Linear Interpolation Algorithm Research Articles

Linearly interpolating missing values in time series helps little for land cover classification using recurrent or attention networks

Satellite time series data, widely used for land cover classification, often contain missing values due to cloud contamination, which can negatively affect classification. Numerous strategies have been developed to reconstruct the missing values to produce regular time series for machine learning classifiers, among which the compositing followed by the linear interpolation is most widely used. However, the classification improvement of linear interpolation for land cover classification has not been examined. Recently developed deep learning models such as long short term memory (LSTM) and Transformer allow such examination as they can classify time series with missing values. In this study, we compared the time series composites with missing values (without linear interpolation) and the linearly interpolated time series composites (without missing values) for land cover classification. About 18 thousand Harmonized Landsat Sentinel-2 (HLS) images acquired over Amur River Basin of China (890,308 km2) in 2021 were composited to 14 16-day periods. Two time series composites were classified, i.e., (i) the 16-day composites without interpolation that have on average 15.35% 16-day periods with missing values and (ii) the linearly interpolated 16-day composites with no missing values. The classifications showed that (1) between classifications with and without linear interpolation there was < 0.2% overall accuracy differences for the bidirectional LSTM (Bi-LSTM) and < 0.5% for the Transformer both of which were smaller than model training randomness; and (2) the computation time can be saved using composites without linear interpolation. The findings suggested that it is unnecessary to use the time-consuming linear interpolation in Bi-LSTM and Transformer-based land cover classifications. The findings were confirmed by experiments for sensitivity to the number of cloud-free composites and to different classification legends using crop type classifications. It implied the linear interpolation algorithm cannot reconstruct reliable time series for land cover classifications and historical use of such method is more about mitigating the inability of traditional classifiers to handle missing values rather than improving classifications. Linear interpolation is not necessary for LSTM and Transformer with capability to handle missing values. The training datasets and developed codes in this study are made publicly available.

Integration of conventional well logs and core samples to predict porosity of tight reservoir: a case study from Ordos Basin

To the reservoirs of the oil wells with no cored data, predicting porosity from wireline logs and core samples is an effective approach. Integration of conventional well logs and core samples to predict porosity with large accuracy is a particularly challenging work due to complex logging responses of tight sandstone. Therefore, a novel predicting workflow based on linear interpolation algorithm (LIA) is described to estimate porosity from well logs in the present study. Based on core reposition, porosity correction under overburden pressure, core-log data matching, and calculation of shale content, two multi regression formulas to estimate porosity values are obtained by nearest neighbor algorithm and linear interpolation algorithm respectively. The formulas are applied to the tight sandstone in Chang 9 member of Yanchang Formation in northeast Wuqi Oilfield, Ordos Basin. The comparison results indicate that the porosity predicted from the formula obtained by LIA is in better agreement with the measured porosity, showing a better prediction effect. The application example demonstrates that the LIA formula is of good applicability for the core porosity prediction in the study region. This methodology can further be applied for porosity prediction in other oil regions that have similarities in geological background.

A frost ice formation method of horizontal axis wind turbine blade based on polynomial fitting

In this paper, a new direct calculation method of frost ice shape on the blade surface of horizontal axis wind turbine is proposed. Using linear interpolation algorithm, the airfoil ice shape obtained by LEWICE 2D software or ice wind tunnel experiment was fitted with equidistant step length in the first and fourth quadrants and equidistant step length in the second and third quadrants. The key point coordinates of ice shapes on cross-sections along the span-wise were mapped into lagging and flapping surfaces through the mathematical dimension reduction, respectively. The polynomial fitting was used to deal with ice projection points of multiple sections in lagging and flapping surfaces, and then the blade’s frost ice shape was obtained. By calculating the sum of squared residuals of the polar diameter at the same polar angle, the errors between experimental and airfoil frost shape fitting methods, experimental and FENSAP, and blade frost shape formation methods and FENSAP were analyzed. The results show that the new method is in good agreement with the ice shape of FENSAP simulation results and experimental results. The residual sum of squares is small. This method makes the analysis of frost ice morphology of wind turbine blades do not need to consider interdisciplinary. The calculation process is simple and reliable.

Industrial design of 3D printing technology combined with assisted medical service robots

AbstractWith the continuous development and maturation of the era of intelligent manufacturing, there is a perpetual emergence of new information technology, control technology, and material technology, which are constantly accelerating 3D printing technology to advance to an unprecedented level. To achieve the safety perception interaction ability of auxiliary medical service robots, this study develops a direct write hybrid 3D printing auxiliary medical service robot system, enabling it to achieve temperature sensing function. Moreover, combined with linear interpolation algorithms, 3D printing technology has been improved to achieve improvement of system control accuracy. The results indicate that the apparent viscosity of the printing material Ag‐TPU is still greater than 2000 Pa s at a rate of 87 s−1. The change in resistance during 20% stretching is within 1.2 Ω, and the change is around 3 Ω during 30% stretching. When the preset temperature is 39.2°C, the absolute deviation is the smallest, about 0.03. When the preset temperature is 41.7°C, the maximum value is approximately 0.17. The absolute error of real‐time temperature collection for auxiliary medical service robots is less than 0.2°C at temperatures ranging from 38 to 42°C. Over the past 30 days of overall operation, the system has had 970 users, 3270 interactions, and 99.4% availability. This system improves the perception and interaction ability of auxiliary medical service robots, which has certain practical potential in the field of medical services.

Multi-Network Latency Prediction for IoT and WSNs

The domain of Multi-Network Latency Prediction for IoT and Wireless Sensor Networks (WSNs) confronts significant challenges. However, continuous research efforts and progress in areas such as machine learning, edge computing, security technologies, and hybrid modelling are actively influencing the closure of identified gaps. Effectively addressing the inherent complexities in this field will play a crucial role in unlocking the full potential of latency prediction systems within the dynamic and diverse landscape of the Internet of Things (IoT). Using linear interpolation and extrapolation algorithms, the study explores the use of multi-network real-time end-to-end latency data for precise prediction. This approach has significantly improved network performance through throughput and response time optimization. The findings indicate prediction accuracy, with the majority of experimental connection pairs achieving over 95% accuracy, and within a 70% to 95% accuracy range. This research provides tangible evidence that data packet and end-to-end latency time predictions for heterogeneous low-rate and low-power WSNs, facilitated by a localized database, can substantially enhance network performance, and minimize latency. Our proposed JosNet model simplifies and streamlines WSN prediction by employing linear interpolation and extrapolation techniques. The research findings also underscore the potential of this approach to revolutionize the management and control of data packets in WSNs, paving the way for more efficient and responsive wireless sensor networks.

Interpolation on the unit hypersphere using the n-dimensional generalized Kuramoto model

Computer graphics, robotics, and physics are one of the many domains where interpolation on the unit sphere S n (often called a unit hypersphere or unit n-sphere) plays a crucial role. In this paper, we introduce a novel approach for achieving smooth and precise interpolation on the unit sphere S n−1 using the n-dimensional generalized Kuramoto model. The proposed algorithm finds the shortest and most direct path between two points on that non-Euclidean manifold. Our simulation results demonstrate that it achieves performance comparable to that of a Spherical Linear Interpolation algorithm. Also, the paper proposes the application of our algorithm in the interpolation of rotations that are presented in the form of four-dimensional data.

A Spatio-Temporal Dynamic Visualization Method of Time-Varying Wind Fields Based on Particle System

The particle system is widely used in vector field feature visualization due to its dynamics and simulation. However, there are some defects of the vector field visualization method based on the Euler fields, such as unclear feature expression and discontinuous temporal expression, so the method cannot effectively express the characteristics of wind field on the temporal scale. We propose a Lagrangian visualization method based on spatio-temporal interpolation to solve these problems, which realizes the fusion and expression of the particle system and the time-varying wind data based on the WebGL shader. Firstly, the linear interpolation algorithm is used to interpolate to obtain continuous and dense wind field data according to the wind field data at adjacent moments. Then, we introduce the Lagrangian analysis method to study the structure of the wind field and optimize the visualization effect of the particle system based on Runge–Kutta algorithms. Finally, we adopt the nonlinear color mapping method with double standard deviation (2SD) to improve the expression effect of wind field features. The experimental results indicate that the wind visualization achieves a comprehensive visual effect and the rendering frame rate is greater than 45. The methods can render the particles smoothly with stable and outstanding uniformity when expressing continuous spatio-temporal dynamic visualization characteristics of the wind field.

Application of a linear interpolation algorithm in radiation therapy dosimetry for 3D dose point acquisition

Air-vented ion chambers are generally used in radiation therapy dosimetry to determine the absorbed radiation dose with superior precision. However, in ion chamber detector arrays, the number of array elements and their spacing do not provide sufficient spatial sampling, which can be overcome by interpolating measured data. Herein, we investigated the potential principle of the linear interpolation algorithm in volumetric dose reconstruction based on computed tomography images in the volumetric modulated arc therapy (VMAT) technique and evaluated how the ion chamber spacing and anatomical mass density affect the accuracy of interpolating new data points. Plane measurement doses on 83 VMAT treatment plans at different anatomical sites were acquired using Octavius 729, Octavius1500, and MatriXX ion chamber detector arrays, followed by the linear interpolation to reconstruct volumetric doses. Dosimetric differences in planning target volumes (PTVs) and organs at risk (OARs) between treatment planning system and reconstruction were evaluated by dose volume histogram metrics. The average percentage dose deviations in the mean dose (Dmean) of PTVs reconstructed by 729 and 1500 arrays ranged from 4.7 to 7.3% and from 1.5 to 2.3%, while the maximum dose (Dmax) counterparts ranged from 2.3 to 5.5% and from 1.6 to 7.6%, respectively. The average percentage dose/volume deviations of mixed PTVs and OARs in the abdomen/gastric and pelvic sites were 7.6%, 3.5%, and 7.2%, while mediastinum and lung plans showed slightly larger values of 8.7%, 5.1%, and 8.9% for 729, 1500, and MatriXX detector arrays, respectively. Our findings indicated that the smaller the spacing between neighbouring detectors and the more ion chambers present, the smaller the error in interpolating new data points. Anatomical regions with small local mass density inhomogeneity were associated with superior dose reconstruction. Given a large mass density difference in the various human anatomical structures and the characteristics of the linear interpolation algorithm, we suggest that an alternative data interpolation method should be used in radiotherapy dosimetry.

Physics-informed sinogram completion for metal artifact reduction in CT imaging

Objective. Metal artifacts in the computed tomography (CT) imaging are unavoidably adverse to the clinical diagnosis and treatment outcomes. Most metal artifact reduction (MAR) methods easily result in the over-smoothing problem and loss of structure details near the metal implants, especially for these metal implants with irregular elongated shapes. To address this problem, we present the physics-informed sinogram completion (PISC) method for MAR in CT imaging, to reduce metal artifacts and recover more structural textures. Approach. Specifically, the original uncorrected sinogram is firstly completed by the normalized linear interpolation algorithm to reduce metal artifacts. Simultaneously, the uncorrected sinogram is also corrected based on the beam-hardening correction physical model, to recover the latent structure information in metal trajectory region by leveraging the attenuation characteristics of different materials. Both corrected sinograms are fused with the pixel-wise adaptive weights, which are manually designed according to the shape and material information of metal implants. To furtherly reduce artifacts and improve the CT image quality, a post-processing frequency split algorithm is adopted to yield the final corrected CT image after reconstructing the fused sinogram. Main results. We qualitatively and quantitatively evaluated the presented PISC method on two simulated datasets and three real datasets. All results demonstrate that the presented PISC method can effectively correct the metal implants with various shapes and materials, in terms of artifact suppression and structure preservation. Significance. We proposed a sinogram-domain MAR method to compensate for the over-smoothing problem existing in most MAR methods by taking advantage of the physical prior knowledge, which has the potential to improve the performance of the deep learning based MAR approaches.

Norm and interpolation based absolute stress evaluation in steel members using the Lcr wave

The uniaxial absolute stress in steel members is a vital indicator for evaluating the performance of steel structures. Non-destructive ultrasonic detection is widely applied in the absolute stress measurement of steel members with the advantages of easy implementation, no limits on materials and high precision. The acquisition algorithm of time-of-flight (TOF), the sampling rate of ultrasonic signal and the processing of outliers determine the accuracy of the stress measurement. In this study, we proposed a norm and interpolation based algorithm for absolute stress measurement using the longitudinal critically refracted (Lcr) waves, which improves the accuracy and velocity of the absolute stress measurement. Firstly, the norm was designed to obtain the TOF of the ultrasonic signal. When only the Lcr wave arose in the ultrasonic signal, the two-norm similarity algorithm was found to be consistent with the cross-correlation-based algorithm and the corresponding theoretical expressions were derived. With the non-Lcr wave in the ultrasonic echo, the two-norm similarity algorithm has stronger anti-interference ability. Secondly, the pchipslopes, spline and linear interpolation algorithms were introduced via which the TOF of ultrasound with low sampling rate reaches the same accuracy as that of high sampling rate signal. Finally, a special class of outliers in TOF was explored, and a threshold based on the number of shift points and the stress feature was proposed to calibrate these outliers to normal values. Experiments demonstrate the proposed algorithm improves the accuracy and velocity of uniaxial absolute stress measurement in structural steel members.

An adaptive infrared image denoising method based on two-dimensional empirical mode decomposition for distribution network inspection UAV

An adaptive denoising method based on 2D empirical mode decomposition (EMD) is proposed to improve the infrared image quality of inspection UNMANNED aerial vehicles (UAVs) and provide guarantee for improving the inspection level of distribution network. Through rapid adaptive two-dimensional empirical mode decomposition algorithm decomposition of a UAV collected for distribution network inspection original noise of infrared image, get more than the IMF component and the residual amount, a forecast noise dominated the IMF component parameters such as threshold value and the variance of noise, using the estimated parameters in combination with the optimal linear interpolation algorithm of noise threshold function of leading the IMF component implementation of threshold denoising. After the denoised IMF component is obtained, the denoised infrared image is obtained after reconstruction with the signal-dominated IMF component, and the adaptive denoising of the infrared image of the distribution network inspection UAV is realized. The experimental results show that the method in this paper can maintain the details of the image, improve the definition, significantly improve the visual effect, the overall denoising performance is stable and feasible, and ensure the quality inspection UAV to collect infrared images.

Optimal linear interpolation algorithm with fixed delay

One of the most important characteristics of satellite navigation receiving equipment is the accuracy of navigation definitions. To improve this characteristic, the paper considers the possibility of using optimal interpolation algorithms with a fixed delay. A new interpolation algorithm with a fixed delay in discrete time was deduced. The use of the proposed interpolation algorithm instead of filtering made it possible to reduce the standard deviation by 2 times with a delay in the output of the result by 10 cycles. The advantage of the proposed algorithm is the reduction in computational complexity compared to the optimal two-way algorithm. The proposed algorithm requires 2 matrix accesses per cycle of operation, subject to the use of additional memory resources, to save pre-calculated elements.

МЕТОД КОДОВОЇ ЛІНІЙНОЇ ІНТЕРПОЛЯЦІЇ ДЛЯ ФОРМУВАННЯ ВІДРІЗКІВ ПРЯМИХ

Graphic images are formed using graphic primitives. These are the smallest, indivisible from the point of view of application programs, graphic elements used as the basis for building more complex images. Among the graphic primitives, the segments of straight lines, for the formation of which provides linear interpolation, have the highest specific weight. The performance of forming a graphic scene depends on the time of vector generation, therefore the question of increasing the performance of linear interpolation is relevant, especially for dynamic images. With the use of matrix screens and matrix executive bodies in registration devices, the possibility of one-stroke reproduction of row or column elements appears, which allows you to significantly increase the speed of these devices. This mode of operation is promising. Its organization requires the development of interpolation methods that allow in one interpolation cycle to receive the increment code in a row or column (code interpolation). A coded linear interpolation method is proposed, the feature of which is determined in each interpolation clock of digital segments, which includes the number of increments of the same type with the same ordinate (abscissa). For this cycle, prepare for interpolation a larger increment of the line segment to a smaller one. In the future, this ratio and the remainder of the division are used to determine digital segments. In the proposed method, the remainder of the division of a larger increment by a smaller one is accumulated, which is equal to the smaller increment. This allows you to eliminate the accumulation of error and ensure that the end point of the straight line segment is reached. The maximum interpolation error in this case does not exceed half of the discretization step, which is due to the symmetry of the error. The code linear interpolation algorithm is proposed. The research carried out in the work can be used to build high-performance computer graphics tools.

Improvement and Comparison of Multi-Reference Station Regional Tropospheric Delay Modeling Method Considering the Effect of Height Difference

Tropospheric delay information is particularly important for network RTK (Network Real-time Kinematic) positioning. Conventionally, tropospheric delay information at a virtual reference station (VRS) is obtained using the linear interpolation method (LIM). However, the conventional LIM cannot work well when there is a substantial height difference between the rover station and the reference station. Consequently, we propose a modified linear interpolation method (MLIM) by carefully handling the height difference between the rover station and the reference station. The new MLIM method first corrects the systematic error of the double-difference (DD) tropospheric delay in the elevation direction caused by the height difference, and then utilizes the linear interpolation algorithm to obtain the tropospheric delay of the VRS station. To determine the parameters of the low-order surface model (LSM), we also propose a modified LSM (MLSM) interpolation method in the triangular network and evaluate it in the positioning domains. The two new interpolation methods are evaluated using two regional GNSS networks with obvious height disparities. Results show that the DD tropospheric delay interpolation accuracy obtained by the new MLIM and MLSM is improved by 56.5% and 78.7% on average in the two experiments compared to the conventional method. The new MLIM and MLSM are more accurate than the traditional LIM (TLIM) in cases with low elevation satellites. Additionally, the positioning accuracies are improved by using the MLIM and MLSM methods. The MLIM and MLSM outperform TLIM in the up-component by an average of 72.8% and 80.7%, respectively.

Detection of surface water temperature variations of Mongolian lakes benefiting from the spatially and temporally gap-filled MODIS data

• We applied an effective spatiotemporal gap-filled method to improve the LSWT from MODIS. • The long-term variations of the annual and seasonal mean LSWTs of Mongolian lakes were provided. • The LSWT of Mongolian lakes confirmed the phenomenon “global warming hiatus”. Lakes provide critical water resources for human activities and ecosystems, particularly in the Mongolian Plateau (MP), which is characterized by a dry climate and a harsh environment. As a region that is sensitive to anthropogenic warming, tracking lake surface water temperature (LSWT) changes in Mongolian lakes is crucial for understanding the consequences of a warming climate on lake ecosystems. However, the long-term monitoring of LSWT is restricted by the spatiotemporal gaps in the raw imagery of remote sensing-based land surface temperature (LST), e.g., the commonly used Moderate Resolution Imaging Spectroradiometer (MODIS) LST products. This study applied an improved gap-filling method by utilizing the discrete cosine transform-based penalized least squares (DCT-PLS) strategy in the spatial domain combined with the linear interpolation (LI) algorithm in the temporal domain. The method was applied to fill gaps in the LSWT imagery of 12 representative lakes across MP. The randomly sampled high-quality MODIS LSWT values in the spatial and temporal domains were excavated as false data gaps and considered “virtual true” validation datasets. The spatial validation results showed that the estimated LSWT for all the lake cases were comparable with the “virtual true” LSWT values, with the average values of the coefficient of determination, mean absolute error, mean square error, and root mean square error being 0.98, 0.38 °C, 0.45 °C, and 0.59 °C, respectively. Meanwhile, the error of nighttime LSWT results was relatively lower than that of daytime LSWT. For temporal interpolation validation, the LI algorithm exhibited relatively better performance and could more objectively indicate the variation in LSWT. Benefiting from the spatially and temporally well-constrained data, we analyzed the interannual and intra-annual change characteristics of the LSWTs of the 12 lakes. The long-term variations of annual and seasonal mean LSWTs in the 12 selected lakes exhibited no evident trends in 2000–2020, while presented apparent interannual fluctuations. The slight changes in the average LSWTs of the 12 selected lakes were in excellent synchronization with the surrounding LST derived from the reanalysis datasets, confirming the widely reported phenomenon of “global warming hiatus” that occurred in the early 21st century. This study improves the understanding of the LSWT variations in Mongolian lakes in response to global climate change. It has the potential to provide an effective approach for monitoring LSWT changes in other large-scale studies.

Design optimization of a blunt trailing-edge airfoil for wind turbines under rime ice conditions

A new optimization method is developed for the design of a blunt trailing-edge airfoil for wind turbines under rime ice conditions. The parametric representation of the airfoil is given using the profile integration theory and B-spline function. The rime ice shape from wind tunnel tests is fitted using a linear interpolation algorithm with equidistant and equiangular steps to preserve the same number of ice shape feature points. The blunt trailing-edge thickness and distribution ratio on the upper side of the middle arc are included in the design variables. The optimizer, based on the Bare-Bones Multi-Objective Particle Swarm Optimization (BB-MOPSO) algorithm integrated with ICEM-CFD and FLUENT software, seeks the solutions maximizing the lift coefficient and lift-drag ratio. A new airfoil NACA0012BT (with BT denoting the blunt trailing-edge) with the trailing-edge thickness of 2.2004 %c (with c denoting the chord length) and distribution ratio of 1:55.8482 is obtained, and the lift and drag coefficients, lift-drag ratios, pressure distributions, and flow characteristics are investigated using the Computational Fluid Dynamics (CFD) method. Significant improvements of the NACA0012 airfoil design are achieved in this process, confirming that the developed method constitutes a valuable tool for designing and optimizing the wind turbine airfoil operating in icing conditions.

A Non-Local Tensor Completion Algorithm Based on Weighted Tensor Nuclear Norm

In this paper, we proposed an image inpainting algorithm, including an interpolation step and a non-local tensor completion step based on a weighted tensor nuclear norm. Specifically, the proposed algorithm adopts the triangular based linear interpolation algorithm firstly to preform the observation image. Second, we extract the non-local similar patches in the image using the patch match algorithm and rearrange them to a similar tensor. Then, we use the tensor completion algorithm based on the weighted tensor nuclear norm to recover the similar tensors. Finally, we regroup all these recovered tensors to obtain the final recovered image. From the image inpainting experiments on color RGB images, we can see that the performance of the proposed algorithm on the peak signal-to-noise ratio (PSNR) and the relative standard error (RSE) are significantly better than other image inpainting methods.

Needle Tip Pose Estimation for Ultrasound-Guided Steerable Flexible Needle With a Complicated Trajectory in Soft Tissue

Visualization of the surgical needle is critical for an image guided insertion. However, it is difficult to obtain a 3D tip pose of a steerable flexible needle under 2D US images. In this letter, an image processing method is used to extract the needle shaft radial cross-sectional centroid coordinates (NSRCCCs). Based on the NSRCCCs, two methods are proposed to estimate the needle tip pose in real-time, which are based on Rodrigues’ rotation formula, and Fourier expansion with three-terms and linear interpolation algorithm (FEWT&LIA), respectively. In order to validate the methods proposed in this letter, the insertion experiments with different depths are conducted in phantom tissues, and the results show that the mean errors are 0.68 mm and 0.65 mm with using the Rodrigues’ rotation formula based method and the FEWT&LIA based method, respectively, showing clinically acceptable accuracy.

Interpolation Methods with Phase Control for Backprojection of Complex-Valued SAR Data.

Time-domain backprojection algorithms are widely used in state-of-the-art synthetic aperture radar (SAR) imaging systems that are designed for applications where motion error compensation is required. These algorithms include an interpolation procedure, under which an unknown SAR range-compressed data parameter is estimated based on complex-valued SAR data samples and backprojected into a defined image plane. However, the phase of complex-valued SAR parameters estimated based on existing interpolators does not contain correct information about the range distance between the SAR imaging system and the given point of space in a defined image plane, which affects the quality of reconstructed SAR scenes. Thus, a phase-control procedure is required. This paper introduces extensions of existing linear, cubic, and sinc interpolation algorithms to interpolate complex-valued SAR data, where the phase of the interpolated SAR data value is controlled through the assigned a priori known range time that is needed for a signal to reach the given point of the defined image plane and return back. The efficiency of the extended algorithms is tested at the Nyquist rate on simulated and real data at THz frequencies and compared with existing algorithms. In comparison to the widely used nearest-neighbor interpolation algorithm, the proposed extended algorithms are beneficial from the lower computational complexity perspective, which is directly related to the offering of smaller memory requirements for SAR image reconstruction at THz frequencies.

Calculating the grain boundary inclination of voxelated grain structures using a smoothing algorithm

We have developed a flexible method for calculating the grain boundary (GB) inclinations of voxelated grain structure data using smoothing algorithms. We compared the performance of four algorithms: the linear interpolation, Allen–Cahn, level-set, and vertex algorithms. We assessed their accuracy using 2D and 3D cases with known inclinations. The vertex algorithm provided the best balance between accuracy and efficiency for 2D structures while the linear interpolation algorithm provided the best balance for 3D structures. We compared the GB inclinations calculated using our smoothing method on a 3D high energy X-ray diffraction microscopy (HEDM) dataset to those determined by meshing the GBs. The two approaches determined similar GB plane distributions, though they varied significantly at triple junctions. Finally, the smoothing method was demonstrated for two sources of 3D voxelated grain structures: HEDM data and results from Monte Carlo Potts grain growth simulations.