Classical Interpolation Research Articles

XCycles Backprojection Acoustic Super-Resolution.

The computer vision community has paid much attention to the development of visible image super-resolution (SR) using deep neural networks (DNNs) and has achieved impressive results. The advancement of non-visible light sensors, such as acoustic imaging sensors, has attracted much attention, as they allow people to visualize the intensity of sound waves beyond the visible spectrum. However, because of the limitations imposed on acquiring acoustic data, new methods for improving the resolution of the acoustic images are necessary. At this time, there is no acoustic imaging dataset designed for the SR problem. This work proposed a novel backprojection model architecture for the acoustic image super-resolution problem, together with Acoustic Map Imaging VUB-ULB Dataset (AMIVU). The dataset provides large simulated and real captured images at different resolutions. The proposed XCycles BackProjection model (XCBP), in contrast to the feedforward model approach, fully uses the iterative correction procedure in each cycle to reconstruct the residual error correction for the encoded features in both low- and high-resolution space. The proposed approach was evaluated on the dataset and showed high outperformance compared to the classical interpolation operators and to the recent feedforward state-of-the-art models. It also contributed to a drastically reduced sub-sampling error produced during the data acquisition.

An isoperimetric problem with a competing nonlocal singular term

In this paper, we investigate the minimization of a functional in which the usual perimeter is competing with a nonlocal singular term comparable (but not necessarily equal to) a fractional perimeter. The motivation for this problem is a cell motility model introduced in some previous work by the first author. We establish several facts about global minimizers with a volume constraint. In particular we prove that minimizers exist and are radially symmetric for small mass, while minimizers cannot be radially symmetric for large mass. For large mass, we prove that the minimizing sequences either split into smaller sets that drift to infinity or must develop intricate non-symmetrical shape. Finally, we connect these two alternatives to a related minimization problem for the optimal constant in a classical interpolation inequality (a Gagliardo–Nirenberg type inequality for fractional perimeter).

On the Ra\u015fa Inequality for Higher Order Convex Functions

We study the following (q-1)th convex ordering relation for qth convolution power of the difference of probability distributions mu and nu (ν-μ)∗q≥(q-1)cx0,q≥2,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} (\\nu -\\mu )^{*q}\\ge _{(q-1)cx} 0 , \\quad q\\ge 2, \\end{aligned}$$\\end{document}and we obtain the theorem providing a useful sufficient condition for its verification. We apply this theorem for various families of probability distributions and we obtain several inequalities related to the classical interpolation operators. In particular, taking binomial distributions, we obtain a new, very short proof of the inequality given recently by Abel and Leviatan (2020).

A partitioned solution approach for the fluid–structure interaction of thin-walled structures and high-Reynolds number flows using RANS and hybrid RANS–LES turbulence models

In this work the partitioned solution approach for the fluid–structure interaction (FSI) of thin-walled structures and high-Reynolds number (Re) flows modeled using Reynolds–Averaged Navier–Stokes (RANS) and hybrid Reynolds–Averaged Navier–Stokes – Large Eddy Simulation (RANS–LES) turbulence models are described. The advanced turbulence modeling is needed to capture very complex fluid phenomena which triggers instabilities of thin–walled structures present in supersonic flow regimes. The finite element (FE) updated Lagrangian formulation (ULF) for the nonlinear elastic solids is used to predict its dynamical behavior. The main contribution addresses to the linear stress–strain relation Laplacian members, which are solved implicitly, on that way decreasing required memory resources and improving solution stability in the same time. The structures of the interest include the vast variety of membranes, curved shells and plates. The instabilities encountering these structures include limit cycle oscillations (LCO), flutter and buckling of the panels. The phenomena appear in everyday engineering practice and a need for the powerful tools to handle such problems is a common goal. Utilization of the unstructured non-regular meshes allows the precise distribution of computational nodes at the physical boundaries of the fluid and solid domains. It is naturally allowing application of the common approach for the fluid–solid interface coupling, as well as classical data interpolation schemes between fluid and solid on the FSI interface. High-Re flows, both 2D (benchmark) and 3D turbulent FSI case are chosen for the validation. Two numerical methods are coupled via a moving boundary treatment, in a staggered way. The proposed coupling method showed a good agreement with the reference test cases. The current FSI framework is developed to serve as a tool for the liquid rocket engine development.

Matrix Factorization-Based RSS Interpolation for Radio Environment Prediction

This letter proposes matrix factorization (MF) based interpolation of received signal strength (RSS) from a transmitter mounted on moving robot in factory environment. For realizing the reliable wireless communication, machine learning based channel prediction methods have been intensively studied in the past decade. However, some traffic models will make the observation of RSS sequence be intermittent, and the missing values must be interpolated before input to the predictor. Classical interpolation such as linear interpolation cannot appropriately estimate the missing values because the result of the interpolation depends on the observation time. In this letter, we propose to apply an MF-based interpolation technique to RSS interpolation in order to restore the true RSS variation pattern. Moreover, the basic MF-based interpolation is improved by introducing a smoothing term in an objective function to represent the smooth variation of the RSS sequence. The simulation results show that the MF-based interpolation can improve the prediction accuracy of the machine learning based channel prediction method.

Otsu’s Image Segmentation Algorithm with Memory-Based Fruit Fly Optimization Algorithm

In this paper, the most common pepper noise in grayscale image noise is investigated in depth in the median filtering algorithm, and the improved median filtering algorithm, adaptive switching median filtering algorithm, and adaptive polar median filtering algorithm are applied to the OTSU algorithm. Two improved OTSU algorithms such as the adaptive switched median filter-based OTSU algorithm and the polar adaptive median filter-based OTSU algorithm are obtained. The experimental results show that the algorithm can better cope with grayscale images contaminated by pretzel noise, and the segmented grayscale images are not only clear but also can better retain the detailed features of grayscale images. A genetic algorithm is a kind of search algorithm with high adaptive, fast operation speed, and good global space finding ability, and it will have a good effect when applied to the threshold finding of the OTSU algorithm. However, the traditional genetic algorithm will fall into the local optimal solution in different degrees when finding the optimal threshold. The advantages of the two interpolation methods proposed in this paper are that one is the edge grayscale image interpolation algorithm using OTSU threshold adaptive segmentation and the other is the edge grayscale image interpolation algorithm using local adaptive threshold segmentation, which can accurately divide the grayscale image region according to the characteristics of different grayscale images and effectively improve the loss of grayscale image edge detail information and jagged blur caused by the classical interpolation algorithm. The visual effect of grayscale images is enhanced by selecting grayscale images from the standard grayscale image test set and interpolating them with bilinear interpolation, bucolic interpolation, NEDI interpolation, and FEOI interpolation for interpolation simulation validation. The subjective evaluation and objective evaluation, as well as the running time, are compared, respectively, showing that the method of this paper can effectively improve the quality of grayscale image interpolation.

Analysis and Comparison of Three Classical Color Image Interpolation Algorithms

With the development of society, the progress of science and technology, the level of computer has been improved. Digital imaging technology has also achieved unprecedented development. The most commonly used digital cameras are Charge-coupled device (CCD) and Complementary Metal Oxide Semiconductor (CMOS) image sensors. Bayer filters are often used to help sensors capture color information. However, only one component can be achieved per pixel. Full-color images must be obtained by color interpolation. Therefore, the color interpolation algorithm has attracted the attention of the academic community. In this paper, three classical color interpolation algorithms- Bilinear interpolation algorithm, COK Algorithm, Gradient Interpolation Algorithm- are compared and studied. The Optimal Algorithm is selected by comparing false color, noise and color moire fringe. After comparison, it is found that the three basic algorithms will have color moire fringes appear. But the gradient interpolation algorithm has the least moire fringe and noise. COK Algorithm, second of all. But the COK Algorithm has color overflows.

A representation of the interpolation polynomial

We provide a decomposition formula for the classical polynomial interpolation operator and obtain the generalized Hermite interpolant through a limiting process. As a consequence of our results, we obtain/reobtain known and new identities related to interpolation theory.

MEMC-Net: Motion Estimation and Motion Compensation Driven Neural Network for Video Interpolation and Enhancement.

Motion estimation (ME) and motion compensation (MC) have been widely used for classical video frame interpolation systems over the past decades. Recently, a number of data-driven frame interpolation methods based on convolutional neural networks have been proposed. However, existing learning based methods typically estimate either flow or compensation kernels, thereby limiting performance on both computational efficiency and interpolation accuracy. In this work, we propose a motion estimation and compensation driven neural network for video frame interpolation. A novel adaptive warping layer is developed to integrate both optical flow and interpolation kernels to synthesize target frame pixels. This layer is fully differentiable such that both the flow and kernel estimation networks can be optimized jointly. The proposed model benefits from the advantages of motion estimation and compensation methods without using hand-crafted features. Compared to existing methods, our approach is computationally efficient and able to generate more visually appealing results. Furthermore, the proposed MEMC-Net architecture can be seamlessly adapted to several video enhancement tasks, e.g., super-resolution, denoising, and deblocking. Extensive quantitative and qualitative evaluations demonstrate that the proposed method performs favorably against the state-of-the-art video frame interpolation and enhancement algorithms on a wide range of datasets.

Non-Polynomial Interpolation of Functions with Large Gradients and Its Application

Interpolation of a function of one variable with large gradients in the boundary layer region is studied. The problem is that the use of classical polynomial interpolation formulas on a uniform mesh to functions with large gradients can lead to errors of the order of $$O(1)$$ , despite a small mesh size. An interpolation formula based on fitting to the component that defines the boundary-layer growth of the function is investigated. An error estimate, which depends on the number of interpolation nodes and is uniform over the boundary layer component and its derivatives, is obtained. It is shown how the interpolation formula derived can be used to construct formulas for numerical differentiation and integration and in the two-dimensional case. The corresponding error estimates are obtained.

An Adaptive Inverse-Distance Weighting Interpolation Method Considering Spatial Differentiation in 3D Geological Modeling

The inverse-distance weighting interpolation is widely used in 3D geological modeling and directly affects the accuracy of models. With the development of “smart” or “intelligent” geology, classical inverse-distance weighting interpolation cannot meet the accuracy, reliability, and efficiency requirements of large-scale 3D geological models in these fields. Although the improved inverse-distance weighting interpolation can basically meet the requirements of accuracy and reliability, it cannot meet the requirements of efficiency at the same time. In response to these limitations, the adaptive inverse-distance weighting interpolation method based on geological attribute spatial differentiation and geological attribute feature adaptation was proposed. This method takes into account the spatial differentiation of geological attributes to improve the accuracy and considers the first-order neighborhood selection strategy to adaptively improve efficiency to meet above requirements of large-scale geological modeling. The proposed method was applied to an area in eastern China, and the results of the proposed method, compared to the results of classical inverse-distance weighting interpolation and improved inverse-distance weighting interpolation, suggest that the problems encountered above in large-scale geological modeling can be solved with the proposed method. The method can provide effective support for large-scale 3D geological modeling in smart geology.

Nearest Neighbors Strategy, $${\mathbb {P}}_1$$ Lagrange Interpolation, and Error Estimates for a Learning Neural Network

Approximating a function with a learning neural network (LNN) has been considered for a long time by many authors and is known as “the universal approximation property”. The smaller the precision, the more neurons in the hidden layer one should take to reach the required precision. An other challenge for learning neural networks is understanding how to reduce the computational expense and how to select the training examples. Our contribution here is to consider an LNN as a discretization of the time-dependent transport of the identity function, Id, towards the function T to be approximated by the LNN. Using classical interpolation properties for $${\mathbb {P}}_1$$ Lagrange functions, we are able to give space-time error estimates for the simple LNN we introduce.

Bergman–Weil expansion for holomorphic functions

Using a modified Cauchy–Weil representation formula in a Weil polyhedron $${\varvec{D}}_f\subset U\subset \mathbb {C}^n$$ , we prove a generalized version of Lagrange interpolation formula (at any order) with respect to a discrete set defined by $$V_{{\varvec{D}}_f}(f):=\{f_1=\cdots = f_m =0\}\, \cap {\varvec{D}}_f$$ , when $$m>n$$ and $$\{f_1,\ldots ,f_m\}$$ is minimal as a defining system. Thus the set $$V_{{\varvec{D}}_f}(f)$$ fails to be a complete intersection. We present our result as an averaged version of the classic Lagrange interpolation formula in the case $$m=n$$ . We invoke to that purpose Crofton’s formula, which plays a key role in the construction of Vogel generalized cycles as proposed in Andersson et al. (J Reine Angew Math 728: 105–136, 2017; Math Ann, 2020. https://doi.org/10.1007/s00208-020-01973-y). This leads us naturally to the construction of Bochner–Martinelli kernels. We also introduce $$f^{-1}(\{0\})$$ -Lagrange interpolators (at any order) subordinate to the choice of a smooth hermitian metric on the trivial m-bundle $$\mathbb {C}_U^m=U\times \mathbb {C}^m$$ , while the mapping $$f = (f_1,\ldots ,f_m)$$ is considered as its section.

“Interpolated Factored Green Function” method for accelerated solution of scattering problems

This paper presents a novel Interpolated Factored Green Function method (IFGF) for the accelerated evaluation of the integral operators in scattering theory and other areas. Like existing acceleration methods in these fields, the IFGF algorithm evaluates the action of Green function-based integral operators at a cost of O(Nlog⁡N) operations for an N-point surface mesh. The IFGF strategy, which leads to an extremely simple algorithm, capitalizes on slow variations inherent in a certain Green function analytic factor, which is analytic up to and including infinity, and which therefore allows for accelerated evaluation of fields produced by groups of sources on the basis of a recursive application of classical interpolation methods. Unlike other approaches, the IFGF method does not utilize the Fast Fourier Transform (FFT), and is thus better suited than other methods for efficient parallelization in distributed-memory computer systems. Only a serial implementation of the algorithm is considered in this paper, however, whose efficiency in terms of memory and speed is illustrated by means of a variety of numerical experiments—including a 43 min., single-core operator evaluation (on 10 GB of peak memory), with a relative error of 1.5×10−2, for a problem of acoustic size of 512 wavelengths.

Approximation of the Lebesgue Constant of a Lagrange Polynomial by a Logarithmic Function with Shifted Argument

Well-known two-sided estimates for the Lebesgue constants of two classical trigonometric interpolation Lagrange polynomials are improved. Approximations of these Lebesgue constants are based on logarithmic functions with shifted arguments.

Determination of AIS Position Accuracy and Evaluation of Reconstruction Methods for Maritime Observation Data

Abstract Automatic Identification System (AIS) increasingly serves as the primary sensor for decision making. But the provided information can be manipulated, and the signal can even be manually disabled to hide illegal activities. In addition, the quality of the AIS data depends strongly on the quality of the sensors installed on the vessel. To determine the accuracy of AIS position data, the paper will compare AIS data with associated radar tracks. The radar data is assumed to be ground truth so that the accuracy of AIS can be determined. In addition, various existing AIS interpolation approaches are discussed and executed on a common representative dataset with over 6.2 million data points of the German Bight. This allows to make meaningful statements about the quality of the methods when interpolating AIS data. Overall, it was revealed that the mean discrepancy between AIS and radar data measures about 97.72m. Moreover, some presented approaches can approximate AIS data with higher accuracy than the classical linear interpolation. However, this accuracy is also associated with an increased computation time, so that it should be weighed which interpolation method should be used depending on the use case.

THE PROBLEM OF THE COMBINED USE OF FILTRATION THEORY AND MACHINE LEARNING ELEMENTS FOR SOLVING THE INVERSE PROBLEM OF RESTORING THE HYDRAULIC CONDUCTIVITY OF AN OIL FIELD

This article presents the methodology involving the combined use of machine learning elements and a physically meaningful filtration model. The authors propose using a network of radial basis functions for solving the problem of restoring hydraulic conductivity in the interwell space for an oil field. The advantage of the proposed approach in comparison with classical interpolation methods as applied to the problems of reconstructing the filtration-capacitive properties of the interwell space is shown. The paper considers an algorithm for the interaction of machine learning methods, a filtration model, a mechanism for separating input data, a form of a general objective function, which includes physical and expert constraints. The research was carried out on the example of a symmetrical element of an oil field. The proposed procedure for finding a solution includes solving a direct and an adjoint problem.

Extreme Rainfall over Complex Terrain: An Application of the Linear Model of Orographic Precipitation to a Case Study in the Italian Pre-Alps

Intense meteorological events are the primary cause of geohazard phenomena in mountain areas. In this paper, we present a study of the intense rainfall event that occurred in the provinces of Lecco and Sondrio from 11 to 12 June 2019. The aim of our work is to understand the effect of local topography on the spatial distribution of rainfall and to attempt the reconstruction of a realistic rainfall field relative to that extreme event. This task represents a challenge in the context of complex orography. Classical rain-gauge interpolation techniques, such as Kriging, may be too approximate, while meteorological models can be complex and often unable to accurately predict rainfall extremes. For these reasons, we tested the linear upslope model (LUM) designed for estimating rainfall records in orographic precipitation. This model explicitly addresses the dependence of rainfall intensification caused by the terrain elevation. In our case study, the available radio sounding data identified the convective nature of the event with a sustained and moist southern flow directed northward across the Pre-Alps, resulting in an orographic uplift. The simulation was conducted along a smoothed elevation profile of the local orography. The result was a reliable reconstruction of the rainfall field, validated with the ground-based rain gauge data. The error analysis revealed a good performance of the LUM with a realistic description of the interaction between the airflow and local orography. The areas subjected to rainfall extremes were correctly identified, confirming the determinant role of complex terrain in precipitation intensification.

Optimal design for kernel interpolation: Applications to uncertainty quantification

The paper is concerned with classic kernel interpolation methods, in addition to approximation methods that are augmented by gradient measurements. To apply kernel interpolation using radial basis functions (RBFs) in a stable way, we propose a type of quasi-optimal interpolation points, searching from a large set of candidate points, using a procedure similar to designing Fekete points or power function maximizing points that use pivot from a Cholesky decomposition. The proposed quasi-optimal points results in smaller condition number, and thus mitigates the instability of the interpolation procedure when the number of points becomes large. Applications to parametric uncertainty quantification are presented, and it is shown that the proposed interpolation method can outperform sparse grid methods in many interesting cases. We also demonstrate the new procedure can be applied to constructing gradient-enhanced Gaussian process emulators.

A Novel Adaptive Directional Interpolation Algorithm for Digital Video Resolution Enhancement

In this paper, a novel digital video resolution enhancement algorithm based on adaptive directional interpolation is proposed, where the directionality of the edge structure and the nonlocal self-similarity prior within the current frame as well as its adjacent frames are both considered. First, we establish the regularization equation that conforms to the prior model of a video frame and then take the classic bicubic interpolation result as the initial estimation to iteratively solve the restoration equation, in which the edge structures and contours in low resolution (LR) input are reconstructed to estimate and refine the desired high resolution (HR) output. Experimental results show that the proposed algorithm can effectively enhance the clarity of a video frame, with satisfying subjective visual quality and PSNR value.