Natural Interpolation Research Articles

Estimation of Noisy Data: The Case of Partially Missing Information

We wish to estimate each random vector xω(t) in the set Kx from the corresponding large set of noisy observations. The conceptual foundation of the proposed filter is an optimal least squares linear estimate of the incremental change to the p signal pairs, extended by a natural interpolation to an estimated value of each reference signal.

Lobatto IIIA methods, direct transcription, and DAEs with delays

Recent results in the literature have shown that LobattoIIIA methods may have some undesirable stability properties when used to numerically integrate differential algebraic equations (DAEs) which have delays. In these studies Lagrange interpolation was used to compute the delayed terms. Here we show that this undesirable behavior of LobattoIIIA methods reported in the literature is tied to the use of Lagrange interpolation and their use in an ODE integrator. With these and other natural interpolations some LobattoIIIA methods do not exhibit this undesirable behavior when used within a direct transcription framework.

Combined DG--CG Time Stepping for Wave Equations

The continuous and discontinuous Galerkin time stepping methodologies are combined to develop approximations of second order time derivatives of arbitrary order. This eliminates the doubling of the number of variables that results when a second order problem is written as a first order system. Stability, convergence, and accuracy, of these schemes is established in the context of the wave equation. It is shown that natural interpolation of nonhomogeneous boundary data can degrade accuracy, and that this problem can be circumvented using interpolants matched with the time stepping scheme.

Local and Nonlocal Regularization to Image Interpolation

This paper presents an image interpolation model with local and nonlocal regularization. A nonlocal bounded variation (BV) regularizer is formulated by an exponential function including gradient. It acts as the Perona-Malik equation. Thus our nonlocal BV regularizer possesses the properties of the anisotropic diffusion equation and nonlocal functional. The local total variation (TV) regularizer dissipates image energy along the orthogonal direction to the gradient to avoid blurring image edges. The derived model efficiently reconstructs the real image, leading to a natural interpolation which reduces blurring and staircase artifacts. We present experimental results that prove the potential and efficacy of the method.

Volumetric three-dimensional intravascular ultrasound visualization using shape-based nonlinear interpolation

BackgroundIntravascular ultrasound (IVUS) is a standard imaging modality for identification of plaque formation in the coronary and peripheral arteries. Volumetric three-dimensional (3D) IVUS visualization provides a powerful tool to overcome the limited comprehensive information of 2D IVUS in terms of complex spatial distribution of arterial morphology and acoustic backscatter information. Conventional 3D IVUS techniques provide sub-optimal visualization of arterial morphology or lack acoustic information concerning arterial structure due in part to low quality of image data and the use of pixel-based IVUS image reconstruction algorithms. In the present study, we describe a novel volumetric 3D IVUS reconstruction algorithm to utilize IVUS signal data and a shape-based nonlinear interpolation.MethodsWe developed an algorithm to convert a series of IVUS signal data into a fully volumetric 3D visualization. Intermediary slices between original 2D IVUS slices were generated utilizing the natural cubic spline interpolation to consider the nonlinearity of both vascular structure geometry and acoustic backscatter in the arterial wall. We evaluated differences in image quality between the conventional pixel-based interpolation and the shape-based nonlinear interpolation methods using both virtual vascular phantom data and in vivo IVUS data of a porcine femoral artery. Volumetric 3D IVUS images of the arterial segment reconstructed using the two interpolation methods were compared.ResultsIn vitro validation and in vivo comparative studies with the conventional pixel-based interpolation method demonstrated more robustness of the shape-based nonlinear interpolation algorithm in determining intermediary 2D IVUS slices. Our shape-based nonlinear interpolation demonstrated improved volumetric 3D visualization of the in vivo arterial structure and more realistic acoustic backscatter distribution compared to the conventional pixel-based interpolation method.ConclusionsThis novel 3D IVUS visualization strategy has the potential to improve ultrasound imaging of vascular structure information, particularly atheroma determination. Improved volumetric 3D visualization with accurate acoustic backscatter information can help with ultrasound molecular imaging of atheroma component distribution.

A Bidirectional Flow Joint Sobolev Gradient for Image Interpolation

An energy functional with bidirectional flow is presented to sharpen image by reducing its edge width, which performs a forward diffusion in brighter lateral on edge ramp and backward diffusion that proceeds in darker lateral. We first consider the diffusion equations asL2gradient flows on integral functionals and then modify the inner product fromL2to a Sobolev inner product. The experimental results demonstrate that our model efficiently reconstructs the real image, leading to a natural interpolation with reduced blurring, staircase artifacts and preserving better the texture features of image.

Optimum 2-D Nonuniform Spatial Sampling for Microwave SAR-Based NDE Imaging Systems

Microwave and millimeter-wave synthetic aperture radar (SAR)-based imaging techniques, which are used for nondestructive evaluation (NDE), have shown tremendous usefulness for the inspection of a wide variety of complex composite materials and structures. An important practical issue associated with these imaging techniques is the required criteria associated with the physical gathering of the imaging data. In previous work on uniform sampling optimization, it was shown that the (uniform) spatial sampling density should be higher than the Nyquist density if preservation of spatial resolution, as defined by the half-power width of an ideal point target, is of interest. Conversely, nonuniform sampling has shown to provide effective signal reconstruction even for average spatial sampling densities below the Nyquist density-a distinct advantage over uniform sampling. This paper presents a comprehensive study into the optimization of nonuniform sampling for microwave SAR-based NDE imaging using three typical reconstruction techniques for nonuniformly sampled data, including natural interpolation, area-weighted Fourier integration, and conjugate gradient residual error minimization methods. To study the efficacy of these reconstruction methods, simulations of a point target were performed for a range of target distances and a range of average spatial sample separations. Resulting SAR images from the reconstruction techniques are then analyzed and compared according to two metrics: error between an image and an ideal image and resolution as determined by the half-power width of a point target. This is followed by experimental results corroborating the simulation results. Finally, nonuniform sampling requirements, given a minimum metric performance, are generalized for a given imager aperture size.

Combination of Newmark method and natural element method for elastodynamics

The natural element method (NEM) is a meshless method. The trial and test functions of the NEM are constructed using natural neighbor interpolations which are based on the Voronoi tessellation of a set of nodes. The NEM interpolation is linear between adjacent nodes on the boundary of the convex hull, which makes imposition of essential boundary conditions easy to implement. We investigate the performance of the NEM combined with the Newmark method for problems of elastodynamics in this article. Applications are considered for a cantilever beam with different initial load conditions. The NEM numerical results are compared with the finite element method. NEM shows promise for these applications.

Variational implementation of immersed finite element methods

Dirac-δ distributions are often crucial components of the solid–fluid coupling operators in immersed solution methods for fluid–structure interaction (FSI) problems. This is certainly so for methods like the immersed boundary method (IBM) or the immersed finite element method (IFEM), where Dirac-δ distributions are approximated via smooth functions. By contrast, a truly variational formulation of immersed methods does not require the use of Dirac-δ distributions, either formally or practically. This has been shown in the finite element immersed boundary method (FEIBM), where the variational structure of the problem is exploited to avoid Dirac-δ distributions at both the continuous and the discrete level.In this paper, we generalize the FEIBM to the case where an incompressible Newtonian fluid interacts with a general hyperelastic solid. Specifically, we allow (i) the mass density to be different in the solid and the fluid, (ii) the solid to be either viscoelastic of differential type or purely elastic, and (iii) the solid to be either compressible or incompressible. At the continuous level, our variational formulation combines the natural stability estimates of the fluid and elasticity problems. In immersed methods, such stability estimates do not transfer to the discrete level automatically due to the non-matching nature of the finite dimensional spaces involved in the discretization. After presenting our general mathematical framework for the solution of FSI problems, we focus in detail on the construction of natural interpolation operators between the fluid and the solid discrete spaces, which guarantee semi-discrete stability estimates and strong consistency of our spatial discretization.

Freeform vector graphics with controlled thin-plate splines

Recent work defines vector graphics using diffusion between colored curves. We explore higher-order fairing to enable more natural interpolation and greater expressive control. Specifically, we build on thin-plate splines which provide smoothness everywhere except at user-specified tears and creases (discontinuities in value and derivative respectively). Our system lets a user sketch discontinuity curves without fixing their colors, and sprinkle color constraints at sparse interior points to obtain smooth interpolation subject to the outlines. We refine the representation with novel contour and slope curves, which anisotropically constrain interpolation derivatives. Compound curves further increase editing power by expanding a single curve into multiple offsets of various basic types (value, tear, crease, slope, and contour). The vector constraints are discretized over an image grid, and satisfied using a hierarchical solver. We demonstrate interactive authoring on a desktop CPU.

Tri-cubic polynomial natural spline interpolation for scattered data

In this paper an interpolation problem for 3D scattered data defined on a rectangular parallelepiped with natural boundary conditions is considered. By using spline function theory in Hilbert space...

A remark on supercloseness and extrapolation of the quadrilateral Han element for the Stokes equations

We analyze the supercloseness properties of the nonconforming quadrilateral Han finite element for the Stokes equations. It is shown that the difference between the discrete solution and the natural interpolation of the continuous solution does not have the supercloseness property. Based on this analysis, we propose a modified interpolation operator which allows for such a result. It is then used to obtain a simple third-order extrapolation scheme.

Interpolating sequences for analytic selfmappings of the disc

Schwarz's Lemma leads to a natural interpolation problem for analytic functions from the disc into itself. The corresponding interpolating sequences are geometrically described in terms of a certain hyperbolic density.

Modelling tensile properties of jute fibres

This short communication extends earlier modelling of the tensile strength and failure strain of jute technical fibres. A maximum likelihood estimate (MLE) model, a linear model and a natural logarithmic interpolation model (NLIM) are compared. The NLIM model is found to give superior predictions.

The Nonlocal p‐Laplacian Evolution for Image Interpolation

This paper presents an image interpolation model with nonlocal p‐Laplacian regularization. The nonlocal p‐Laplacian regularization overcomes the drawback of the partial differential equation (PDE) proposed by Belahmidi and Guichard (2004) that image density diffuses in the directions pointed by local gradient. The grey values of images diffuse along image feature direction not gradient direction under the control of the proposed model, that is, minimal smoothing in the directions across the image features and maximal smoothing in the directions along the image features. The total regularizer combines the advantages of nonlocal p‐Laplacian regularization and total variation (TV) regularization (preserving discontinuities and 1D image structures). The derived model efficiently reconstructs the real image, leading to a natural interpolation, with reduced blurring and staircase artifacts. We present experimental results that prove the potential and efficacy of the method.

The natural neighbour Petrov–Galerkin method for thick plates

In this paper, a meshless natural neighbour Petrov–Galerkin method (NNPG) is presented for a plate described by the Mindlin theory. The discrete model of the domain Ω consists of a set of distinct nodes N, and a polygonal description of the boundary. In the NNPG, the trial functions on a local domain are constructed using natural neighbour interpolation and the three-node triangular FEM shape functions are taken as test functions. The natural neighbour interpolants are strictly linear between adjacent nodes on the boundary of the convex hull, which facilitate imposition of essential boundary conditions. The local weak forms of the equilibrium equations and the boundary conditions are satisfied in local polygonal sub-domains in the mean surface of the plate. These sub-domains are constructed with Delaunay tessellations and domain integrals are evaluated over included Delaunay triangles by using Gaussian quadrature scheme. Both elasto-static and dynamic problems are considered. The numerical results show the presented method is easy to implement and very accurate for these problems.

Clustering with shallow trees

We propose a new method for obtaining hierarchical clustering based on the optimizationof a cost function over trees of limited depth, and we derive a message-passing method thatallows one to use it efficiently. The method and the associated algorithm canbe interpreted as a natural interpolation between two well-known approaches,namely that of single linkage and the recently presented affinity propagation. Weanalyse using this general scheme three biological/medical structured data sets(human population based on genetic information, proteins based on sequencesand verbal autopsies) and show that the interpolation technique provides newinsight.

Tensile properties of jute fibres

One hundred tensile tests were undertaken at each of five distinct fibre lengths (6, 10, 20, 30 and 50 mm) on a single batch of jute fibres from South Asia. The Young's moduli were found to be independent of length. The ultimate stress (fracture strength) and fracture strains were found to decrease with increasing fibre length. The variation in mechanical properties at each fibre length was characterised using Weibull statistics based on a maximum likelihood estimate; referred to as point estimates. Two empirical based models (a linear and a natural logarithmic interpolation model) have been developed to estimate the fracture properties at any length between 6 and 50 mm. These two interpolation models were also developed based on maximum likelihood estimates. The point estimates were used to benchmark the performance of the two models. The natural logarithmic model was found to be superior to the linear model.

Electronic and magnetic properties ofRMnO3/AMnO3heterostructures

The ground-state properties of $R{\text{MnO}}_{3}/A{\text{MnO}}_{3}$ (RMO/AMO) heterostructures (with $R=\text{La},\text{Pr},\dots{}$, a trivalent rare-earth cation and $A=\text{Sr},\text{Ca},\dots{}$, a divalent alkaline cation) are studied using a two-orbital double-exchange model including the superexchange coupling and Jahn-Teller lattice distortions. To describe the charge transfer across the interface, the long-range Coulomb interaction is taken into account at the mean-field level, by self-consistently solving the Poisson's equation. The calculations are carried out numerically on finite clusters. We find that the state stabilized near the interface of the heterostructure is similar to the state of the bulk compound $(R,A)\text{MO}$ at electronic density close to 0.5. For instance, a charge and orbitally ordered CE state is found at the interface if the corresponding bulk $(R,A)\text{MO}$ material is a narrow-to-intermediate bandwidth manganite. But instead the interface regime accommodates an A-type antiferromagnetic state with a uniform ${x}^{2}\ensuremath{-}{y}^{2}$ orbital order, if the bulk $(R,A)\text{MO}$ corresponds to a wide bandwidth manganite. We argue that these results explain some of the properties of long-period ${(R\text{MO})}_{m}/{(A\text{MO})}_{n}$ superlattices, such as ${({\text{PrMnO}}_{3})}_{m}/{({\text{CaMnO}}_{3})}_{n}$ and ${({\text{LaMnO}}_{3})}_{m}/{({\text{SrMnO}}_{3})}_{n}$. We also remark that the intermediate states in between the actual interface and the bulklike regimes of the heterostructure are dependent on the bandwidth and the screening of the Coulomb interaction. In these regions of the heterostructures, states are found that do not have an analog in experimentally known bulk phase diagrams. These new states of the heterostructures provide a natural interpolation between magnetically ordered states that are stable in the bulk at different electronic densities.

Bipolarization of posets and natural interpolation

The Choquet integral w.r.t. a capacity can be seen in the finite case as a parsimonious linear interpolator between vertices of [ 0 , 1 ] n . We take this basic fact as a starting point to define the Choquet integral in a very general way, using the geometric realization of lattices and their natural triangulation, as in the work of Koshevoy. A second aim of the paper is to define a general mechanism for the bipolarization of ordered structures. Bisets (or signed sets), as well as bisubmodular functions, bicapacities, bicooperative games, as well as the Choquet integral defined for them can be seen as particular instances of this scheme. Lastly, an application to multicriteria aggregation with multiple reference levels illustrates all the results presented in the paper.