Isotropic Kernel Research Articles

Expressions for the calculation of isotropic Gaussian filter kernels in the spherical harmonic domain

Expressions for the calculation of isotropic Gaussian filter kernels in the spherical harmonic domain

Anisotropic Phase Stretch Transform-Based Algorithm for Segmentation of Activated Sludge Phase-Contrast Microscopic Image

The activated sludge (AS) process is a biological treatment process of wastewater used in sewage treatment plants, in which settling of AS is vitally important for wastewater treatment. In AS, however, bulking caused by filamentous bacteria will significantly reduce the settling capacity of activated sludge. Traditionally, the physicochemical method has been used to monitor the status or performance of filamentous bacteria in activated sludge, while it is a very compromising means in modern digital quality control for wastewater treatment when digital image processing and analysis technology is used to determine the status of filamentous bacteria in activated sludge. In order to avoid the disadvantages of deficient direction selectivity in the translation vector field for isotropic phase stretch transform (PST) and high noise susceptibility in determining the status of filamentous bacteria in AS and low accuracy, an anisotropic PST (APST) method for filamentous bacteria test in AS was proposed in this paper; specifically speaking, by analyzing and deriving the disadvantages of the traditional isotropic PST kernel function, we designed an APST kernel function, put forward a non-maximum suppression strategy for processing of images from different aspects and combined with the relative total variation theory to form APST-based algorithm for segmentation of activated sludge phase-contrast microscopic image. According to a large number of experimental results, in terms of the overall segmentation effect or the subjective and objective evaluation indicators, the resultant algorithm in this paper is superior to the latest traditional PST segmentation algorithm, Canny image segmentation algorithm and Sobel image segmentation algorithm. Also, the segmentation results for filamentous bacteria showed that probability rand index (PRI) and global consistency error (GCE) indicators were all improved by about 30%, supporting the effectiveness and superiority of the algorithm in this paper.

Benchmarking the performance of Bayesian optimization across multiple experimental materials science domains

Bayesian optimization (BO) has been leveraged for guiding autonomous and high-throughput experiments in materials science. However, few have evaluated the efficiency of BO across a broad range of experimental materials domains. In this work, we quantify the performance of BO with a collection of surrogate model and acquisition function pairs across five diverse experimental materials systems. By defining acceleration and enhancement metrics for materials optimization objectives, we find that surrogate models such as Gaussian Process (GP) with anisotropic kernels and Random Forest (RF) have comparable performance in BO, and both outperform the commonly used GP with isotropic kernels. GP with anisotropic kernels has demonstrated the most robustness, yet RF is a close alternative and warrants more consideration because it is free from distribution assumptions, has smaller time complexity, and requires less effort in initial hyperparameter selection. We also raise awareness about the benefits of using GP with anisotropic kernels in future materials optimization campaigns.

Cortical Surface-Informed Volumetric Spatial Smoothing of fMRI Data via Graph Signal Processing.

Conventionally, as a preprocessing step, functional MRI (fMRI) data are spatially smoothed before further analysis, be it for activation mapping on task-based fMRI or functional connectivity analysis on resting-state fMRI data. When images are smoothed volumetrically, however, isotropic Gaussian kernels are generally used, which do not adapt to the underlying brain structure. Alternatively, cortical surface smoothing procedures provide the benefit of adapting the smoothing process to the underlying morphology, but require projecting volumetric data on to the surface. In this paper, leveraging principles from graph signal processing, we propose a volumetric spatial smoothing method that takes advantage of the gray-white and pial cortical surfaces, and as such, adapts the filtering process to the underlying morphological details at each point in the cortex.

Examining the Influence of Spatial Smoothing on Spatiotemporal Features of Intrinsic Connectivity Networks at Low ICA Model Order.

Using a relatively high model order of independent component analysis (ICA with 75 ICs) of functional magnetic resonance imaging (fMRI) data, we have reported a clear effect of spatial smoothing Gaussian kernel size on spatiotemporal properties of intrinsic connectivity networks (ICNs). However, many if not the majority of ICA fMRI studies are usually performed at low model order, e.g., 20-IC decomposition, as such low order is generally enough to extract the few networks of interest such as the default-mode network (DMN). The aim of this study is to investigate if we can replicate the spatial smoothing effects on spatiotemporal features of ICNs at low ICA model order. Same resting state fMRI data that we used with 75-IC analysis were used here. Spatial smoothing using an isotropic Gaussian filter kernel with full width at half maximum (FWHM) of 4, 8, and 12 mm was applied during preprocessing. ICNs were identified from 20-IC decomposition and evaluated in terms of three primary features: spatial map intensity, functional network connectivity (FNC), and power spectra. The results identified similar effects of spatial smoothing on spatial map intensities and power spectra at p < 0.01, false discovery rate (FDR) corrected for multiple comparisons. Reduced spatial smoothing kernel size resulted in decreased spatial map intensities as well as a generally decreased low-frequency power (0.01 - 0.10 Hz) but increased high-frequency power (0.15 - 0.25 Hz). FNC, however, did not show a uniform change in correlation values with the size of smoothing kernel. Notably, FNC between DMNs decreased but FNC between central executive and visual networks increased with an increase in smoothing kernel size. These preliminary findings confirm spatial smoothing influences ICN features regardless of model order. The discussion focuses on differences between observed changes at low and high ICA model orders.

The core-radius approach to supercritical fractional perimeters, curvatures and geometric flows

We consider a core-radius approach to nonlocal perimeters governed by isotropic kernels having critical and supercritical exponents, extending the nowadays classical notion of s-fractional perimeter, defined for 0<s<1, to the case s≥1. We show that, as the core-radius vanishes, such core-radius regularized s-fractional perimeters, suitably scaled, Γ-converge to the standard Euclidean perimeter. Under the same scaling, the first variation of such nonlocal perimeters gives back regularized s-fractional curvatures which, as the core radius vanishes, converge to the standard mean curvature; as a consequence, we show that the level set solutions to the corresponding nonlocal geometric flows, suitably reparametrized in time, converge to the standard mean curvature flow. Furthermore, we show the same asymptotic behavior as the core-radius vanishes and s→s̄≥1 simultaneously. Finally, we prove analogous results in the case of anisotropic kernels with applications to dislocation dynamics.

Propriety of the reference posterior distribution in Gaussian process modeling

In a seminal article, Berger, De Oliveira and Sansó [J. Amer. Statist. Assoc. 96 (2001) 1361–1374] compare several objective prior distributions for the parameters of Gaussian process models with isotropic correlation kernel. The reference prior distribution stands out among them insofar as it always leads to a proper posterior. They prove this result for rough correlation kernels: Spherical, Exponential with power ρ<2, Matérn with smoothness ν<1. This paper provides a proof for smooth correlation kernels: Exponential with power ρ=2, Matérn with smoothness ν≥1, Rational Quadratic, along with tail rates of the reference prior for these kernels.

Strong uniform consistency with rates for kernel density estimators with general kernels on manifolds

Abstract When analyzing modern machine learning algorithms, we may need to handle kernel density estimation (KDE) with intricate kernels that are not designed by the user and might even be irregular and asymmetric. To handle this emerging challenge, we provide a strong uniform consistency result with the $L^\infty $ convergence rate for KDE on Riemannian manifolds with Riemann integrable kernels (in the ambient Euclidean space). We also provide an $L^1$ consistency result for kernel density estimation on Riemannian manifolds with Lebesgue integrable kernels. The isotropic kernels considered in this paper are different from the kernels in the Vapnik–Chervonenkis class that are frequently considered in statistics society. We illustrate the difference when we apply them to estimate the probability density function. Moreover, we elaborate the delicate difference when the kernel is designed on the intrinsic manifold and on the ambient Euclidian space, both might be encountered in practice. At last, we prove the necessary and sufficient condition for an isotropic kernel to be Riemann integrable on a submanifold in the Euclidean space.

Mark-release-recapture of male Aedes aegypti (Diptera: Culicidae): Use of rhodamine B to estimate movement, mating and population parameters in preparation for an incompatible male program.

Rapid advances in biological and digital support systems are revolutionizing the population control of invasive disease vectors such as Aedes aegypti. Methods such as the sterile and incompatible insect techniques (SIT/IIT) rely on modified males to seek out and successfully mate with females, and in doing so outcompete the wild male population for mates. Currently, these interventions most frequently infer mating success through area-wide population surveillance and estimates of mating competitiveness are rare. Furthermore, little is known about male Ae. aegypti behaviour and biology in field settings. In preparation for a large, community scale IIT program, we undertook a series of mark- release-recapture experiments using rhodamine B to mark male Ae. aegypti sperm and measure mating interactions with females. We also developed a Spatial and Temporally Evolving Isotropic Kernel (STEIK) framework to assist researchers to estimate the movement of individuals through space and time. Results showed that ~40% of wild females captured daily were unmated, suggesting interventions will need to release males multiple times per week to be effective at suppressing Ae. aegypti populations. Males moved rapidly through the landscape, particularly when released during the night. Although males moved further than what is typically observed in females of the species, survival was considerably lower. These unique insights improve our understanding of mating interactions in wild Ae. aegypti populations and lay the foundation for robust suppression strategies in the future.

A one-sided direct forcing immersed boundary method using moving least squares

This paper presents a one-sided immersed boundary (IB) method using kernel functions constructed via a moving least squares (MLS) method. The resulting kernels effectively couple structural degrees of freedom to fluid variables on only one side of the fluid-structure interface. This reduces spurious feedback forcing and internal flows that are typically observed in IB models that use isotropic kernel functions to couple the structure to fluid degrees of freedom on both sides of the interface. The method developed here extends the original MLS methodology introduced by Vanella and Balaras (2009) [27]. Prior IB/MLS methods have used isotropic kernel functions that coupled fluid variables on both sides of the boundary to the interfacial degrees of freedom. The original IB/MLS approach converts the cubic spline weights typically employed in MLS reconstruction into an IB kernel function that satisfies particular discrete moment conditions. This paper shows that the same approach can be used to construct one-sided kernel functions (kernel functions are referred to as generating functions in the MLS literature). We also examine the performance of the new approach for a family of kernel functions introduced by Peskin. It is demonstrated that the one-sided MLS construction tends to generate non-monotone interpolation kernels with large over- and undershoots. We present two simple weight shifting strategies to construct generating functions that are positive and monotone, which enhances the stability of the resulting IB methodology. Benchmark cases are used to test the order of accuracy and verify the one-sided IB/MLS simulations in both two and three spatial dimensions. This new IB/MLS method is also used to simulate flow over the Ahmed car model, which highlights the applicability of this methodology for modeling complex engineering flows.

Efficient projection kernels for discontinuous Galerkin simulations of disperse multiphase flows on arbitrary curved elements

In this work, we develop projection kernels for Euler-Lagrange point-particle simulations of disperse multiphase flows on arbitrary curved elements. These kernels are employed in a high-order discontinuous Galerkin framework for projecting the action of the particles to the Eulerian mesh. Instead of commonly used isotropic kernels, such as a Gaussian-type kernel, we construct an anisotropic polynomial-based smoothing function that preserves the compactness of the discontinuous Galerkin method on high-aspect-ratio elements and maintains an acceptable computational cost. At the same time, it mitigates inaccuracies associated with larger particles and numerical instabilities arising from the Dirac delta function and low-order kernels. Specifically, the geometric mapping of a physical element to the reference element is exploited to construct a kernel that is elliptical in 2D and ellipsoidal in 3D. We also develop a strategy to conserve interphase transfer near boundaries, particularly on curved elements. This strategy employs a high-order polynomial approximation to appropriately rescale the interphase source terms in an efficient manner. The compatibility of the proposed methodology with different types of meshes is investigated. We then apply it to a number of particle-laden flow configurations, including supersonic dusty flow over a flat plate, moving shocks interacting with clouds of particles, and hypersonic dusty flows over blunt bodies.

How isotropic kernels perform on simple invariants

We investigate how the training curve of isotropic kernel methods depends on the symmetry of the task to be learned, in several settings. (i) We consider a regression task, where the target function is a Gaussian random field that depends only on variables, fewer than the input dimension d. We compute the expected test error ϵ that follows where p is the size of the training set. We find that β ∼ 1/d independently of , supporting previous findings that the presence of invariants does not resolve the curse of dimensionality for kernel regression. (ii) Next we consider support-vector binary classification and introduce the stripe model, where the data label depends on a single coordinate , corresponding to parallel decision boundaries separating labels of different signs, and consider that there is no margin at these interfaces. We argue and confirm numerically that, for large bandwidth, , where ξ ∈ (0, 2) is the exponent characterizing the singularity of the kernel at the origin. This estimation improves classical bounds obtainable from Rademacher complexity. In this setting there is no curse of dimensionality since as . (iii) We confirm these findings for the spherical model, for which . (iv) In the stripe model, we show that, if the data are compressed along their invariants by some factor λ (an operation believed to take place in deep networks), the test error is reduced by a factor .

High-resolution synthetic aperture radar image classification using multi-scale anisotropic convolutional sparse coding

The classification of high-resolution (HR) synthetic aperture radar (SAR) image is of great significance to SAR scene interpretation and understanding. However, the HR SAR image contains rich ground information and complex spatial structural features. The extraction of effective distinguishing features under the influence of coherent speckle is still a challenging task. A multi-scale anisotropic convolution sparse coding (MACSC) algorithm is proposed for HR SAR image classification. First, the low-order statistical features are extracted to improve the richness of SAR image. Then an unsupervised MACSC model is utilized to learn a set of sparse feature maps and convolution filters for the above statistical features by solving the designed objective function. In MACSC, the multi-scale anisotropic Gaussian kernels are utilized to initialize the dictionary of MACSC. Compared with the isotropic Gaussian kernel of the original CSC, these kernels can more accurately describe the details of different directions and scales in the SAR scene. Meanwhile, the adaptive sparse control factor is introduced into the MACSC model, which can make the learned sparse feature maps suppress the speckle and capture the abundant edge and texture information of the SAR image. After that, the aggregation operation is conducted on multi-scale and multi-direction sparse feature maps to integrate the neighbor pixels and reduce the computing burden. Finally, the obtained feature vector is input into the support vector machine classifier to realize classification. Experiments on three challenging SAR data sets demonstrate that MACSC achieves better classification performance than related sparse representation methods.

Gaussian smoothing and modified histogram normalization methods to improve neural-biomarker interpretations for dyslexia classification mechanism.

Achieving biologically interpretable neural-biomarkers and features from neuroimaging datasets is a challenging task in an MRI-based dyslexia study. This challenge becomes more pronounced when the needed MRI datasets are collected from multiple heterogeneous sources with inconsistent scanner settings. This study presents a method of improving the biological interpretation of dyslexia’s neural-biomarkers from MRI datasets sourced from publicly available open databases. The proposed system utilized a modified histogram normalization (MHN) method to improve dyslexia neural-biomarker interpretations by mapping the pixels’ intensities of low-quality input neuroimages to range between the low-intensity region of interest (ROIlow) and high-intensity region of interest (ROIhigh) of the high-quality image. This was achieved after initial image smoothing using the Gaussian filter method with an isotropic kernel of size 4mm. The performance of the proposed smoothing and normalization methods was evaluated based on three image post-processing experiments: ROI segmentation, gray matter (GM) tissues volume estimations, and deep learning (DL) classifications using Computational Anatomy Toolbox (CAT12) and pre-trained models in a MATLAB working environment. The three experiments were preceded by some pre-processing tasks such as image resizing, labelling, patching, and non-rigid registration. Our results showed that the best smoothing was achieved at a scale value, σ = 1.25 with a 0.9% increment in the peak-signal-to-noise ratio (PSNR). Results from the three image post-processing experiments confirmed the efficacy of the proposed methods. Evidence emanating from our analysis showed that using the proposed MHN and Gaussian smoothing methods can improve comparability of image features and neural-biomarkers of dyslexia with a statistically significantly high disc similarity coefficient (DSC) index, low mean square error (MSE), and improved tissue volume estimations. After 10 repeated 10-fold cross-validation, the highest accuracy achieved by DL models is 94.7% at a 95% confidence interval (CI) level. Finally, our finding confirmed that the proposed MHN method significantly outperformed the normalization method of the state-of-the-art histogram matching.

Enlightening behavioral disturbances in myotonic dystrophy type 1 through neuropsychology and imaging correlations: Insights from the frontal and temporal lobe functions

AbstractBackgroundMyotonic Dystrophy Type 1 (MD1) is a neuromuscular disease in which subtle behavioral disturbance is often described. Few studies evaluated frontal lobe function and behavioral scales in relation with brain pathology.Method27 MD1 adult patients (mean age 46), were included. All patients performed an exhaustive neuropsychological assessment with a specific focus on cognitive and behavioral frontal functions. 18 MD1 patients and 18 healthy controls had a brain MRI with T1 and T2 Flair sequences. We performed a voxel‐based analysis with SPM12 via Clinica Software. The normalized data were smoothed with an isotropic Gaussian kernel of 8 mm. Statistical analysis was performed using general linear model with age sex and total intra cranial volume as covariates. Statistics were corrected for multiple comparisons with family‐wise error (FWE) correction at the cluster level with a height threshold of 0.001. T2 white matter lesions were segmented with VolBrain software.ResultMore than 60 % of the patients showed significant impairment in executive frontal functions (inhibition, mental flexibility, contextualization and theory of mind) with relative sparing of motivation, emotion recognition and initiation/planification.MD1 patients MRI compared to healthy controls showed grey matter atrophy in medial temporal lobe, (parahippocampus mostly) and in a smaller extent in the middle frontal gyrus and the right caudate nucleus and excessive white matter hyperintensities, mostly in temporal lobes.Frontal cognitive and behavioral impairment on neuropsychological testing was correlated with medial temporal and in a smaller extent frontal localization.ConclusionBehavioral disturbance in MD1 might be related to cognitive impairment in mental flexibility and contextualization rather than a specific social cognition deficit. This impairment might be related to white matter lesions or grey matter disease (including frontal‐subcortical network or parahippocampus, known to apprehend contextualization in emotion detection). Altogether, these findings confirm the hypothesis of a specific central nervous system involvement in MD1 patients.

A general framework of kernel-driven modeling in the thermal infrared domain

Radiometric measurements in the Thermal Infrared (TIR) domain exhibit an angular variation over most surface types, known as the Thermal Radiation Directionality (TRD) phenomenon. A primary objective of the ongoing development of TRD physical models is to perform a correction of the angular effects to obtain comparable land surface temperature products. In practice, it is advised to handle only the models having a limited number of input parameters for the purpose of operational applications. The use of semi-empirical kernel-driven models (KDMs) appears to be a good tradeoff between physical accuracy and computational efficiency as it was already demonstrated through a broad usage in the optical domain. It remains that the existing state-of-the-art 3-parameter TIR KDMs (RossThick-LiSparseR, LiStrahlerFriedl-LiDenseR, Vinnikov, and RoujeanLagouarde) underestimate the hotspot phenomenon, especially for continuous canopies marked by a narrow peak. In this study, a new general framework of TIR kernel-driven modeling is proposed to overcome such issue. It is a linear combination of three kernels (including a base shape kernel, a hotspot kernel with adjustable width and an isotropic kernel) with the ability to simulate the bowl, dome and bell shapes in the solar principal plane. Four specific 4-parameter models (Vinnikov-RoujeanLagouarde, LiStrahlerFriedl-RoujeanLagouarde, Vinnikov-Chen, and LiStrahlerFriedl-Chen, named “base shape kernel - hotspot kernel”) within the new framework were studied to assess their abilities to mimic the patterns of the directional brightness temperature for both continuous and discrete vegetation canopies. These four 4-parameter KDMs and four 3-parameter KDMs were comprehensively evaluated with 306 groups of simulated multi-angle datasets generated by a modernized analytical 4-stream radiative transfer model based on the Scattering by Arbitrarily Inclined Leaves (4SAIL), and a Discrete Anisotropic Radiative Transfer (DART) model considering different solar zenith angles (SZA), canopy architectures and component temperatures, and 2 groups of airborne measured multi-angle datasets over continuous maize and discrete pine forest. Results show that the four 4-parameter KDMs behave better than the four existing 3-parameter KDMs over continuous canopies (e.g. R2 increases from 0.661~0.970 to 0.940~0.997 and RMSE decreases from 0.17~0.71 to 0.07~0.16 when SZA = 30°) and discrete canopies (e.g. R2 increases from 0.791~0.989 to 0.976~0.996 and RMSE decreases from 0.10~0.84 to 0.08~0.21 when SZA = 30°). The new general framework with four parameters (three kernel coefficients and an adjustable hotspot width) improves the fitting ability significantly, compared to the four existing three-parameter KDMs, given the addition of one more degree of freedom. Results show that the coefficients of the base shape kernel, hotspot kernel and isotropic kernel are related to the temperature difference between leaf and background, temperature difference between sunlit component and shaded component, and the nadir brightness temperature, respectively. However, the estimated hotspot width depends on vegetation structure. The new kernel-driven modeling framework has the potential to be a tool for angular correction of multi-angle satellite observations and angular optimization of future multi-angle TIR sensors.

A unifying theory for two-dimensional spatial redistribution kernels with applications in population spread modelling.

When building models to explain the dispersal patterns of organisms, ecologists often use an isotropic redistribution kernel to represent the distribution of movement distances based on phenomenological observations or biological considerations of the underlying physical movement mechanism. The Gaussian, two-dimensional (2D) Laplace and Bessel kernels are common choices for 2D space. All three are special (or limiting) cases of a kernel family, the Whittle-Matérn-Yasuda (WMY), first derived by Yasuda from an assumption of 2D Fickian diffusion with gamma-distributed settling times. We provide a novel derivation of this kernel family, using the simpler assumption of constant settling hazard, by means of a non-Fickian 2D diffusion equation representing movements through heterogeneous 2D media having a fractal structure. Our derivation reveals connections among a number of established redistribution kernels, unifying them under a single, flexible modelling framework. We demonstrate improvements in predictive performance in an established model for the spread of the mountain pine beetle upon replacing the Gaussian kernel by the Whittle-Matérn-Yasuda, and report similar results for a novel approximation, the product-Whittle-Matérn-Yasuda, that substantially speeds computations in applications to large datasets.

Automatic feature scaling and selection for support vector machine classification with functional data

FunctionalData Analysis (FDA) has become a very important field in recent years due to its wide range of applications. However, there are several real-life applications in which hybrid functional data appear, i.e., data with functional and static covariates. The classification of such hybrid functional data is a challenging problem that can be handled with the Support Vector Machine (SVM). Moreover, the selection of the most informative features may yield to drastic improvements in the classification rates. In this paper, an embedded feature selection approach for SVM classification is proposed, in which the isotropic Gaussian kernel is modified by associating a bandwidth to each feature. The bandwidths are jointly optimized with the SVM parameters, yielding an alternating optimization approach. The effectiveness of our methodology was tested on benchmark data sets. Indeed, the proposed method achieved the best average performance when compared to 17 other feature selection and SVM classification approaches. A comprehensive sensitivity analysis of the parameters related to our proposal was also included, confirming its robustness.

A consistent parallel isotropic unstructured mesh generation method based on multi-phase SPH

In this paper, we propose a consistent parallel unstructured mesh generator based on a multi-phase SPH method. A set of physics-motivated modeling equations are developed to achieve the targets of domain decomposition, communication volume optimization and high-quality unstructured mesh generation simultaneously. A unified density field is defined as the target function for both partitioning the geometry and distributing the mesh-vertexes. A multi-phase Smoothing Particle Hydrodynamics (SPH) method is employed to solve the governing equations. All the optimization targets are achieved implicitly and consistently by the particle relaxation procedure without constructing triangulation/tetrahedralization explicitly. The target of communication reduction is achieved by introducing a surface tension model between distinct partitioning sub-domains, which are characterized by colored SPH particles. The resulting partitioning diagram features physically localized sub-domains and optimized interface communication. The target of optimizing the mesh quality is achieved by introducing a tailored equation-of-state (EOS) and a smooth isotropic kernel function. The mesh quality near the interface of neighboring sub-domains is improved by gradually removing the surface-tension force once a steady state is achieved. The proposed method is developed basing on a new parallel environment for multi-resolution SPH to exploit both coarse- and fine-grained parallelism. A set of benchmarks are conducted to verify that all the optimization targets are achieved consistently within the current framework.

Certified and fast computations with shallow covariance kernels

Many techniques for data science and uncertainty quantification demand efficient tools to handle Gaussian random fields, which are defined in terms of their mean functions and covariance operators. Recently, parameterized Gaussian random fields have gained increased attention, due to their higher degree of flexibility. However, especially if the random field is parameterized through its covariance operator, classical random field discretization techniques fail or become inefficient. In this work we introduce and analyze a new and certified algorithm for the low-rank approximation of a parameterized family of covariance operators which represents an extension of the adaptive cross approximation method for symmetric positive definite matrices. The algorithm relies on an affine linear expansion of the covariance operator with respect to the parameters, which needs to be computed in a preprocessing step using, e.g., the empirical interpolation method. We discuss and test our new approach for isotropic covariance kernels, such as Mat\'ern kernels. The numerical results demonstrate the advantages of our approach in terms of computational time and confirm that the proposed algorithm provides the basis of a fast sampling procedure for parameter dependent Gaussian random fields.