of the quadrature formula for the Poisson integral are non-negative, which greatly simplifies the study of the approximate solution. An analysis of the quadrature sum for stability is carried out. An example of solving a boundary value problem is considered.

Metric Growth of the Discrete $q$-Laplacian

The metric growth of Laplacian $\Delta u$ in the theory of potentials on Riemannian manifolds by [13] means the growth of Green potential of $(\Delta u)^2$. This notion was introduced to obtain a Riesz representation for a biharmonic functions. On a hyperbolic network, we investigated a similar problem in [7], by taking the discrete Laplacian $\Delta$ and the (discrete) Green function. Let $q$ be a non-negative function such that $q \not\equiv 0$. Then $- \Delta + q$ is a Schrödinger operator. Let us put $\Delta _q := \Delta - q$ and call it $q$-Laplacian. The $q$-Green function $g_a$ of the network with pole at $a$ always exists. In this paper, we estimate the metric growth of $\Delta _q u$ by $q$-Green potential of $|\Delta _q u|$ and $|\Delta _q u|^2$. As an application, we prove Riesz representation theorem for $q$-biharmonic functions.

Modeling the Deformation Sources in Volcanic Environments Through Multi-Scale Analysis of DInSAR Measurements

In this work, we propose a multi-scale approach for modeling the ground deformation field of volcanic sources. The methodology is based on the use of Multiridge and ScalFun methods to analyze the elastic deformation fields by providing source information, such as the depth, the horizontal position, and its morphological features. This strategy is alternative with respect to the classical inverse approaches since it allows overcoming some aspects of the ambiguities related to the interpretation of DInSAR measurements. Indeed, the multi-scale method does not need a priori constraints on the model parameters to achieve an unambiguous solution. First, we argue on the general physical conditions so that the deformation field, generally represented by biharmonic functions, also satisfies Laplace’s equation and the law of homogeneity. This occurs in the case of sources with hydrostatic pressure-change embedded in an elastic half-space. In these conditions, the properties of harmonic and homogeneous functions can be employed to model the ground deformation fields using multi-scale procedures. Then, we demonstrate the soundness of the proposed approach through the application of Multiridge and ScalFun methods to synthetic tests. We analyze the fields generated by spherical, pipes- and sills-like sources and consider different model settings, as the layered half-space, the noisy, and the multi-source scenarios. For all these simulations, we achieve unconstrained information related to the source geometry with satisfying accuracy. Finally, in order to show the flexibility of the multi-scale approach in different volcanic environments, we use Multiridge and ScalFun methods for analyzing DInSAR measurements relevant to Uturuncu, Okmok, and Fernandina volcanoes, retrieving information about their volcanic systems. In the first case, we point out a transient source at a depth of 4.5 km b.s.l., which is activated during 2006–2007 together with the stable and well-known Altiplano Puna Magma Body (APMB); in the second one, we retrieve a concentrated body at a depth of 3.1 km b.s.l. during 2003–2004; and in the last case, we interpret the 2013 unrest episode as caused by a pipe-like source located at a depth of 1.7 km b.s.l.

On the Cauchy Problem for the Biharmonic Equation

The work is devoted to the study of continuation and stability estimation of the solution of the Cauchy problem for the biharmonic equation in the domain G from its known values on the smooth part of the boundary @G. The problem under consideration belongs to the problems of mathematical physics in which there is no continuous dependence of solutions on the initial data. In this work, using the Carleman function, not only the biharmonic function itself, but also its derivatives are restored from the Cauchy data on a part of the boundary of the region. The stability estimates for the solution of the Cauchy problem in the classical sense are obtained

Bayes-Probabilistic-Based Fusion Method for Image Inpainting

Image inpainting removes unwanted objects from the image, signifying the original image restoration. Even though several techniques are introduced for image inpainting, but still, there are several challenging issues associated with the conventional methods regarding data loss, which are effectively handled based on the proposed approach. In this paper, we propose an effective hybrid image inpainting method that is termed as ALGDKH, which is the hybridization of Ant Lion–Gray Wolf Optimizer (ALG)-based Markov random field (MRF) modeling, deep learning, [Formula: see text]-nearest neighbors (KNN) and the harmonic functions. The crack input image is forwarded as an input to Markov random field modeling to obtain image inpainting, where the MRF energy is minimized based on the ALG. Then, the same crack image is subjected to the Whale–MBO-based DCNN, KNN with Bhattacharya distance and Bi-harmonic function modules to obtain the inpainting results. Finally, the results from the proposed ALG-based Markov random field modeling, Whale–MBO-based DCNN, KNN with Bhattacharya distance and Bi-harmonic function modules are fused through Bayes-probabilistic fusion for the final inpainting results. The proposed method produces the maximal PSNR of 38.14[Formula: see text]dB, maximal SDME of 75.70[Formula: see text]dB and the maximal SSIM of 0.983.

Mean value theorems for polyharmonic functions

The problem of characterization of polyharmonic functions by mean value expressions is analyzed. A sufficient condition for the biharmonic function is formulated and proved in the paper. Mean value theorems and their operators have various applications in function theory (approximation of functions, description of functional spaces) and in the qualitative theory of linear differential equations with partial derivatives (boundary properties, elimination of features, differential properties of solutions, etc.). The main properties of polyharmonic functions, in particular, properties with mean value are investigated in the work. The results of L. Salcman and V. Volchkov, which generalize the classical mean value theorems, are analyzed. These results also reveal deep connections between complex analysis, the theory of linear differential equations with partial derivatives, harmonic analysis, integral geometry and the theory of special functions. The new directions of research in the theory of polyharmonic functions, differential equations, as well as in computational mathematics are established with the mean value theorems (algorithms "walk on spheres"\ , application of the Monte Carlo method). Particular attention is paid to the case of biharmonic functions. It is found that when performing the equality with the mean value on the circle from the set on the complex plane, the function of class $C^2$ on the specified set satisfies the Laplace equation of the second order, i.e. the function is polyharmonic of the second order. By using a smooth function with a compact support, the required function is obtained as a weak solution of the corresponding Laplace equation. In the future, this result can be used to establish a sufficient condition of biharmonicity of a twice continuously differentiated function on a plane that satisfies the above-mentioned relation only for a pair of positive values of $r_1$, $r_2$. Thus, it is possible to formulate a new theorem on two radii.

Numerical integration to obtain second moment of inertia of axisymmetric heterogeneous body

The second moment of inertia of a continuous axisymmetric object with an arbitrary shape made of a nonhomogeneous material is usually calculated by dividing it into small domains. However, it is a burdensome process to specify the density of the small domains. In this paper, a technique of easily calculating the second moment of inertia of an axisymmetric nonhomogeneous material using boundary integral equations is proposed. The calculations of the mass, primary moment, and center of mass of an arbitrarily shaped object made of a nonhomogeneous material are also shown. A formulization of the boundary element method is utilized, and a technique for the direct numerical integration of the axisymmetric domain using an axisymmetric interpolation method without the need to carry out domain division is proposed. Heterogeneous materials include laminated material composites, in which the density distribution is discontinuous. Axisymmetric interpolation using harmonic and biharmonic functions has a weak point that can be easily overcome using a scale method.

Avkhadiev–Backer type p-valent conditions for biharmonic functions

This paper is devoted to locally univalent complex-valued biharmonic functions. We obtain subclasses of biharmonic multivalet functions and get certain sufficient conditions for functions to be in these classes. We construct new families of biharmonic univalent and p-valent mappings of the unit disk similar to Avkhadiev–Becker type conditions for analytic functions. We also consider biharmonic functions defined in the exterior of the unit disk. New univalent and p-valent conditions for harmonic mappings of the unit disk are obtained.

Concentration phenomena for the fractional Q‐curvature equation in dimension 3 and fractional Poisson formulas

We study the compactness properties of metrics of prescribed fractional $Q$-curvature of order $3$ in $\R^3$. We will use an approach inspired from conformal geometry, seeing a metric on a subset of $\R^3$ as the restriction of a metric on $\R^4_+$ with vanishing fourth-order $Q$-curvature. We will show that a sequence of such metrics with uniformly bounded fractional $Q$-curvature can blow up on a large set (roughly, the zero set of the trace of a nonpositive biharmonic function $\Phi$ in $\R^4_+$), in analogy with a $4$-dimensional result of Adimurthi-Robert-Struwe, and construct examples of such behaviour. In doing so, we produce general Poisson-type representation formulas (also for higher dimension), which are of independent interest.

A Riemann jump problem for biharmonic functions in fractal domains

Biharmonic functions are the solutions of the fourth order partial differential equation $$\Delta \Delta \omega =0$$ . The purpose of this paper is to solve a kind of Riemann boundary value problem for biharmonic functions assuming higher order Lipschitz boundary data. We approach this problem making use of generalized Teodorescu transforms for obtaining the explicit expression of its solution in a Jordan domain $$\Omega \subset \mathbb {R}^2$$ with fractal boundary.