Classical Interpolation Methods Research Articles

A robust interpolation method for constructing digital elevation models from remote sensing data

A digital elevation model (DEM) derived from remote sensing data often suffers from outliers due to various reasons such as the physical limitation of sensors and low contrast of terrain textures. In order to reduce the effect of outliers on DEM construction, a robust algorithm of multiquadric (MQ) methodology based on M-estimators (MQ-M) was proposed. MQ-M adopts an adaptive weight function with three-parts. The weight function is null for large errors, one for small errors and quadric for others. A mathematical surface was employed to comparatively analyze the robustness of MQ-M, and its performance was compared with those of the classical MQ and a recently developed robust MQ method based on least absolute deviation (MQ-L). Numerical tests show that MQ-M is comparative to the classical MQ and superior to MQ-L when sample points follow normal and Laplace distributions, and under the presence of outliers the former is more accurate than the latter. A real-world example of DEM construction using stereo images indicates that compared with the classical interpolation methods, such as natural neighbor (NN), ordinary kriging (OK), ANUDEM, MQ-L and MQ, MQ-M has a better ability of preserving subtle terrain features. MQ-M replaces thin plate spline for reference DEM construction to assess the contribution to our recently developed multiresolution hierarchical classification method (MHC). Classifying the 15 groups of benchmark datasets provided by the ISPRS Commission demonstrates that MQ-M-based MHC is more accurate than MQ-L-based and TPS-based MHCs. MQ-M has high potential for DEM construction.

CONVOLUTIONAL NEURAL NETWORK BASED DEM SUPER RESOLUTION

Abstract. DEM super resolution is proposed in our previous publication to improve the resolution for a DEM on basis of some learning examples. Meanwhile, the nonlocal algorithm is introduced to deal with it and lots of experiments show that the strategy is feasible. In our publication, the learning examples are defined as the partial original DEM and their related high measurements due to this way can avoid the incompatibility between the data to be processed and the learning examples. To further extent the applications of this new strategy, the learning examples should be diverse and easy to obtain. Yet, it may cause the problem of incompatibility and unrobustness. To overcome it, we intend to investigate a convolutional neural network based method. The input of the convolutional neural network is a low resolution DEM and the output is expected to be its high resolution one. A three layers model will be adopted. The first layer is used to detect some features from the input, the second integrates the detected features to some compressed ones and the final step transforms the compressed features as a new DEM. According to this designed structure, some learning DEMs will be taken to train it. Specifically, the designed network will be optimized by minimizing the error of the output and its expected high resolution DEM. In practical applications, a testing DEM will be input to the convolutional neural network and a super resolution will be obtained. Many experiments show that the CNN based method can obtain better reconstructions than many classic interpolation methods.

CONVOLUTIONAL NEURAL NETWORK BASED DEM SUPER RESOLUTION

DEM super resolution is proposed in our previous publication to improve the resolution for a DEM on basis of some learning examples. Meanwhile, the nonlocal algorithm is introduced to deal with it and lots of experiments show that the strategy is feasible. In our publication, the learning examples are defined as the partial original DEM and their related high measurements due to this way can avoid the incompatibility between the data to be processed and the learning examples. To further extent the applications of this new strategy, the learning examples should be diverse and easy to obtain. Yet, it may cause the problem of incompatibility and unrobustness. To overcome it, we intend to investigate a convolutional neural network based method. The input of the convolutional neural network is a low resolution DEM and the output is expected to be its high resolution one. A three layers model will be adopted. The first layer is used to detect some features from the input, the second integrates the detected features to some compressed ones and the final step transforms the compressed features as a new DEM. According to this designed structure, some learning DEMs will be taken to train it. Specifically, the designed network will be optimized by minimizing the error of the output and its expected high resolution DEM. In practical applications, a testing DEM will be input to the convolutional neural network and a super resolution will be obtained. Many experiments show that the CNN based method can obtain better reconstructions than many classic interpolation methods.

Cuckoo search-designated fractal interpolation functions with winner combination for estimating missing values in time series

Reliable data have always played a vital role in time series analysis and research. Nevertheless, missing data, which can bias the original properties of the time series if the pattern of missed data is systematic, is also a very common phenomenon in observed processes. Thus, how to address missing values is a very important challenge, especially in the upcoming big data era. This paper proposes the cuckoo search-designated fractal interpolation functions (CS-DFIFs) method and the CS-DFIFs-winners combining (CS-DFIFs-WC) method for estimating missing values. The former method skillfully transforms Fractal Interpolation Functions (FIFs) to make it possible to calculate a specified point's missing value, which is difficult to obtain with the traditional approach. Then, to optimize the parameters that transforming process generates, the cuckoo search algorithm (CS) and DFIFs are synthesized into a novel model, CS-DFIFs. Considering classical interpolation methods, such as Linear, Cubic Spline and Piecewise Cubic Hermite Interpolation Polynomial (PCHIP), having some born advantages, the inspiration of winners combining is sparked. CS algorithm is used to obtain the best weights of winners in this combined model, CS-DFIFs-WC. Two databases, electricity demands and prices, are chosen to be the numerical testing object at 7 missing levels in this paper. The results show that CS-DFIFs-WC overcomes the deficiency of FIFs that cannot calculate a specified point's missing value, and outperforms benchmarks at almost every level by four criteria.

A total error-based multiquadric method for surface modeling of digital elevation models

Digital elevation model (DEM) source data are subject to both horizontal and vertical errors owing to improper instrument operation, physical limitations of sensors, and bad weather conditions. These factors may bring a negative effect on some DEM-based applications requiring low levels of positional errors. Although classical smoothing interpolation methods have the ability to handle vertical errors, they are prone to omit horizontal errors. Based on the statistical concept of the total least squares method, a total error-based multiquadric (MQ-T) method is proposed in this paper to reduce the effects of both horizontal and vertical errors in the context of DEM construction. In nature, the classical multiquadric (MQ) method is a vertical error regression procedure, whereas MQ-T is an orthogonal error regression model. Two examples, including a numerical test and a real-world example, are employed in a comparative performance analysis of MQ-T for surface modeling of DEMs. The numerical test indicates that MQ-T performs better than the classical MQ in terms of root mean square error. The real-world example of DEM construction with sample points derived from a total station instrument demonstrates that regardless of the sample interval and DEM resolution, MQ-T is more accurate than classical interpolation methods including inverse distance weighting, ordinary kriging, and Australian National University DEM. Therefore, MQ-T can be considered as an alternative interpolator for surface modeling with sample points subject to both horizontal and vertical errors.

A step by step recovery of spectral data from colorimetric information

In this paper, a technique named step by step progressive method is devised for recovery of spectral data from the corresponding colorimetric information. The method consists of several stages and employs different techniques, each of which is used to enhance the performance of the recovery as much as possible. The method firstly tests a series of 6 to 3 dimensional linear interpolation techniques, successively; however, there are some samples lying outside the interpolation color gamut, which is the weakest point of the interpolation method, limiting the performance of the first solution. To overcome this problem, a modified version of weighted principal component analysis (wPCA) is employed for the spectral recovery of the samples that have been failed with linear interpolation approach. The employed weighting function takes the spectral distances of the target data and the learning set into accounts. The efficiency of the method is then evaluated by the spectral and colorimetric deviations between the actual and recovered reflectance spectra and is compared with those obtained by classical 3D linear interpolation and PCA methods. As results show that the proposed progressive approach achieves significant improvement in terms of both spectral and colorimetric errors in comparison to the classical methods.

Metamodel for Mathematical Modelling Surfaces of Celestial Bodies on the Base of Radiolocation Data

The paper proposes for the mathematical modelling surfaces of celestial bodies by the given radar data use the models, produced from the geometrical metamodel G. The metamodel G consists of the corresponding to the dimensions of a space geometrical objects (points, lines, surfaces), the mathematical methods of interpolation, interlinations, interflatation, and the set of rules for producing mathematical models. Using the metamodelling approach allows us to consider from a unique point of view the different methods for the modelling surfaces of bodies and also to develop a computer tool which accelerates and simplify the modelling process, starting from the problem specification and finishing visualization and interpretation of the obtained solution. The correctness of the derived from the metamodel G the set of the models M1, M2 ... MN results from the correspondence of the structure of the model objects of G to the structure of experimental data, recorded in the process of radiolocation as values of the functions in the points, and traces of the functions on the given lines and surfaces. Widely used in cartography Digital Elevation Model (DEM), can be also produced from G by using for the description of the bodies the surfaces in the form of triangles. Using the metamodel G allows us to integrate the DEM method for the data specification with setting data in points and on lines (which are the basic objects of the metamodel G) and so apply more precise (comparatively with the classical interpolation) methods of interlination and interflatation of functions. Another advantage of the proposed metamodelling approach is a possibility of development of complex geometrical models by composition of the basic elements of the metamodel. As an example, the paper proposes a new method for the reconstruction of the surface of a celestial body by the data, given on the system of strips – interstripation (form the inter – in between – of the strips).

A Method of DTM Construction Based on Quadrangular Irregular Networks and Related Error Analysis.

A new method of DTM construction based on quadrangular irregular networks (QINs) that considers all the original data points and has a topological matrix is presented. A numerical test and a real-world example are used to comparatively analyse the accuracy of QINs against classical interpolation methods and other DTM representation methods, including SPLINE, KRIGING and triangulated irregular networks (TINs). The numerical test finds that the QIN method is the second-most accurate of the four methods. In the real-world example, DTMs are constructed using QINs and the three classical interpolation methods. The results indicate that the QIN method is the most accurate method tested. The difference in accuracy rank seems to be caused by the locations of the data points sampled. Although the QIN method has drawbacks, it is an alternative method for DTM construction.

A Robust Algorithm of Multiquadric Method Based on an Improved Huber Loss Function for Interpolating Remote-Sensing-Derived Elevation Data Sets

Remote-sensing-derived elevation data sets often suffer from noise and outliers due to various reasons, such as the physical limitations of sensors, multiple reflectance, occlusions and low contrast of texture. Outliers generally have a seriously negative effect on DEM construction. Some interpolation methods like ordinary kriging (OK) are capable of smoothing noise inherent in sample points, but are sensitive to outliers. In this paper, a robust algorithm of multiquadric method (MQ) based on an Improved Huber loss function (MQ-IH) has been developed to decrease the impact of outliers on DEM construction. Theoretically, the improved Huber loss function is null for outliers, quadratic for small errors, and linear for others. Simulated data sets drawn from a mathematical surface with different error distributions were employed to analyze the robustness of MQ-IH. Results indicate that MQ-IH obtains a good balance between efficiency and robustness. Namely, the performance of MQ-IH is comparative to those of the classical MQ and MQ based on the Classical Huber loss function (MQ-CH) when sample points follow a normal distribution, and the former outperforms the latter two when sample points are subject to outliers. For example, for the Cauchy error distribution with the location parameter of 0 and scale parameter of 1, the root mean square errors (RMSEs) of MQ-CH and the classical MQ are 0.3916 and 1.4591, respectively, whereas that of MQ-IH is 0.3698. The performance of MQ-IH is further evaluated by qualitative and quantitative analysis through a real-world example of DEM construction with the stereo-images-derived elevation points. Results demonstrate that compared with the classical interpolation methods, including natural neighbor (NN), OK and ANUDEM (a program that calculates regular grid digital elevation models (DEMs) with sensible shape and drainage structure from arbitrarily large topographic data sets), and two versions of MQ, including the classical MQ and MQ-CH, MQ-IH has a better ability of maximally reducing the impact of outliers, while faithfully preserving terrain features. Theoretically, MQ-IH is not a promising interpolation method, and some side effects can be found from its simulation results. For example, the contours of MQ-IH are coarser than ANUDEM in some locations of the real-world study site, and its hill shade may not strictly agree with the real-world surface at rough terrain. Furthermore, the computing cost of MQ-IH is much bigger than that of the classical interpolation methods. However, compared with the classical interpolation methods, MQ-IH has significant potential for interpolating remote-sensing-derived elevation data sets.

Metric tensor recovery for adaptive meshing

Adaptive computation is now recognized as essential for solving complex PDE problems. Conceptually, such a computation requires at each step the definition of a continuous metric field (mesh size and direction) to govern the generation of adapted meshes. In practice, in the adaptive computation, an appropriate a posteriori error estimation is used and an upper-bounding of the error is expressed in terms of discrete metrics associated with the element vertices. In order to obtain a continuous metric field, the discrete field is recovered in the whole domain mesh using an appropriate interpolation method on each element. In this paper, a new method for interpolating discrete metric fields, based on a so-called “natural decomposition” of metrics, is introduced. The proposed method uses known matrix decompositions and is computationally robust and efficient. Classical interpolation methods are recalled and, from numerical examples on simplicial mesh elements, some qualitative comparisons against the new methodology are made to show its relevance.

Parallel algorithm of a modified surface modeling method and its application in digital elevation model construction

High accuracy surface modeling (HASM) has been proved to be a superior method for surface simulation compared to classical interpolation methods. However, the fact that HASM is time consuming combined with its dependence on its driving field restricts its application in large area problems. This research develops a modified HASM which can get rid of the driving field in the surface simulation, and the parallel version of the modified HASM is also proposed with the purpose of improving its computational efficiency. Light detection and ranging (LIDAR) data are used as an optimum constraint to construct digital elevation model (DEM). Tests show that the modified HASM can perform surface simulation successfully without the driving field. And it also shows that the simulation accuracy of the modified HASM is almost the same as the old HASM and the classical interpolation methods when the sampling rate is larger than 0.5 %, while the modified HASM shows significantly increased simulation accuracy as the sampling rate decreases. This characteristic indicates that the modified HASM no longer relies on the driving field in the surface simulation. And it also improves the simulation accuracy compared to the old HASM and the classical interpolation methods. Tests of parallel efficiency show that the master–slave mode used in the parallel algorithm obtains a satisfactory result, indicating that the HASM can be applied to surface simulation of large area problems. And it also shows that the modified HASM would have great potential where applied in high-resolution DEM and digital surface model (DSM) construction from LIDAR data.

Interpolation via weighted ℓ1 minimization

Functions of interest are often smooth and sparse in some sense, and both priors should be taken into account when interpolating sampled data. Classical linear interpolation methods are effective under strong regularity assumptions, but cannot incorporate nonlinear sparsity structure. At the same time, nonlinear methods such as ℓ1 minimization can reconstruct sparse functions from very few samples, but do not necessarily encourage smoothness. Here we show that weightedℓ1 minimization effectively merges the two approaches, promoting both sparsity and smoothness in reconstruction. More precisely, we provide specific choices of weights in the ℓ1 objective to achieve approximation rates for functions with coefficient sequences in weighted ℓp spaces with p≤1. We consider implications of these results for spherical harmonic and polynomial interpolation, in the univariate and multivariate setting. Along the way, we extend concepts from compressive sensing such as the restricted isometry property and null space property to accommodate weighted sparse expansions; these developments should be of independent interest in the study of structured sparse approximations and continuous-time compressive sensing problems.

The Generalized Empirical Interpolation Method: Stability theory on Hilbert spaces with an application to the Stokes equation

The Generalized Empirical Interpolation Method (GEIM) is an extension first presented by Maday and Mula in [1] in 2013 of the classical empirical interpolation method (presented in 2004 by Barrault, Maday, Nguyen and Patera in [2]) where the evaluation at interpolating points is replaced by the more practical evaluation at interpolating continuous linear functionals on a class of Banach spaces. As outlined in [1], this allows to relax the continuity constraint in the target functions and expand both the application domain and the stability of the approach. In this paper, we present a thorough analysis of the concept of stability condition of the generalized interpolant (the Lebesgue constant) by relating it to an inf-sup problem in the case of Hilbert spaces. In the second part of the paper, it will be explained how GEIM can be employed to monitor in real time physical experiments by providing an online accurate approximation of the phenomenon that is computed by combining the acquisition of a minimal number, optimally placed, measurements from the processes with their mathematical models (parameter-dependent PDEs). This idea is illustrated through a parameter dependent Stokes problem in which it is shown that the pressure and velocity fields can e ciently be reconstructed with a relatively low-dimensional interpolation space.

An improvement EMD method based on the optimized rational Hermite interpolation approach and its application to gear fault diagnosis

A demodulation technique based on improvement empirical mode decomposition (EMD) is investigated in this paper. Firstly, the problem of the envelope line in EMD is introduced and the drawbacks of two classic interpolation methods, cubic spline interpolation method and cubic Hermite interpolation method are discussed; then a new envelope interpolation method called optimized rational Hermite interpolation method (O-EMD) is proposed, which has a shape controlling parameter compared with the cubic Hermite interpolation algorithm. At the same time, in order to improve the envelope approximation accuracy of local mean, the parameter determining criterion is put forward and an optimization with Genetic Algorithm (GA) is applied to automatic select the suitable shape controlling parameter in each sifting process. The effectiveness of O-EMD method is validated by the numerical simulations and an application to gear fault diagnosis. Results demonstrate that O-EMD method can improve the reliability and accuracy significantly compared with traditional EMD method.

A rational iterated function system for resolution of univariate constrained interpolation

Iterated Function Systems (IFSs) provide a standard framework for generating Fractal Interpolation Functions (FIFs) that yield smooth or non-smooth approximants. Nevertheless, the most widely studied FIFs so far in the literature that are obtained through polynomial IFSs are, in general, incapable of reproducing important shape properties inherent in a given data set. Abandoning the polynomiality of the functions defining the IFS, we introduce a new class of rational IFS that generates fractal functions (self-referential functions) for solving constrained interpolation problems. Suitable values of the rational IFS parameters are identified so that: (i) the corresponding FIF inherits positivity and/or monotonicity properties present in the data set, and (ii) the attractor of the IFS lies within an axis-aligned rectangle. The proposed IFS schemes for the shape preserving interpolation generalize some of the classical non-recursive interpolation methods, and expand the interpolation/approximation, including approximants for which functions themselves or the first derivatives can even be non-differentiable in a dense set of points of the domain. For appropriate values of the IFS parameters, the resulting rational quadratic FIF converges uniformly to the original function $$\varPhi \in \mathcal {C}^3[x_1, x_n]$$ with $$h^3$$ order of convergence, where $$h$$ denotes the norm of the partition. We also provide a number of examples intended to demonstrate the proposed schemes and to suggest how these schemes outperform their classical counterparts.

Interpolation of Morrey Spaces on Metric Measure Spaces

AbstractIn this article, via the classical complex interpolation method and some interpolation methods traced to Gagliardo, the authors obtain an interpolation theorem for Morrey spaces on quasimetric measure spaces, which generalizes some known results on ℝn.

Smooth Surface Modeling of DEMs Based on a Regularized Least Squares Method of Thin Plate Spline

Thin plate spline (TPS) has been widely accepted as a method for smooth fitting of noisy data. However, the classical TPS always has an ill-conditioning problem when two sample points are very close. Although the modified orthogonal least squares-based TPS (TPS-M) avoids this ill-conditioning problem, it is not completely immune to over-fitting when sample points are noisy. In this paper, a regularized least squares method of thin plate spline (TPS-RLS) was developed, which adds a weight penalty term to the error criterion of orthogonal least squares (OLS). TPS-RLS combines the advantages of both regularization and OLS, which avoid the over-fitting and the ill-conditioning problems simultaneously. Numerical tests indicate that irrespective of the standard deviation of sampling errors and the number of knots, TPS-RLS is always more accurate than TPS-M for smooth fitting of noisy data, whereas TPS-M would have a serious over-fitting problem if the optimal number of knots were not determined in advance. The real-world example of fitting total station instrument data shows that among the classical interpolation methods including IDW, natural neighbor and ordinary kriging, TPS-RLS has the highest accuracy for a series of DEMs with different resolutions, especially for the coarse one. Surface modeling of DEMs with contour lines demonstrate that TPS-RLS has a better performance than the classical methods in terms of both root mean squared error and relief shaded map appearance.

Collaborative patch-based super-resolution for diffusion-weighted images

In this paper, a new single image acquisition super-resolution method is proposed to increase image resolution of diffusion weighted (DW) images. Based on a nonlocal patch-based strategy, the proposed method uses a non-diffusion image (b0) to constrain the reconstruction of DW images. An extensive validation is presented with a gold standard built on averaging 10 high-resolution DW acquisitions. A comparison with classical interpolation methods such as trilinear and B-spline demonstrates the competitive results of our proposed approach in terms of improvements on image reconstruction, fractional anisotropy (FA) estimation, generalized FA and angular reconstruction for tensor and high angular resolution diffusion imaging (HARDI) models. Besides, first results of reconstructed ultra high resolution DW images are presented at 0.6×0.6×0.6mm3 and 0.4×0.4×0.4mm3 using our gold standard based on the average of 10 acquisitions, and on a single acquisition. Finally, fiber tracking results show the potential of the proposed super-resolution approach to accurately analyze white matter brain architecture.

A Robust Multiquadric Method for Digital Elevation Model Construction

The multiquadric method (MQ) with high interpolation accuracy has been widely used for interpolating spatial data. However, MQ is an exact interpolation method, which is improper to interpolate noisy sampling data. Although the least squares MQ (LSMQ) has the ability to smooth out sampling errors, it is inherently not robust to outliers due to the least squares criterion in estimating the weights of sampling knots. In order to reduce the impact of outliers on the accuracy of digital elevation models (DEMs), a robust method of MQ (MQ-R) has been developed. MQ-R includes two independent procedures: knot selection and the solution of the system of linear equations. The two independent procedures were respectively achieved by the space-filling design and the least absolute deviation, both of which are very robust to outliers. Gaussian synthetic surface, which is subject to a series of errors with different distributions, was employed to compare the performance of MQ-R with that of LSMQ. Results indicate that LSMQ is seriously affected by outliers, whereas MQ-R performs well in resisting outliers, and can construct satisfactory surfaces even though the data are contaminated by severe outliers. A real-world example of DEM construction was employed to evaluate the robustness of MQ-R, LSMQ, and the classical interpolation methods including inverse distance weighting method, thin plate spline, and ANUDEM. Results showed that compared with the classical methods, MQ-R has the highest accuracy in terms of root mean square error. In conclusion, when sampling data is subject to outliers, MQ-R can be considered as an alternative method for DEM construction.

Interpolating point spread function anisotropy

Planned wide-field weak lensing surveys are expected to reduce the statistical errors on the shear field to unprecedented levels. In contrast, systematic errors like those induced by the convolution with the point spread function (PSF) will not benefit from that scaling effect and will require very accurate modeling and correction. While numerous methods have been devised to carry out the PSF correction itself, modeling of the PSF shape and its spatial variations across the instrument field of view has, so far, attracted much less attention. This step is nevertheless crucial because the PSF is only known at star positions while the correction has to be performed at any position on the sky. A reliable interpolation scheme is therefore mandatory and a popular approach has been to use low-order bivariate polynomials. In the present paper, we evaluate four other classical spatial interpolation methods based on splines (B-splines), inverse distance weighting (IDW), radial basis functions (RBF) and ordinary Kriging (OK). These methods are tested on the Star-challenge part of the GRavitational lEnsing Accuracy Testing 2010 (GREAT10) simulated data and are compared with the classical polynomial fitting (Polyfit). We also test all our interpolation methods independently of the way the PSF is modeled, by interpolating the GREAT10 star fields themselves (i.e., the PSF parameters are known exactly at star positions). We find in that case RBF to be the clear winner, closely followed by the other local methods, IDW and OK. The global methods, Polyfit and B-splines, are largely behind, especially in fields with (ground-based) turbulent PSFs. In fields with non-turbulent PSFs, all interpolators reach a variance on PSF systematics $\sigma_{sys}^2$ better than the $1\times10^{-7}$ upper bound expected by future space-based surveys, with the local interpolators performing better than the global ones.