Abstract: Interpolation between pairs of values, typically vectors, is a fundamental operation in many computer graphics applications. In some cases simple linear interpolation yields meaningful results without requiring domain knowledge. However, interpolation between pairs of distributions or pairs of functions often demands more care because features may exhibit translational motion between exemplars. This property is not captured by linear interpolation. This paper develops the use of displacement interpolation for this class of problem, which provides a generic method for interpolating between distributions or functions based on advection instead of blending. The functions can be non-uniformly sampled, high-dimensional, and defined on non-Euclidean manifolds, e.g., spheres and tori. Our method decomposes distributions or functions into sums of radial basis functions (RBFs). We solve a mass transport problem to pair the RBFs and apply partial transport to obtain the interpolated function. We describe practical methods for computing the RBF decomposition and solving the transport problem. We demonstrate the interpolation approach on synthetic examples, BRDFs, color distributions, environment maps, stipple patterns, and value functions.

Abstract: Machine learning methods, like random forest (RF), have shown their superior performance in various disciplines, but have not been previously applied to the spatial interpolation of environmental variables. In this study, we compared the performance of 23 methods, including RF, support vector machine (SVM), ordinary kriging (OK), inverse distance squared (IDS), and their combinations (i.e., RFOK, RFIDS, SVMOK and SVMIDS), using mud content samples in the southwest Australian margin. We also tested the sensitivity of the combined methods to input variables and the accuracy of averaging predictions of the most accurate methods. The accuracy of the methods was assessed using a 10-fold cross-validation. The spatial patterns of the predictions of the most accurate methods were also visually examined for their validity. This study confirmed the effectiveness of RF, in particular its combination with OK or IDS, and also confirmed the sensitivity of RF and its combined methods to the input variables. Averaging the predictions of the most accurate methods showed no significant improvement in the predictive accuracy. Visual examination proved to be an essential step in assessing the spatial predictions. This study has opened an alternative source of methods for spatial interpolation of environmental properties.

Abstract: This paper presents bilinear and bicubic interpolation methods tailored for the division of focal plane polarization imaging sensor. The interpolation methods are targeted for a 1-Mega pixel polarization imaging sensor operating in the visible spectrum. The five interpolation methods considered in this paper are: bilinear, weighted bilinear, bicubic spline, an approximated bicubic spline and a bicubic interpolation method. The modulation transfer function analysis is applied to the different interpolation methods, and test images as well as numerical error analyses are also presented. Based on the comparison results, the full frame bicubic spline interpolation achieves the best performance for polarization images.

Abstract: The interpolation algorithm plays an essential role in the digital image correlation (DIC) technique for shape, deformation, and motion measurements with subpixel accuracies. At the present, little effort has been made to improve the interpolation methods used in DIC. In this Letter, a family of recursive interpolation schemes based on B-spline representation and its inverse gradient weighting version is employed to enhance the accuracy of DIC analysis. Theories are introduced, and simulation results are presented to illustrate the effectiveness of the method as compared with the common bicubic interpolation.

Abstract: The linear regression model is a very attractive tool to design effective image interpolation schemes. Some regression-based image interpolation algorithms have been proposed in the literature, in which the objective functions are optimized by ordinary least squares (OLS). However, it is shown that interpolation with OLS may have some undesirable properties from a robustness point of view: even small amounts of outliers can dramatically affect the estimates. To address these issues, in this paper we propose a novel image interpolation algorithm based on regularized local linear regression (RLLR). Starting with the linear regression model where we replace the OLS error norm with the moving least squares (MLS) error norm leads to a robust estimator of local image structure. To keep the solution stable and avoid overfitting, we incorporate the l2-norm as the estimator complexity penalty. Moreover, motivated by recent progress on manifold-based semi-supervised learning, we explicitly consider the intrinsic manifold structure by making use of both measured and unmeasured data points. Specifically, our framework incorporates the geometric structure of the marginal probability distribution induced by unmeasured samples as an additional local smoothness preserving constraint. The optimal model parameters can be obtained with a closed-form solution by solving a convex optimization problem. Experimental results on benchmark test images demonstrate that the proposed method achieves very competitive performance with the state-of-the-art interpolation algorithms, especially in image edge structure preservation.

Abstract: . Hydrological modelling of floods relies on precipitation data with a high resolution in space and time. A reliable spatial representation of short time step rainfall is often difficult to achieve due to a low network density. In this study hourly precipitation was spatially interpolated with the multivariate geostatistical method kriging with external drift (KED) using additional information from topography, rainfall data from the denser daily networks and weather radar data. Investigations were carried out for several flood events in the time period between 2000 and 2005 caused by different meteorological conditions. The 125 km radius around the radar station Ummendorf in northern Germany covered the overall study region. One objective was to assess the effect of different approaches for estimation of semivariograms on the interpolation performance of short time step rainfall. Another objective was the refined application of the method kriging with external drift. Special attention was not only given to find the most relevant additional information, but also to combine the additional information in the best possible way. A multi-step interpolation procedure was applied to better consider sub-regions without rainfall. The impact of different semivariogram types on the interpolation performance was low. While it varied over the events, an averaged semivariogram was sufficient overall. Weather radar data were the most valuable additional information for KED for convective summer events. For interpolation of stratiform winter events using daily rainfall as additional information was sufficient. The application of the multi-step procedure significantly helped to improve the representation of fractional precipitation coverage.

Abstract: Interpolation procedures are widely used in science, especially in sciences that involve spatial data and continuous phenomena that can be depicted on a continuous spatial surface. Interpolation makes use of accurate and qualitative sampling data in order to produce a continuous representation of the phenomenon in question. The accuracy of the data used for interpolation directly affects the results. This research examines error propagation within the Inverse Distance Weighted (IDW) method, applied as a means of representing the earth's relief. Interpolation of a DEM within contours on a topographical map is considered to be a three-stage procedure. The first stage is the digitising of the contours depicted on the analogue map. Errors involved in this stage are propagated to the second stage, the geometric transformation of coordinates of these digitised contours. Additional errors due to the application of the transformation model are embedded within the results thus obtained. Finally, the errors are pro...

Abstract: . Recent studies highlight that spatial interpolation techniques of point data can be effectively applied to the problem of regionalization of hydrometric information. This study compares two innovative interpolation techniques for the prediction of low-flows in ungauged basins. The first one, named Physiographical-Space Based Interpolation (PSBI), performs the spatial interpolation of the desired streamflow index (e.g., annual streamflow, low-flow index, flood quantile, etc.) in the space of catchment descriptors. The second technique, named Topological kriging or Top-kriging, predicts the variable of interest along river networks taking both the area and nested nature of catchments into account. PSBI and Top-kriging are applied for the regionalization of Q355 (i.e., a low-flow index that indicates the streamflow that is equalled or exceeded 355 days in a year, on average) over a broad geographical region in central Italy, which contains 51 gauged catchments. The two techniques are cross-validated through a leave-one-out procedure at all available gauges and applied to a subregion to produce a continuous estimation of Q355 along the river network extracted from a 90m elevation model. The results of the study show that Top-kriging and PSBI present complementary features. Top-kriging outperforms PSBI at larger river branches while PSBI outperforms Top-kriging for headwater catchments. Overall, they have comparable performances (Nash-Sutcliffe efficiencies in cross-validation of 0.89 and 0.83, respectively). Both techniques provide plausible and accurate predictions of Q355 in ungauged basins and represent promising opportunities for regionalization of low-flows.

Abstract: Inverse distance weighting (IDW) is a simple method for multivariate interpolation but has poor prediction accuracy. In this article we show that the prediction accuracy of IDW can be substantially improved by integrating it with a linear regression model. This new predictor is quite flexible, computationally efficient, and works well in problems having high dimensions and/or large datasets. We also develop a heuristic method for constructing confidence intervals for prediction. This article has supplementary material online.

Abstract: One of the aims of regional Environmental Authorities is to provide citizens information about the quality of the atmosphere over a certain region. To reach this objective Environmental Authorities need suitable tools to interpolate the data coming from monitoring networks to domain locations where no measures are available. In this work a spatial interpolation system has been developed to estimate 8-h mean daily maximum ozone concentrations and daily mean PM10 concentrations over a domain, starting from measured concentration values. The presented approach is based on a cokriging technique, using the results of a deterministic Chemical Transport Model (CTM) simulation as secondary variable. The developed methodology has been tested over a 60 x 60 km^2 domain located in Northern Italy, including Milan metropolitan area, one of the most polluted areas in Europe.

Abstract: This study proposes network-based spatial interpolation methods to help predict unknown spatial values along networks more accurately. It expands on two of the commonly used spatial interpolation methods, IDW (inverse distance weighting) and OK (ordinary kriging), and applies them to analyze spatial data observed on a network. The study first provides the methodological framework, and it then examines the validity of the proposed methods by cross-validating elevations from two contrasting patterns of street network and comparing the MSEs (Mean Squared Errors) of the predicted values measured with the two proposed network-based methods and their conventional counterparts. The study suggests that both network-based IDW and network-based OK are generally more accurate than their existing counterparts, with network-based OK constantly outperforming the other methods. The network-based methods also turn out to be more sensitive to the edge effect, and their performance improves after edge correction. Furthermore, the MSEs of standard OK and network-based OK improve as more sample locations are used, whereas those of standard IDW and network-based IDW remain stable regardless of the number of sample locations. The two network-based methods use a similar set of sample locations, and their performance is inherently affected by the difference in their weight distribution among sample locations.

Abstract: Inverse distance weighting (IDW) is a simple method for multivariate interpolation but has poor prediction accuracy. In this article we show that the prediction accuracy of IDW can be substantially improved by integrating it with a linear regression model. This new predictor is quite flexible, computationally efficient, and works well in problems having high dimensions and/or large datasets. We also develop a heuristic method for constructing confidence intervals for prediction. This article has supplementary material online.

Abstract: It was not unusual in soil and environmental studies that the distribution of data is severely skewed with several high peak values, which causes the difficulty for Kriging with data transformation to make a satisfied prediction. This paper tested an approach that integrates kriging and triangular irregular network interpolation to make predictions. A data set consisting of total Copper (Cu) concentrations of 147 soil samples, with a skewness of 4.64 and several high peak values, from a copper smelting contaminated site in Zhejiang Province, China. The original data were partitioned into two parts. One represented the holistic spatial variability, followed by lognormal distribution, and then was interpolated by lognormal ordinary kriging. The other assumed to show the local variability of the area that near to high peak values, and triangular irregular network interpolation was applied. These two predictions were integrated into one map. This map was assessed by comparing with rank-order ordinary kriging and normal score ordinary kriging using another data set consisting of 54 soil samples of Cu in the same region. According to the mean error and root mean square error, the approach integrating lognormal ordinary kriging and triangular irregular network interpolation could make improved predictions over rank-order ordinary kriging and normal score ordinary kriging for the severely skewed data with several high peak values.

Abstract: A two-phase research was implemented to determine the effect of topography on climate parameters by using spatial interpolation and conventional statistical procedures in non-homogeneous topography. The primary set of climate data for the Central Anatolia Region includes monthly mean global solar radiation, sunshine duration, surface air temperature, relative humidity, wind speed and rainfall, recorded from 1976 to 2005. In the first phase, the effect of elevation on climate parameters was evaluated. For this purpose, kriging and co-kriging geostatistical interpolation techniques were compared to determine which one of the two techniques was more successful in determining the spatial distribution of climate parameters in variable topography. The inclusion of elevation as a covariate resulted in reduction of errors on sunshine duration, temperature and wind speed. On the basis of these error values, there is a relationship between elevation and sunshine duration, temperature and wind speed. In the second phase, multiple regression equations were developed to determine the effect of topography on annual mean values of climate factors. The highest correlation (−0.76) was found between solar radiation and latitude. The most effective factors were latitude and elevation. They alone explain 57% of the variability for sunshine duration and 56% for temperature, respectively. The multiple regression results were more significant than were the individual, pairwise correlation relationships. The mostly explained factor was temperature. Its variability was explained by latitude, elevation, aspect and slope as a ratio of 81.7%. Separate regression models for each data set and both response variables varied in their ability to explain variability in the response, with R2 values between 0.125 and 0.817. Copyright © 2010 Royal Meteorological Society

Abstract: Information available for mapping continuous soil attributes often includes point field data and choropleth maps (e.g. soil or geological maps) that model the spatial distribution of soil attributes as the juxtaposition of polygons (areas) with constant values. This paper presents two approaches to incorporate both point and areal data in the spatial interpolation of continuous soil attributes. In the first instance, area-to-point kriging is used to map the variability within soil units while ensuring the coherence of the prediction so that the average of disaggregated estimates is equal to the original areal datum. The resulting estimates are then used as local means in residual kriging. The second approach proceeds in one step and capitalizes on: 1) a general formulation of kriging that allows the combination of both point and areal data through the use of area-to-area, area-to-point, and point-to-point covariances in the kriging system, 2) the availability of GIS to discretize polygons of irregular shape and size, and 3) knowledge of the point-support variogram model that can be inferred directly from point measurements, thereby eliminating the need for deconvolution procedures. The two approaches are illustrated using the geological map and heavy metal concentrations recorded in the topsoil of the Swiss Jura. Sensitivity analysis indicates that the new procedures improve prediction over ordinary kriging and traditional residual kriging based on the assumption that the local mean is constant within each mapping unit.

Abstract: Geographically weighted regression (GWR) and Multiple Linear Regression (MLR) models have been applied to derive the spatial structure of urban heat island (UHI) in Wroclaw, SW Poland and compared. It was found that GWR is better suited for spatial modeling of UHI than MLR, as it takes into account non-stationarity of the spatial process. Both local and global models were extended by the interpolation of regression residuals, and used for spatial interpolation of the UHI structure. The combined: GWR + interpolated regression residuals (GWRK) approach is recommended for spatial modeling of UHI © 2010 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Spatial Statistics2011

Abstract: Areal interpolation is used to transfer attribute information from the initial set of source units with known values to the target units with unknown values before subsequent spatial analysis can occur. The areal units with unknown attribute information can be either at a finer scale or misaligned with respect to the source data layer. This article presents and describes a geographically weighted regression (GWR) method for solving areal interpolation problems for nested areal units and misaligned areal units. Population data, selected as the attribute information, are interpolated from census tracts to block groups (a finer scale) and pseudo-tracts (misaligned from tracts but at the same approximate scale). Root mean square error, adjusted root mean square error, and mean absolute error are calculated to evaluate the performance of the interpolation methods. The land cover data derived from Landsat Thematic Mapper Satellite Imagery with a 30×30 m spatial resolution are applied to as the ancillary data to...

Abstract: Provided are a method of interpolating an image by determining interpolation filter coefficients, and an apparatus for performing the same. The method includes: differently selecting an interpolation filter, from among interpolation filters for generating at least one sub-pel-unit pixel value located between integer-pel-unit pixels, based on a sub-pel-unit interpolation location and a smoothness; and generating the at least one sub-pel-unit pixel value by interpolating, using the selected interpolation filter, pixel values of the integer-pel-unit pixels.

Abstract: Soil pH has a major effect on plant nutrient availability by controlling the chemical structure of the nutrient. Adjusting soil acidity or alkalinity improves soil nutrition without adding extra fertilizers. Soil nutrients needed by plants in the largest amount are referred to as macronutrients. In addition to macronutrients, plants also need trace nutrients and both macro and trace nutrient availability is controlled by soil pH. Understanding of spatial variability of soil properties is important in site-specific management. Analysis of spatial variation of soil properties is fundamental to sustainable agricultural and rural development. The special variability of soil property is often measured using various interpolation methods resulting in map generation. Selecting a proper spatial interpolation method is crucial in surface analysis, since different methods of interpolation can lead to different surface results. Among statistical methods, geo-statistical kriging-based techniques have been frequently used for spatial analysis and surface mapping. In this work, three common interpolation methods are used to study the spatial distributions of soil pH in a vineyard. Interpolation techniques were used to estimate the pH measurement in unsampled points and create a continuous dataset that could be represented over a map of the entire study area. The method investigated includes; Inverse Distance Weighting (IDW), Radial base Function (RBF) and Ordinary Kriging (OK). The performance of conventional statistics showed that soil pH had a law variation in this study. Experimental anisotropic semivariograms were fitted with the Spherical, Exponential, Gaussian and Exponential models and the Exponential model was found as the best fitted model using the cross- validation method. The performances of interpolation methods were evaluated and compared using the cross-validation. The results showed that RBF method performed better than IDW and OK for prediction of the spatial distribution of topsoil pH (Figure 1).

Abstract: Methods of interpolation, whether based on regressions or on kriging, are global methods in which all the available data for a given study area are used. But the quality of results is affected when the study area is spatially very heterogeneous. To overcome this difficulty, a method of local interpolation is proposed and tested here with temperature in France. Starting from a set of weather stations spread across the country and digitized as 250 m-sided cells, the method consists in modelling local spatial variations in temperature by considering each point of the grid and the n weather stations that are its nearest neighbours. The procedure entails a series of steps: recognition of the n stations closest to the cell to be evaluated and subdivision of the study area into polygons defined by a neighbourhood rule, elaboration of a local model by multiple regression for each polygon, and application of the parameter estimate from the regression to obtain a predicted value of temperature at each point of the polygon under consideration. These results are compared with results from three global interpolation methods: (1) regression, (2) ordinary kriging, and (3) regression with kriging of residuals. We then develop the original results from local interpolation such as mapping of the coefficients of determination and of the parameter estimate related to altitude and to distance to the sea. These developments highlight the processes that dictate the spatial variation of climate

Abstract: Estimation of radio environments is a novel method for efficient management of spectrum resources in future wireless networks. It allows insight into the radio field, the interference and the possible geo-locations of various field transmitters. This paper compares three spatial interpolation techniques capable of estimating radio environments using a limited number of field measurements. Comparisons are conducted with extensive simulations for indoor and outdoor scenarios.

Abstract: The use of daily climate data in agriculture has increased considerably over the past two decades due to the rapid development of information technology and the need to better assess impacts and risks from extreme weather and accelerating climate change. While daily station data is now regularly used as an input to biophysical and biogeochemical models for the study of climate, agriculture, and forestry, questions still remain on the level of uncertainty in using daily data, especially for predictions made by spatial interpolation models. We evaluate the precision of three models (i.e., spline, weighted-truncated Gaussian filter, and hybrid inverse-distance/natural-neighbor) for interpolating daily precipitation and temperature at 10 km across the Canadian landmass south of 60o latitude (encompassing Canada's agricultural region). We compute daily, weekly, and monthly-aggregated bias and root-mean-square (RMSE) validation statistics, examining how error varies with orography and topography, and proximity to large water. Our findings show the best approach for interpolating daily temperature and precipitation across Canada requires a mixed-model/Bayesian approach. Further application of interpolation methods that consider non-stationary spatial covariance, alongside measurement of spatial correlation range would aid considerably in reducing interpolation prediction uncertainty. Copyright © 2010 Crown in Right of Canada.

Abstract: The estimation of commercial property prices in a touristic city can be explored through spatial interpolation methods, but in the presence of small sample sizes, auxiliary stochastic processes tha...