Abstract: Standard approaches to computing interpolated (in-between) frames in a video sequence require accurate pixel correspondences between images e.g. using optical flow. We present an efficient alternative by leveraging recent developments in phase-based methods that represent motion in the phase shift of individual pixels. This concept allows in-between images to be generated by simple per-pixel phase modification, without the need for any form of explicit correspondence estimation. Up until now, such methods have been limited in the range of motion that can be interpolated, which fundamentally restricts their usefulness. In order to reduce these limitations, we introduce a novel, bounded phase shift correction method that combines phase information across the levels of a multi-scale pyramid. Additionally, we propose extensions for phase-based image synthesis that yield smoother transitions between the interpolated images. Our approach avoids expensive global optimization typical of optical flow methods, and is both simple to implement and easy to parallelize. This allows us to interpolate frames at a fraction of the computational cost of traditional optical flow-based solutions, while achieving similar quality and in some cases even superior results. Our method fails gracefully in difficult interpolation settings, e.g., significant appearance changes, where flow-based methods often introduce serious visual artifacts. Due to its efficiency, our method is especially well suited for frame interpolation and retiming of high resolution, high frame rate video.

Abstract: Image super-resolution (SR) aims to recover high-resolution images from their low-resolution counterparts for improving image analysis and visualization. Interpolation methods, widely used for this purpose, often result in images with blurred edges and blocking effects. More advanced methods such as total variation (TV) retain edge sharpness during image recovery. However, these methods only utilize information from local neighborhoods, neglecting useful information from remote voxels. In this paper, we propose a novel image SR method that integrates both local and global information for effective image recovery. This is achieved by, in addition to TV, low-rank regularization that enables utilization of information throughout the image. The optimization problem can be solved effectively via alternating direction method of multipliers (ADMM). Experiments on MR images of both adult and pediatric subjects demonstrate that the proposed method enhances the details in the recovered high-resolution images, and outperforms methods such as the nearest-neighbor interpolation, cubic interpolation, iterative back projection (IBP), non-local means (NLM), and TV-based up-sampling.

Abstract: In structural reliability, an important challenge is to reduce the number of calling the performance function, especially a finite element model in engineering problem which usually involves complex computer codes and requires time-consuming computations. To solve this problem, one of the metamodels, Kriging is then introduced as a surrogate for the original model. Kriging presents interesting characteristics such as exact interpolation and a local index of uncertainty on the prediction which can be used as an active learning method. In this paper, a new learning function based on information entropy is proposed. The new learning criterion can help select the next point effectively and add it to the design of experiments to update the metamodel. Then it is applied in a new method constructed in this paper which combines Kriging and Line Sampling to estimate the reliability of structures in a more efficient way. In the end, several examples including non-linearity, high dimensionality and engineering problems are performed to demonstrate the efficiency of the methods with the proposed learning function.

Abstract: Analysis that is based on geostatistics eliminates many corresponding defects and limitations compared to classical statistics which have been developed by random distribution theory of processes and variables. Interpolation is important for local analysis by GIS, because many maps used for GIS operation are made by interpolation. In this research, two different methods which is Kriging method and Inverse Distance Weighted (IDW) method was examined for developing Digital Elevation Model image. Each method’s advantages and disadvantages were considered. The study are, Kepil, is within Kulon Progo physiographic and stratigraphic area, located in the western part of Yogyakarta city. This area is located close to the Java Island Subduction Zone, hence influence of tectonic plate movement is relatively dominant. Geological structures become a main factor that shapes the recent morphology. This study area also has many settlements and has high weathering and erosion rate. Lineaments are extracted based on Digital Elevation Model to provide assistance in delineating geological structures. The structural geology analysis and an understanding of tectonic phase of the area provide useful information for geological map- ping. Accuracy of lineament depends on extraction and imagery parameters used. In this study, the extraction was conducted by two different raster methods, namely Kriging and Inverse Distance Weighted (IDW) with the same resolution of 30 meters. Lineament extracted automatically (digitally) with certain parameter settings. Keywords: Kriging, inverse distance weighted, interpolation, lineament, random distribution, digital elevation model.

Abstract: This paper introduces a new framework for constructing the Discrete Empirical Interpolation Method DEIM projection operator. The interpolation node selection procedure is formulated using the QR factorization with column pivoting, and it enjoys a sharper error bound for the DEIM projection error. Furthermore, for a subspace $\mathcal{U}$ given as the range of an orthonormal $U$, the DEIM projection does not change if $U$ is replaced by $U \Omega$ with arbitrary unitary matrix $\Omega$. In a large-scale setting, the new approach allows modifications that use only randomly sampled rows of $U$, but with the potential of producing good approximations with corresponding probabilistic error bounds. Another salient feature of the new framework is that robust and efficient software implementation is easily developed, based on readily available high performance linear algebra packages.

Abstract: Although image scaling algorithms in classical image processing have been extensively studied and widely used as basic image transformation methods, the quantum versions do not exist. Therefore, this paper proposes quantum algorithms and circuits to realize the quantum image scaling based on the improved novel enhanced quantum representation (INEQR) for quantum images. It is necessary to use interpolation in image scaling because there is an increase or a decrease in the number of pixels. The interpolation method used in this paper is nearest neighbor which is simple and easy to realize. First, NEQR is improved into INEQR to represent images sized $$2^{n_{1}} \times 2^{n_{2}}$$2n1×2n2. Based on it, quantum circuits for image scaling using nearest neighbor interpolation from $$2^{n_{1}} \times 2^{n_{2}}$$2n1×2n2 to $$2^{m_{1}} \times 2^{m_{2}}$$2m1×2m2 are proposed. It is the first time to give the quantum image processing method that changes the size of an image. The quantum strategies developed in this paper initiate the research about quantum image scaling.

Abstract: We introduce a new structured kernel interpolation (SKI) framework, which generalises and unifies inducing point methods for scalable Gaussian processes (GPs). SKI methods produce kernel approximations for fast computations through kernel interpolation. The SKI framework clarifies how the quality of an inducing point approach depends on the number of inducing (aka interpolation) points, interpolation strategy, and GP covariance kernel. SKI also provides a mechanism to create new scalable kernel methods, through choosing different kernel interpolation strategies. Using SKI, with local cubic kernel interpolation, we introduce KISSGP, which is 1) more scalable than inducing point alternatives, 2) naturally enables Kronecker and Toeplitz algebra for substantial additional gains in scalability, without requiring any grid data, and 3) can be used for fast and expressive kernel learning. KISS-GP costs O(n) time and storage for GP inference. We evaluate KISS-GP for kernel matrix approximation, kernel learning, and natural sound modelling.

Abstract: Deep learning has recently been introduced to the field of low-level computer vision and image processing. Promising results have been obtained in a number of tasks including super-resolution, inpainting, deconvolution, filtering, etc. However, previously adopted neural network approaches such as convolutional neural networks and sparse auto-encoders are inherently with translation invariant operators. We found this property prevents the deep learning approaches from outperforming the state-of-the-art if the task itself requires translation variant interpolation (TVI). In this paper, we draw on Shepard interpolation and design Shepard Convolutional Neural Networks (ShCNN) which efficiently realizes end-to-end trainable TVI operators in the network. We show that by adding only a few feature maps in the new Shepard layers, the network is able to achieve stronger results than a much deeper architecture. Superior performance on both image in-painting and super-resolution is obtained where our system outperforms previous ones while keeping the running time competitive.

Abstract: Spatially high resolution climate information is required for a variety of applications in but not limited to functional biodiversity research. In order to scale the generally plot-based research findings to a landscape level, spatial interpolation methods of meteorological variables are required. Based on a network of temperature observation plots across the southern slopes of Mt. Kilimanjaro, the skill of 14 machine learning algorithms in predicting spatial temperature patterns is tested and evaluated against the heavily utilized kriging approach. Based on a 10-fold cross-validation testing design, regression trees generally perform better than linear and non-linear regression models. The best individual performance has been observed by the stochastic gradient boosting model followed by Cubist, random forest and model averaged neural networks which except for the latter are all regression tree-based algorithms. While these machine learning algorithms perform better than kriging in a quantitative evaluation, the overall visual interpretation of the resulting air temperature maps is ambiguous. Here, a combined Cubist and residual kriging approach can be considered the best solution.

Abstract: We present a new non‐intrusive model reduction method for the Navier–Stokes equations. The method replaces the traditional approach of projecting the equations onto the reduced space with a radial basis function (RBF) multi‐dimensional interpolation. The main point of this method is to construct a number of multi‐dimensional interpolation functions using the RBF scatter multi‐dimensional interpolation method. The interpolation functions are used to calculate POD coefficients at each time step from POD coefficients at earlier time steps. The advantage of this method is that it does not require modifications to the source code (which would otherwise be very cumbersome), as it is independent of the governing equations of the system. Another advantage of this method is that it avoids the stability problem of POD/Galerkin. The novelty of this work lies in the application of RBF interpolation and POD to construct the reduced‐order model for the Navier–Stokes equations. Another novelty is the verification and validation of numerical examples (a lock exchange problem and a flow past a cylinder problem) using unstructured adaptive finite element ocean model. The results obtained show that CPU times are reduced by several orders of magnitude whilst the accuracy is maintained in comparison with the corresponding high‐fidelity models. Copyright © 2015 John Wiley & Sons, Ltd.

Abstract: Wireless local area network (WLAN)-based fingerprinting using received signal strength (RSS) has been considered to be one solution for indoor positioning. However, one widely recognized problem in fingerprinting is the collection and maintenance of a proper fingerprint database. In this paper, we consider having an incomplete fingerprint database with realistic coverage gaps, and we study the performance of several interpolation and extrapolation methods for recovering the missing fingerprint data. For this purpose, we have collected an extensive set of data at frequency bands of 2.4 and 5 GHz from one university building with four floors. The accuracy of the interpolation and extrapolation methods is studied by artificially removing fingerprints from the database using a randomized procedure and by comparing the estimated fingerprints with the original fingerprints. The average RSS estimation error of different interpolation and extrapolation methods is shown for various percentages of missing fingerprints. In addition, a cumulative RSS error distribution is studied to reveal the dispersion of the error statistics, which affect the user positioning accuracy. Here, the user positioning accuracy is defined in terms of horizontal positioning error and floor detection probability. The user positioning accuracy is also compared in four cases, namely when using the original fingerprints, the partial fingerprints, the interpolated fingerprints, and the interpolated and extrapolated fingerprints. It is shown that both the horizontal positioning accuracy and the floor detection probability can be improved with proper interpolation and extrapolation methods. However, it is also illustrated that the best positioning performance is not necessarily achieved with the best average interpolation and extrapolation accuracy, but it is important to avoid certain types of errors in interpolation and extrapolation.

Abstract: Sparse transforms play an important role in seismic signal processing steps, such as prestack noise attenuation and data reconstruction. Analytic sparse transforms (so-called implicit dictionaries), such as the Fourier, Radon, and curvelet transforms, are often used to represent seismic data. There are situations, however, in which the complexity of the data requires adaptive sparse transform methods, whose basis functions are determined via learning methods. We studied an application of the data-driven tight frame (DDTF) method to noise suppression and interpolation of high-dimensional seismic data. Rather than choosing a model beforehand (for example, a family of lines, parabolas, or curvelets) to fit the data, the DDTF derives the model from the data itself in an optimum manner. The process of estimating the basis function from the data can be summarized as follows: First, the input data are divided into small blocks to form training sets. Then, the DDTF algorithm is applied on the training set...

Abstract: In this paper, we investigate a recently introduced approach for nonlinear model order reduction based on generalized moment matching. Using basic tensor calculus, we propose a computationally efficient way of computing reduced-order models. We further extend the idea of two-sided interpolation methods to this more general setting by employing the tensor structure of the Hessian. We investigate the use of oblique projections in order to preserve important system properties such as stability. We test one-sided and two-sided projection methods for different semi-discretized nonlinear partial differential equations and show their competitiveness when compared to proper orthogonal decomposition (POD).

Abstract: S U M M A R Y Interpolation and random noise removal is a pre-requisite for multichannel techniques because the irregularity and random noise in observed data can affect their performances. Projection Onto Convex Sets (POCS) method can better handle seismic data interpolation if the data’s signal-to-noise ratio (SNR) is high, while it has difficulty in noisy situations because it inserts the noisy observed seismic data in each iteration. Weighted POCS method can weaken the noise effects, while the performance is affected by the choice of weight factors and is still unsatisfactory. Thus, a new weighted POCS method is derived through the Iterative Hard Threshold (IHT) view, and in order to eliminate random noise, a new adaptive method is proposed to achieve simultaneous seismic data interpolation and denoising based on dreamlet transform. Performances of the POCS method, the weighted POCS method and the proposed method are compared in simultaneous seismic data interpolation and denoising which demonstrate the validity of the proposed method. The recovered SNRs confirm that the proposed adaptive method is the most effective among the three methods. Numerical examples on synthetic and real data demonstrate the validity of the proposed adaptive method.

Abstract: Many material and biological samples in scientific imaging are characterized by non-local repeating structures. These are studied using scanning electron microscopy and electron tomography. Sparse sampling of individual pixels in a 2D image acquisition geometry, or sparse sampling of projection images with large tilt increments in a tomography experiment, can enable high speed data acquisition and minimize sample damage caused by the electron beam. In this paper, we present an algorithm for electron tomographic reconstruction and sparse image interpolation that exploits the non-local redundancy in images. We adapt a framework, termed plug-and-play (P&P) priors, to solve these imaging problems in a regularized inversion setting. The power of the P&P approach is that it allows a wide array of modern denoising algorithms to be used as a "prior model" for tomography and image interpolation. We also present sufficient mathematical conditions that ensure convergence of the P&P approach, and we use these insights to design a new non-local means denoising algorithm. Finally, we demonstrate that the algorithm produces higher quality reconstructions on both simulated and real electron microscope data, along with improved convergence properties compared to other methods.

Abstract: In this paper we propose a variational method to adaptively decompose an image into few different modes of separate spectral bands, which are unknown before. A popular method for recursive one dimensional signal decomposition is the Empirical Mode Decomposition algorithm, introduced by Huang in the nineties. This algorithm, as well as its 2D extension, though extensively used, suffers from a lack of exact mathematical model, interpolation choice, and sensitivity to both noise and sampling. Other state-of-the-art models include synchrosqueezing, the empirical wavelet transform, and recursive variational decomposition into smooth signals and residuals. Here, we have created an entirely non-recursive 2D variational mode decomposition (2D-VMD) model, where the modes are extracted concurrently. The model looks for a number of 2D modes and their respective center frequencies, such that the bandlimited modes reproduce the input image (exactly or in least-squares sense). Preliminary results show excellent performance on both synthetic and real images. Running this algorithm on a peptide microscopy image yields accurate, timely, and autonomous segmentation - pertinent in the fields of biochemistry and nanoscience.

Abstract: Recently, phase-field approaches have gained popularity as a versatile tool for simulating fracture in a smeared manner. In this paper we give a numerical assessment of two types of phase-field models. For the case of brittle fracture we focus on the question whether the functional that describes the smeared crack surface approaches the functional for the discrete crack in the limiting case that the internal length scale parameter vanishes. By a one-dimensional example we will show that Γ-convergence is not necessarily attained numerically. Next, we turn attention to cohesive fracture. The necessity to have the crack opening explicitly available as input for the cohesive traction-relative displacement relation requires the independent interpolation of this quantity. The resulting three-field problem can be solved accurately on structured meshes when using a balanced interpolation of the field variables: displacements, phase field, and crack opening. A simple patch test shows that this observation does not necessarily extend to unstructured meshes.

Abstract: We propose a method to design linear deformation subspaces, unifying linear blend skinning and generalized barycentric coordinates. Deformation subspaces cut down the time complexity of variational shape deformation methods and physics-based animation (reduced-order physics). Our subspaces feature many desirable properties: interpolation, smoothness, shape-awareness, locality, and both constant and linear precision. We achieve these by minimizing a quadratic deformation energy, built via a discrete Laplacian inducing linear precision on the domain boundary. Our main advantage is speed: subspace bases are solutions to a sparse linear system, computed interactively even for generously tessellated domains. Users may seamlessly switch between applying transformations at handles and editing the subspace by adding, removing or relocating control handles. The combination of fast computation and good properties means that designing the right subspace is now just as creative as manipulating handles. This paradigm shift in handle-based deformation opens new opportunities to explore the space of shape deformations.

Abstract: Spatial climate datasets of 1981-2010 long-term mean monthly average dew point and minimum and maximum vapor pressure deficit were developed for the conterminous United States at 30-arcsec (~800m) resolution. Interpolation of long-term averages (twelve monthly values per variable) was performed using PRISM (Parameter-elevation Relationships on Independent Slopes Model). Surface stations available for analysis numbered only 4,000 for dew point and 3,500 for vapor pressure deficit, compared to 16,000 for previously-developed grids of 1981-2010 long-term mean monthly minimum and maximum temperature. Therefore, a form of Climatologically-Aided Interpolation (CAI) was used, in which the 1981-2010 temperature grids were used as predictor grids. For each grid cell, PRISM calculated a local regression function between the interpolated climate variable and the predictor grid. Nearby stations entering the regression were assigned weights based on the physiographic similarity of the station to the grid cell that included the effects of distance, elevation, coastal proximity, vertical atmospheric layer, and topographic position. Interpolation uncertainties were estimated using cross-validation exercises. Given that CAI interpolation was used, a new method was developed to allow uncertainties in predictor grids to be accounted for in estimating the total interpolation error. Local land use/land cover properties had noticeable effects on the spatial patterns of atmospheric moisture content and deficit. An example of this was relatively high dew points and low vapor pressure deficits at stations located in or near irrigated fields. The new grids, in combination with existing temperature grids, enable the user to derive a full suite of atmospheric moisture variables, such as minimum and maximum relative humidity, vapor pressure, and dew point depression, with accompanying assumptions. All of these grids are available online at http://prism.oregonstate.edu, and include 800-m and 4-km resolution data, images, metadata, pedigree information, and station inventory files.

Abstract: For the simulation of fractured porous media, a common approach is the use of co-dimension one models for the fracture description. In order to simulate correctly the behavior at fracture crossings, standard models are not sufficient because they either cannot capture all important flow processes or are computationally inefficient. We propose a new concept to simulate co-dimension one fracture crossings and show its necessity and accuracy by means of an example and a comparison to a literature benchmark. From the application point of view, often the pressure is known only at a limited number of discrete points and an interpolation is used to define the boundary condition at the remaining parts of the boundary. The quality of the interpolation, especially in fracture models, influences the global solution significantly. We propose a new method to interpolate boundary conditions on boundaries that are intersected by fractures and show the advantages over standard interpolation methods.

Abstract: G01 codes generated by CAM (Computer Aided Manufacturing) system are the most common form of tool path in CNC (Computer Numerical Control) machining. For the piecewise linear path, tangential and curvature discontinuities bring about large fluctuation of feedrate and acceleration, which produces vibration of machine tool. In recent studies, the methods for G 2 (curvature-continuous) tool-path smoothing and jerk-limiting feedrate scheduling were developed. However there still exist the deficiencies when these methods are employed in CNC machining. It is difficult to simultaneously ensure that the tool path is chord-error-constrained and G01-point-interpolated in real time. In addition, heavy computational load hinders realtime processing in CNC system. Recently the scholars experimentally found the potential of G 3 (curvature-smooth) trajectory and jerk-continuous motion in reducing the vibration of machinery. This work proposes a realtime tool-path smoothing algorithm, generating G 3 interpolative tool path composed by mixed linear and quartic Bezier segments. The purpose of the smoothing scheme is the simultaneous considerations of G 3 continuity, confined chord error, G01 points interpolated, and realtime performance. And the tool path generated is optimized in curvature variation energy (CVE) and analytical curvature extrema is available. To reduce the vibration, a high-efficient algorithm of jerk-continuous (JC) feedrate scheduling for G 3 tool path is provided. Finally, a realtime tool-path processing scheme is developed, including G 3 interpolation and motion planning functions. As shown in the simulation, the contour error, curvature of tool path, feedrate fluctuation and machining time are reduced compared with G 2 transition scheme. The experiment on a machine tool is conducted to demonstrate the advantages of the proposed algorithm in vibration reduction and surface quality, compared with G 2 transition scheme.

Abstract: Quantifying the similarity between two trajectories is a fundamental operation in analysis of spatio-temporal databases. While a number of distance functions exist, the recent shift in the dynamics of the trajectory generation procedure violates one of their core assumptions; a consistent and uniform sampling rate. In this paper, we formulate a robust distance function called Edit Distance with Projections (EDwP) to match trajectories under inconsistent and variable sampling rates through dynamic interpolation. This is achieved by deploying the idea of projections that goes beyond matching only the sampled points while aligning trajectories. To enable efficient trajectory retrievals using EDwP, we design an index structure called TrajTree. TrajTree derives its pruning power by employing the unique combination of bounding boxes with Lipschitz embedding. Extensive experiments on real trajectory databases demonstrate EDwP to be up to 5 times more accurate than the state-of-the-art distance functions. Additionally, TrajTree increases the efficiency of trajectory retrievals by up to an order of magnitude over existing techniques.

Abstract: This paper shows how to carry out batch continuous-time trajectory estimation for bodies translating and rotating in three-dimensional (3D) space, using a very efficient form of Gaussian-process (GP) regression. The method is fast, singularity-free, uses a physically motivated prior (the mean is constant body-centric velocity), and permits trajectory queries at arbitrary times through GP interpolation. Landmark estimation can be folded in to allow for simultaneous trajectory estimation and mapping (STEAM), a variant of SLAM.

Abstract: We show how to enhance the performance of a Smolyak method for solving dynamic economic models. First, we propose a more efficient implementation of the Smolyak method for interpolation, namely, we show how to avoid costly evaluations of repeated basis functions in the conventional Smolyak formula. Second, we extend the Smolyak method to include anisotropic constructions that allow us to target higher quality of approximation in some dimensions than in others. Third, we show how to effectively adapt the Smolyak hypercube to a solution domain of a given economic model. Finally, we argue that in large-scale economic applications, a solution algorithm based on Smolyak interpolation has substantially lower expense when it uses derivative-free fixed-point iteration instead of standard time iteration. In the context of one- and multi-agent optimal growth models, we find that the proposed modifications to the conventional Smolyak method lead to substantial increases in accuracy and speed.(This abstract was borrowed from another version of this item.)

Abstract: In this paper, we study a new class of fractional partial differential equations which are obtained by minimizing variational problems in fractional Sobolev spaces. We introduce a notion of fractional gradient which has the potential to extend many classical results in the Sobolev spaces to the nonlocal and fractional setting in a natural way.