Standard Linear Interpolation Research Articles

A Comparison of Arterial Input Function Interpolation Methods for Patlak Plot Analysis of 68Ga-PSMA-11 PET Prostate Cancer Studies.

Positron emission tomography (PET) imaging enables quantitative assessment of tissue physiology. Dynamic pharmacokinetic analysis of PET images requires accurate estimation of the radiotracer plasma input function to derive meaningful parameter estimates, and small discrepancies in parameter estimation can mimic subtle physiologic tissue variation. This study evaluates the impact of input function interpolation method on the accuracy of Patlak kinetic parameter estimation through simulations modeling the pharmacokinetic properties of [68Ga]-PSMA-11. This study evaluated both trained and untrained methods. Although the mean kinetic parameter accuracy was similar across all interpolation models, the trained node weighting interpolation model estimated accurate kinetic parameters with reduced overall variability relative to standard linear interpolation. Trained node weighting interpolation reduced kinetic parameter estimation variance by a magnitude approximating the underlying physiologic differences between normal and diseased prostatic tissue. Overall, this analysis suggests that trained node weighting improves the reliability of Patlak kinetic parameter estimation for [68Ga]-PSMA-11 PET.

Deep Learning for Multigroup Cross-Section Representation in Two-Step Core Calculations

We investigate using deep learning, a type of machine-learning algorithm employing multiple layers of artificial neurons, for the mathematical representation of multigroup cross sections for use in the Griffin reactor multiphysics code for two-step deterministic neutronics calculations. A three-dimensional fuel element typical of a high-temperature gas reactor as well as a two-dimensional sodium-cooled fast reactor lattice are modeled using the Serpent Monte Carlo code, and multigroup macroscopic cross sections are generated for various state parameters to produce a training data set and a separate validation data set. A fully connected, feedforward neural network is trained using the open-source PyTorch machine-learning framework, and its accuracy is compared against the standard piecewise linear interpolation model. Additionally, we provide in this work a generic technique for propagating the cross-section model errors up to the keff using sensitivity coefficients with the first-order uncertainty propagation rule. Quantifying the eigenvalue error due to the cross-section regression errors is especially practical for appropriately selecting the mathematical representation of the cross sections. We demonstrate that the artificial neural network model produces lower errors and therefore enables better accuracy relative to the piecewise linear model when the cross sections exhibit nonlinear dependencies; especially when a coarse grid is employed, where the errors can be halved by the artificial neural network. However, for linearly dependent multigroup cross sections as found for the sodium-cooled fast reactor case, a simpler linear regression outperforms deeper networks.

Hardware Adaptive High‐Order Interpolation for Real‐Time Graphics

AbstractInterpolation is a core operation that has widespread use in computer graphics. Though higher‐order interpolation provides better quality, linear interpolation is often preferred due to its simplicity, performance, and hardware support.We present a unified refactoring of quadratic and cubic interpolations as standard linear interpolation plus linear interpolations of higher‐order terms and show how they can be applied to regular grids and (triangular/tetrahedral) simplexes Our formulations can provide significant reduction in computation cost, as compared to typical higher‐order interpolations and prior approaches that utilize existing hardware linear interpolation support to achieve higher‐order interpolation. In addition, our formulation allows approximating the results by dynamically skipping some higher order terms with low weights for further savings in both computation and storage. Thus, higher‐order interpolation can be performed adaptively, as needed.We also describe how relatively minor modifications to existing GPU hardware could provide hardware support for quadratic and cubic interpolations using our approach for both texture filtering operations and barycentric interpolation.We present a variety of examples using triangular, rectangular, tetrahedral, and cuboidal interpolations, showing the effectiveness of our higher‐order interpolations in different applications.

Main results of the first Data Assimilation Challenge

This work presents the main results of the first Data Assimilation (DA) challenge, conducted within the framework of the European Union’s Horizon 2020 project HOMER (Holistic Optical Metrology for Aero-Elastic Research), grant agreement number 769237. The challenge was jointly organised by the research groups of DLR, ONERA and TU Delft. The same synthetic test case as in the Lagrangian Particle Tracking (LPT) challenge (also presented in this symposium) was considered, reproducing the flow in the wake of a cylinder in proximity of a flat wall. The participants were provided with three datasets containing the measured particles locations and their trajectories identification numbers, at increasing tracers concentrations from 0.04 to 1.4 particles/mm3 . The requested outputs were the three components of the velocity, the nine components of the velocity gradient and the static pressure, defined on a Cartesian grid at one specific time instant. The results were analysed in terms of errors of the output quantities and their distributions. Additionally, the performances of the different DA algorithms were compared with that of a standard linear interpolation approach. Although the velocity errors were found to be in the same range as those of the linear interpolation algorithm, typically between 3% and 12% of the bulk velocity, the use of the DA algorithms enabled an increase of the measurement spatial resolution by a factor between 3 and 4. The errors of the velocity gradient were of the order of 10-15% of the peak vorticity magnitude. Accurate pressure reconstruction was achieved in the flow field, whereas the evaluation of the surface pressure revealed more challenging.

MorphOT: transport-based interpolation between EM maps with UCSF ChimeraX.

Cryogenic electron microscopy (cryo-EM) offers the unique potential to capture conformational heterogeneity, by solving multiple three-dimensional classes that co-exist within a single cryo-EM image dataset. To investigate the extent and implications of such heterogeneity, we propose to use an optimal-transport-based metric to interpolate barycenters between EM maps and produce morphing trajectories. While standard linear interpolation mostly fails to produce realistic transitions, our method yields continuous trajectories that displace densities to morph one map into the other, instead of blending them. Our method is implemented as a plug-in for ChimeraX called MorphOT, which allows the use of both CPU or GPU resources. The code is publicly available on GitHub (https://github.com/kdd-ubc/MorphOT.git), with documentation containing tutorial and datasets. Supplementary data are available at Bioinformatics online.

Quantum adiabatic algorithms using unitary interpolation**Project supported by the National Natural Science Foundation of China (Grant Nos. 11504430, 11805279, 61501514, and 61502526).

We present two efficient quantum adiabatic algorithms for Bernstein–Vazirani problem and Simon’s problem. We show that the time complexities of the algorithms for Bernstein–Vazirani problem and Simon’s problem are O(1) and O(n), respectively, which are the same complexities as the corresponding algorithms in quantum circuit model. In these two algorithms, the adiabatic Hamiltonians are realized by unitary interpolation instead of standard linear interpolation. Comparing with the adiabatic algorithms using linear interpolation, the energy gaps of our algorithms keep constant. Therefore, the complexities are much easier to analyze using this method.

An Efficient Machine-Learning Approach for PDF Tabulation in Turbulent Combustion Closure

ABSTRACT Probability density function (PDF) based turbulent combustion modeling is limited by the need to store multi-dimensional PDF tables that can take up large amounts of memory. A significant saving in storage can be achieved by using various machine-learning techniques that represent the thermo-chemical quantities of a PDF table using mathematical functions. These functions can be computationally more expensive than the existing interpolation methods used for thermo-chemical quantities. More importantly, the training time can amount to a considerable portion of the simulation time. In this work, we address these issues by introducing an adaptive training algorithm that relies on multi-layer perception (MLP) neural networks for regression and self-organizing maps (SOMs) for clustering data to tabulate using different networks. The algorithm is designed to address both the multi-dimensionality of the PDF table as well as the computational efficiency of the proposed algorithm. SOM clustering divides the PDF table into several parts based on similarities in data. Each cluster of data is trained using an MLP algorithm on simple network architectures to generate ‘local’ functions for thermo-chemical quantities. The algorithm is validated for the so-called DLR-A turbulent jet diffusion flame using both RANS and LES simulations and the results of the PDF tabulation are compared to the standard linear interpolation method. The comparison yields a very good agreement between the two tabulation techniques and establishes the MLP-SOM approach as a viable method for PDF tabulation.

Repeated Look-up Tables.

Efficient hardware implementations routinely approximate mathematical functions with look-up tables, while keeping the error of the approximation under control. For a certain class of commonly occurring 1D functions, namely monotonically increasing or decreasing functions, we found that it is possible to approximate such functions by repeated application of a very low resolution 1D look-up table. There are many advantages to cascading multiple identical LUTs, including the promise of a very simple hardware design and the use of standard linear interpolation. Further, the complexity associated with unequal bin sizes can be avoided. We show that for realistic applications, including gamma correction, high dynamic range encoding and decoding curves, as well as tone mapping and inverse tone mapping applications, multiple cascaded look-up tables can reduce the approximation error by more than 50% compared to a single look-up table with the same total memory footprint.

A Fast Segmentation and Efficient Slice Reconstruction Technique for Head CT Images

AbstractThe three-dimensional (3D) reconstruction of medical images usually requires hundreds of two-dimensional (2D) scan images. Segmentation, an obligatory part in reconstruction, needs to be performed for all the slices consuming enormous storage space and time. To reduce storage space and time, this paper proposes a three-stage procedure, namely, slice selection, segmentation and interpolation. The methodology will have the potential to 3D reconstruct the human head from minimum selected slices. The first stage of slice selection is based on structural similarity measurement, discarding the most similar slices with none or minimal impact on details. The second stage of segmentation of the selected slices is performed using our proposed phase-field segmentation method. Validation of our segmentation results is done via comparison with other deformable models, and results show that the proposed method provides fast and accurate segmentation. The third stage of interpolation is based on modified curvature registration-based interpolation, and it is applied to re-create the discarded slices. This method is compared to both standard linear interpolation and registration-based interpolation in 100 tomographic data sets. Results show that the modified curvature registration-based interpolation reconstructs missing slices with 96% accuracy and shows an improvement in sensitivity (95.802%) on par with specificity (95.901%).

TSR-TVD: Temporal Super-Resolution for Time-Varying Data Analysis and Visualization

We present TSR-TVD, a novel deep learning framework that generates temporal super-resolution (TSR) of time-varying data (TVD) using adversarial learning. TSR-TVD is the first work that applies the recurrent generative network (RGN), a combination of the recurrent neural network (RNN) and generative adversarial network (GAN), to generate temporal high-resolution volume sequences from low-resolution ones. The design of TSR-TVD includes a generator and a discriminator. The generator takes a pair of volumes as input and outputs the synthesized intermediate volume sequence through forward and backward predictions. The discriminator takes the synthesized intermediate volumes as input and produces a score indicating the realness of the volumes. Our method handles multivariate data as well where the trained network from one variable is applied to generate TSR for another variable. To demonstrate the effectiveness of TSR-TVD, we show quantitative and qualitative results with several time-varying multivariate data sets and compare our method against standard linear interpolation and solutions solely based on RNN or CNN.

Shifting Interpolation Kernel Toward Orthogonal Projection

Orthogonal projection offers the optimal solution for many sampling-reconstruction problems in terms of the least square error. In the standard interpolation setting where the sampling is assumed to be ideal, however, the projection is impossible unless the interpolation kernel is related to the sinc function and the input is bandlimited. In this paper, we propose a notion of shifting kernel toward the orthogonal projection. For a given interpolation kernel, we formulate optimization problems whose solutions lead to shifted interpolations that, while still being interpolatory, are closest to the orthogonal projection in the sense of the minimax regret. The quality of interpolation is evaluated in terms of the average approximation error over input shift. For the standard linear interpolation, we obtain several values of optimal shift, dependent on a priori information on input signals. For evaluation, we apply the new shifted linear interpolations to a Gaussian signal, an ECG signal, a speech signal, a two-dimensional signal, and three natural images. Significant improvements are observed over the standard and the 0.21-shifted linear interpolation proposed early.

Accurate signal reconstruction for higher order Lagrangian–Eulerian back-coupling in multiphase turbulence

Multiphase flow simulation serves a vital purpose in applications as diverse as engineering design, natural disaster prediction, and even study of astrophysical phenomena. In these scenarios, it can be very difficult, expensive, or even impossible to fully represent the physical system under consideration. Even still, many such real-world applications can be modeled as a two-phase flow containing both continuous and dispersed phases. Consequentially, the continuous phase is thought of as a fluid and the dispersed phase as particles. The continuous phase is typically treated in the Eulerian frame of reference and represented on a fixed grid, while the dispersed phase is treated in the Lagrangian frame and represented by a sample distribution of Lagrangian particles that approximate a cloud. Coupling between the phases requires interpolation of the continuous phase properties at the locations of the Lagrangian particles. This interpolation step is straightforward and can be performed at higher order accuracy. The reverse process of projecting the Lagrangian particle properties from the sample points to the Eulerian grid is complicated by the time-dependent non-uniform distribution of the Lagrangian particles. In this paper we numerically examine three reconstruction, or projection, methods: (i) direct summation (DS), (ii) least-squares, and (iii) sparse approximation. We choose a continuous representation of the dispersed phase property that is systematically varied from a simple single mode periodic signal to a more complex artificially constructed turbulent signal to see how each method performs in reconstruction. In these experiments, we show that there is a link between the number of dispersed Lagrangian sample points and the number of structured grid points to accurately represent the underlying functional representation to machine accuracy. The least-squares method outperforms the other methods in most cases, while the sparse approximation method is able to capture physically important flow features when under-sampled but at an increased cost. Interestingly, the DS method has been used in the past but in comparison to the other two methods it offers only first order convergence. The above three methods were also compared against standard linear and cubic interpolation techniques from non-uniform Lagrangian points to the Eulerian grid. The performance of these standard interpolation methods were only of the order of DS.

Analysis of Accuracy of Interpolation Methods in Estimating the Output Factors for Square Fields in Medical Linear Accelerator

Introduction: To estimate the accuracy levels of Lagrange, Newton backward interpolation, and linear interpolation methods in estimating the output factors for square fields used in linear accelerator for 6 MV photons at various depths. Materials and Methods: Ionization measurements were carried out in radiation field analyzer in linear accelerator for 6 MV beams at the depths of 5 and 10 cm by 0.6 cc Farmer-type ionisation chamber. Dosimetry was performed by ion collection method with 0.5 cm2  interval for square fields from 4 × 4 cm2 to 40 × 40 cm2 field sizes. The measured output factor values for 10 square field sizes with equal interval were taken for interpolating the intermediate square field size values. The Lagrange and Newton backward methods were used for predicting the intermediate output factors. Results: The percentage of deviation from the measured value was estimated for all the three methods. The calculated output factor values of the two proposed methods were compared with the standard linear interpolation method used in routine clinical practice. It was observed that the Lagrange and Newton backward methods were not significantly different from the measured value (P=0.77). The linear interpolation values were significantly different from the measured value (p <0.01). Conclusion: It is recommended to use the Lagrange and Newton backward interpolation methods to estimate the intermediate output factors to increase accuracy in treatment delivery. The routine linear interpolation method can be applied only for small intervals. This proposed interpolation method is highly associated with the measured values in all the interval levels.

Adaptive flux-based nodeless variable finite element formulation with error estimation for thermal-structural analysis

A posteriori error estimation for the nodeless variable finite element method is presented. A nodeless variable finite element method using flux-based formulation is developed and combined with an adaptive meshing technique to analyze two-dimensional thermal-structural problems. The continuous flux and stresses are determined by using the flux-based formulation while the standard linear element interpolation functions are used to determine the discontinuous flux and stresses. To measure the global error, the L2 norm error is selected to find the root-mean-square error over the entire domain. The finite element formulation and its detailed finite element matrices are presented. Accuracy of the estimated error is measured by the percentage relative error. An adaptive meshing technique, that can generate meshes corresponding to solution behaviors automatically, is implemented to further improve the solution accuracy. Several examples are presented to evaluate the performance and accuracy of the combined method.

Comparison Between the Mean-Variance Optimal and the Mean-Quadratic-Variation Optimal Trading Strategies

ABSTRACTWe compare optimal liquidation policies in continuous time in the presence of trading impact using numerical solutions of Hamilton–Jacobi–Bellman (HJB) partial differential equations (PDEs). In particular, we compare the time-consistent mean-quadratic-variation strategy with the time-inconsistent (pre-commitment) mean-variance strategy. We show that the two different risk measures lead to very different strategies and liquidation profiles. In terms of the optimal trading velocities, the mean-quadratic-variation strategy is much less sensitive to changes in asset price and varies more smoothly. In terms of the liquidation profiles, the mean-variance strategy is much more variable, although the mean liquidation profiles for the two strategies are surprisingly similar. On a numerical note, we show that using an interpolation scheme along a parametric curve in conjunction with the semi-Lagrangian method results in significantly better accuracy than standard axis-aligned linear interpolation. We also demonstrate how a scaled computational grid can improve solution accuracy.

X-ray CT Metal Artifact Reduction Using Wavelet Domain &lt;formula formulatype="inline"&gt;&lt;tex Notation="TeX"&gt;$L_{0}$&lt;/tex&gt;&lt;/formula&gt; Sparse Regularization

X-ray computed tomography (CT) imaging of patients with metallic implants usually suffers from streaking metal artifacts. In this paper, we propose a new projection completion metal artifact reduction (MAR) algorithm by formulating the completion of missing projections as a regularized inverse problem in the wavelet domain. The Douglas-Rachford splitting (DRS) algorithm was used to iteratively solve the problem. Two types of prior information were exploited in the algorithm: 1) the sparsity of the wavelet coefficients of CT sinograms in a dictionary of translation-invariant wavelets and 2) the detail wavelet coefficients of a prior sinogram obtained from the forward projection of a segmented CT image. A pseudo- L0 synthesis prior was utilized to exploit and promote the sparsity of wavelet coefficients. The proposed L0-DRS MAR algorithm was compared with standard linear interpolation and the normalized metal artifact reduction (NMAR) approach proposed by Meyer using both simulated and clinical studies including hip prostheses, dental fillings, spine fixation and electroencephalogram electrodes in brain imaging. The qualitative and quantitative evaluations showed that our algorithm substantially suppresses streaking artifacts and can outperform both linear interpolation and NMAR algorithms.

Impact of Global Geographic Region on Time in Therapeutic Range on Warfarin Anticoagulant Therapy: Data From the ROCKET AF Clinical Trial

BackgroundVitamin K antagonist (VKA) therapy remains the most common method of stroke prevention in patients with atrial fibrillation. Time in therapeutic range (TTR) is a widely cited measure of the quality of VKA therapy. We sought to identify factors associated with TTR in a large, international clinical trial.Methods and ResultsTTR (international normalized ratio [INR] 2.0 to 3.0) was determined using standard linear interpolation in patients randomized to warfarin in the ROCKET AF trial. Factors associated with TTR at the individual patient level (i‐TTR) were determined via multivariable linear regression. Among 6983 patients taking warfarin, recruited from 45 countries grouped into 7 regions, the mean i‐TTR was 55.2% (SD 21.3%) and the median i‐TTR was 57.9% (interquartile range 43.0% to 70.6%). The mean time with INR <2 was 29.1% and the mean time with an INR >3 was 15.7%. While multiple clinical features were associated with i‐TTR, dominant determinants were previous warfarin use (mean i‐TTR of 61.1% for warfarin‐experienced versus 47.4% in VKA‐naïve patients) and geographic region where patients were managed (mean i‐TTR varied from 64.1% to 35.9%). These effects persisted in multivariable analysis. Regions with the lowest i‐TTRs had INR distributions shifted toward lower INR values and had longer inter‐INR test intervals.ConclusionsIndependent of patient clinical features, the regional location of medical care is a dominant determinant of variation in i‐TTR in global studies of warfarin. Regional differences in mean i‐TTR are heavily influenced by subtherapeutic INR values and are associated with reduced frequency of INR testing.Clinical Trial RegistrationURL: ClinicalTrials.gov. Unique identifier: NCT00403767.

Continuous piecewise linear finite elements for the Kirchhoff–Love plate equation

A family of continuous piecewise linear finite elements for thin plate problems is presented. We use standard linear interpolation of the deflection field to reconstruct a discontinuous piecewise quadratic deflection field. This allows us to use discontinuous Galerkin methods for the Kirchhoff-Love plate equation. Three example reconstructions of quadratic functions from linear interpolation triangles are presented: a reconstruction using Morley basis functions, a fully quadratic reconstruction, and a more general least squares approach to a fully quadratic reconstruction. The Morley reconstruction is shown to be equivalent to the Basic Plate Triangle. Given a condition on the reconstruction operator, a priori error estimates are proved in energy norm and $L^2$ norm. Numerical results indicate that the Morley reconstruction/Basic Plate Triangle does not converge on unstructured meshes while the fully quadratic reconstruction show optimal convergence.

Study of the MLB parameterisation for change in surface solar irradiance with sun zenith angle in clear sky

Abstract. The MLB parameterisation (Modified Lambert-Beer, Mueller et al., 2004) describes the change in SSI with sun zenith angle (SZA) in clear-sky conditions. It applies to the direct and global SSI as well as their spectral distribution. We assess its performances by comparing its results to the outputs of the radiative transfer model libRadtran and standard interpolation procedures. The standard two-point fitting MLB function performs very well at SZA between 0° and 60° and fairly bad from 60° to 89.9°. A parameterisation made of four MLBs for four intervals (0°, 60°), (60°, 75°), (75°, 85°) and (85°, 89.9°) is also tested. This piecewise MLB parameterisation exhibits satisfactory performances at any SZA and outperforms standard linear interpolation techniques. 95 % of errors in global SSI are less than 1 W m−2 for each band and less than 5 W m−2 for total irradiance.

A FAST MORPHING-BASED INTERPOLATION FOR MEDICAL IMAGES: APPLICATION TO CONFORMAL RADIOTHERAPY

A method is presented for fast interpolation between medical images. The method is intended for both slice and projective interpolation. It allows offline interpolation between neighboring slices in tomographic data. Spatial correspondence between adjacent images is established using a block matching algorithm. Interpolation of image intensities is then carried out by morphing between the images. The morphing-based method is compared to standard linear interpolation, block-matching-based interpolation and registrationbased interpolation in 3D tomographic data sets. Results show that the proposed method scored similar performance in comparison to registration-based interpolation, and significantly outperforms both linear and block-matching-based interpolation. This method is applied in the context of conformal radiotherapy for online projective interpolation between Digitally Reconstructed Radiographs (DRRs).