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At least 1,135 records · Page 63
All sourcesNASA NTRSDOE OSTI
DOE OSTI2015

Including parameter dependence in the data and covariance for cosmological inference

The final step of most large-scale structure analyses involves the comparison of power spectra or correlation functions to theoretical models. It is clear that the theoretical models have parameter dependence, but frequently the measurements and the covariance matrix depend upon some of the parameters as well. We show that a very simple interpolation scheme from an unstructured mesh allows for an efficient way to include this parameter dependence self-consistently in the analysis at modest computational expense. We describe two schemes for covariance matrices. The scheme which uses the geometric structure of such matrices performs roughly twice as well as the simplest scheme, though both perform very well.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2014

CHROMOSPHERIC AND CORONAL OBSERVATIONS OF SOLAR FLARES WITH THE HELIOSEISMIC AND MAGNETIC IMAGER

We report observations of white-light ejecta in the low corona, for two X-class flares on 2013 May 13, using data from the Helioseismic and Magnetic Imager (HMI) of the Solar Dynamics Observatory. At least two distinct kinds of sources appeared (chromospheric and coronal), in the early and later phases of flare development, in addition to the white-light footpoint sources commonly observed in the lower atmosphere. The gradual emissions have a clear identification with the classical loop-prominence system, but are brighter than expected and possibly seen here in the continuum rather than line emission. We find the HMI flux exceeds the radio/X-ray interpolation of the bremsstrahlung produced in the flare soft X-ray sources by at least one order of magnitude. This implies the participation of cooler sources that can produce free-bound continua and possibly line emission detectable by HMI. One of the early sources dynamically resembles {sup c}oronal rain{sup ,} appearing at a maximum apparent height and moving toward the photosphere at an apparent constant projected speed of 134 ± 8 km s{sup –1}. Not much literature exists on the detection of optical continuum sources above the limb of the Sun by non-coronagraphic instruments and these observations have potential implications for our basic understanding of flare development, since visible observations can in principle provide high spatial and temporal resolution.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

Connect the Dots: In Situ 4-D Seismic Monitoring of CO 2 Storage With Spatio-Temporal CNNs

4-D seismic imaging has been widely used in CO 2 sequestration projects to monitor the fluid flow in the volumetric subsurface region that is not sampled by wells. Ideally, real-time monitoring and near-future forecasting would provide site operators with great insights to understand the dynamics of the subsurface reservoir and assess any potential risks. However, due to obstacles such as high deployment cost, availability of acquisition equipment, exclusion zones around surface structures, only very sparse seismic imaging data can be obtained during monitoring. That leads to an unavoidable and growing knowledge gap over time. The operator needs to understand the fluid flow throughout the project lifetime and the seismic data are only available at a limited number of times. This is insufficient for understanding reservoir behavior. To overcome those challenges, we have developed spatio-temporal neural-network-based models that can produce high-fidelity interpolated or extrapolated images effectively and efficiently. Specifically, our models are built on an autoencoder, and incorporate the long short-term memory (LSTM) structure with a new loss function regularized by optical flow. We validate the performance of our models using real 4-D post-stack seismic imaging data acquired at the Sleipner CO 2 sequestration field. We employ two different strategies in evaluating our models. Numerically, we compare our models with different baseline approaches using classic pixel-based metrics. We also conduct a blind survey and collect a total of 20 responses from domain experts to evaluate the quality of data generated by our models. Finally, via both numerical and expert evaluation, we conclude that our models can produce high-quality 2-D/3-D seismic imaging data at a reasonable cost, offering the possibility of real-time monitoring or even near-future forecasting of the CO 2 storage reservoir.

4-D seismic imaging↗
DOE OSTI2021

A Comparative Study of Straight-Strip and Zigzag-Interleaved Anode Patterns for MPGD Readouts

Due to their simplicity and versatility of design, straight strip or rectangular pad anode structures are frequently employed with micro-pattern gas detectors to reconstruct high precision space points for various tracking applications. The particle impact point is typically determined by interpolating the charge collected by several neighboring strips. However, to effectively extract the inherent positional information, the lateral spacing of the straight strips must be comparable to or preferably smaller than the full extent of the incident charge cloud. In contrast, highly interleaved anode patterns, such as zigzags, can adequately sample the incident charge with a pitch appreciably larger than the charge cloud.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2017

Optimal Allocation of Dynamic Var Sources Using the Voronoi Diagram Method Integrating Linear Programing

Dynamic reactive power (var) sources can effectively mitigate fault-induced delayed voltage recovery (FIDVR) issues. This paper optimizes the sizes of dynamic var sources at given locations against FIDVR issues under severe contingencies. First, the geometric characteristics about the non-convex solution space of this problem are studied. Accordingly, a Voronoi diagram approach integrating linear programming (LP) is proposed, which disperses a number of sample points of potential solutions in the searching space to construct a Voronoi diagram blending the local cost functions over the entire space by Barycentric interpolation in Voronoi regions. New sample points are then recursively added, including the tentative optimal point using LP, the most depopulated area point ensuring global fidelity, and the connecting point, until the stopping criterion is met. The new approach is demonstrated in detail on the WSCC 9-bus system. A case study on the NPCC 140-bus system also validates that the proposed approach can effectively estimate the boundary and the geometry of the feasible solution region in the searching space and find the optimal solution.

barycentric interpolation↗
DOE OSTI2021

Multifractal Characterization of Distribution Synchrophasors for Cybersecurity Defense of Smart Grids

“Source ID Mix” spoofing emerged as a new type of cyber-attack on Distribution Synchrophasors (DS) where adversaries have the capability to swap the source information of DS without changing the measurement values. Accurate detection of such a highly-deceptive attack is a challenging task especially when the spoofing attack happens on short fragments of DS recorded within a relatively small geographical scale. Herein this letter proposes an effective approach to detect this cyber-attack by realizing the multifractal characteristics of DS measurements. First, the multifractal cross-correlation of DS measured at multiple intra-state locations is revealed. Then the derived correlation is integrated with weighted two-dimensional multifractal surface interpolation to reconstruct quasi high-resolution signals. Finally, informative location-specific signatures are extracted from the high-resolution DS and they are integrated with advanced machine learning techniques for source authentication. Experiments using the real-life DS are performed to verify the proposed method.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2022

Deep Hierarchical Super Resolution for Scientific Data

We present a novel technique for hierarchical super resolution (SR) with neural networks (NNs), which upscales volumetric data represented with an octree data structure to a high-resolution uniform gridwith minimal seam artifacts on octree node boundaries. Our method uses existing state-of-the-art SR models and adds flexibility to upscale input data with varying levels of detail across the domain, instead of only uniform grid data that are supported in previous approaches.The key is to use a hierarchy of SR NNs, each trained to perform 2x SR between two levels of detail, with a hierarchical SR algorithm that minimizes seam artifacts by starting from the coarsest level of detail and working up.We show that our hierarchical approach outperforms baseline interpolation and hierarchical upscaling methods, and demonstrate the usefulness of our proposed approach across three use cases including data reduction using hierarchical downsampling+SR instead of uniform downsampling+SR, computation savings for hierarchical finite-time Lyapunov exponent field calculation, and super-resolving low-resolution simulation results for a high-resolution approximation visualization.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Monotonic Gaussian Process for Physics-Constrained Machine Learning With Materials Science Applications

Physics-constrained machine learning is emerging as an important topic in the field of machine learning for physics. One of the most significant advantages of incorporating physics constraints into machine learning methods is that the resulting model requires significantly less data to train. By incorporating physical rules into the machine learning formulation itself, the predictions are expected to be physically plausible. Gaussian process (GP) is perhaps one of the most common methods in machine learning for small datasets. In this paper, we investigate the possibility of constraining a GP formulation with monotonicity on three different material datasets, where one experimental and two computational datasets are used. The monotonic GP is compared against the regular GP, where a significant reduction in the posterior variance is observed. The monotonic GP is strictly monotonic in the interpolation regime, but in the extrapolation regime, the monotonic effect starts fading away as one goes beyond the training dataset. Imposing monotonicity on the GP comes at a small accuracy cost, compared to the regular GP. The monotonic GP is perhaps most useful in applications where data are scarce and noisy, and monotonicity is supported by strong physical evidence.

36 MATERIALS SCIENCE↗
DOE OSTI2013

Heel effect adaptive flat field correction of digital x-ray detectors

Purpose: Anode heel effect renders large-scale background nonuniformities in digital radiographs. Conventional offset/gain calibration is performed at mono source-to-image distance (SID), and disregards the SID-dependent characteristic of heel effect. It results in a residual nonuniform background in the corrected radiographs when the SID settings for calibration and correction differ. In this work, the authors develop a robust and efficient computational method for digital x-ray detector gain correction adapted to SID-variant heel effect, without resorting to physical filters, phantoms, complicated heel effect models, or multiple-SID calibration and interpolation.Methods: The authors present the Duo-SID projection correction method. In our approach, conventional offset/gain calibrations are performed only twice, at the minimum and maximum SIDs of the system in typical clinical use. A fast iterative separation algorithm is devised to extract the detector gain and basis heel patterns from the min/max SID calibrations. The resultant detector gain is independent of SID, while the basis heel patterns are parameterized by the min- and max-SID. The heel pattern at any SID is obtained from the min-SID basis heel pattern via projection imaging principles. The system gain desired at a specific acquisition SID is then constructed using the projected heel pattern and detector gain map.Results: The method was evaluated for flat field and anatomical phantom image corrections. It demonstrated promising improvements over interpolation and conventional gain calibration/correction methods, lowering their correction errors by approximately 70% and 80%, respectively. The separation algorithm was able to extract the detector gain and heel patterns with less than 2% error, and the Duo-SID corrected images showed perceptually appealing uniform background across the detector.Conclusions: The Duo-SID correction method has substantially improved on conventional offset/gain corrections for digital x-ray imaging in an SID-variant environment. The technique is relatively simple, and can be easily incorporated into multiple-point gain calibration/correction techniques. It offers a potentially valuable tool for preprocessing digital x-ray images to boost image quality of mammography, chest and cardiac radiography, as well as automated computer aided diagnostic radiology.

62 RADIOLOGY AND NUCLEAR MEDICINE↗
DOE OSTI2014

SU-E-J-23: An Accurate Algorithm to Match Imperfectly Matched Images for Lung Tumor Detection Without Markers

Purpose: In order to locate lung tumors on projection images without internal markers, digitally reconstructed radiograph (DRR) is created and compared with projection images. Since lung tumors always move and their locations change on projection images while they are static on DRRs, a special DRR (background DRR) is generated based on modified anatomy from which lung tumors are removed. In addition, global discrepancies exist between DRRs and projections due to their different image originations, scattering, and noises. This adversely affects comparison accuracy. A simple but efficient comparison algorithm is reported. Methods: This method divides global images into a matrix of small tiles and similarities will be evaluated by calculating normalized cross correlation (NCC) between corresponding tiles on projections and DRRs. The tile configuration (tile locations) will be automatically optimized to keep the tumor within a single tile which has bad matching with the corresponding DRR tile. A pixel based linear transformation will be determined by linear interpolations of tile transformation results obtained during tile matching. The DRR will be transformed to the projection image level and subtracted from it. The resulting subtracted image now contains only the tumor. A DRR of the tumor is registered to the subtracted image to locate the tumor. Results: This method has been successfully applied to kV fluoro images (about 1000 images) acquired on a Vero (Brainlab) for dynamic tumor tracking on phantom studies. Radiation opaque markers are implanted and used as ground truth for tumor positions. Although, other organs and bony structures introduce strong signals superimposed on tumors at some angles, this method accurately locates tumors on every projection over 12 gantry angles. The maximum error is less than 2.6 mm while the total average error is 1.0 mm. Conclusion: This algorithm is capable of detecting tumor without markers despite strong background signals.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2014

SU-E-T-140: Dynamic Wave Arc Trajectory Verification Using KV X-Ray Fluoroscopy

Purpose: Purpose: This study investigates the geometric accuracy of simultaneous Gantry/Ring rotation during Dynamic Wave Arc (DWA) delivery. Methods: The Vero SBRT system consists of a 6MV LINAC mounted on an O-ring gantry that can rotate around the vertical axis (±60°), similar to couch rotation on C-arm gantries. To provide CBCT and fluoroscopy imaging functionalities, two orthogonal kV imaging units are attached to the O-ring at −45°/+45° from the beam axis.Dynamic Wave Arc maximizes Vero's motion capabilities by employing synchronized gantry and ring motion on a complex non-coplanar trajectory in combination with aperture based optimized MLC segments.Four wave arc trajectories (T1-4) were delivered using a cubic phantom with a configuration of five lead beads. O-ring gantry position information was retrieved through continuous dual-source kV X-ray image acquisition during DWA. An in-house algorithm read in the image set, extracted the projected marker positions and determined the angulation through reconstruction of the beam source position. The geometric error was quantified as the distance between the independently detected positions from kV-images and reference trajectory derived from the treatment plan in the Ring-Gantry space. Results: The average displacement between the 3D gantry/ring positions reconstructed from the fluoroscopy images and the reference trajectory was 0.346 mm (SD 0,171) for T1. A mean offset of 0.348 mm (SD 0,182) and 0.357 mm (SD 0.194) was observed for trajectory T2(2segmens) and T3(4segments), respectively. The saw shape T4 presented a mean geometric error of 0.363 (SD 0.156). The overall systematic error of 0.350 was caused by the difference between planned reference trajectory created by linear interpolation between CP, and the machine delivery following a spline curve. Conclusion: An independent geometric QA approach has been developed for DWA delivery verification, successfully applied on diverse trajectories and disclosed that the Vero system is capable of accurately follow complex trajectories.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2014

SU-E-T-444: Quantitatively Comparison of Low Dose Spillage Outside of PTV Edge in Arc Therapy Modalities

Purpose: To quantitatively compare low dose spillage outside of PTV edge in arc therapy modalities Methods: The machines used in the study are Tomotherapy Hi-Arc and Varian 21EX with millennium120 MLC. TPS are TomoPlaning and RayStation for VMAT, respectively. The phantom is a 30cm diameter cylindrical solid water (TOMOTHERAPY, TOMOPHANTOM ASSY). The PTV is 4cm length with ellipsoidal sectional shape with major axis=5cm, minor axis=3cm in the axial plane and reversed in the coronal plane. The PTV volume is created with interpolation. It is located at the center of the phantom. The prescribed dose is 1000x5 cGy to 95% the PTV. The isocenter is set co-centered with the PTV. EBT-3 film was used to measure iso-dose lines at the center plane. Film dosimetry is performed with the RIT, v6.2. Results: the study shows: (1) dose falloff gradient is usually uneven, depending on the PTV shape in the gantry rotation plane. For an elliptical shape, the low dose spillage is wider in the minor axis direction than that in the major axis direction. The more a shape is closer to circular, the more even gradient is all directions; (2)for a circular shape (CAX plane in this study), the maximum dose in % of Rx dose at 2cm from PTV is 55% for Tomo, vs. 70% for VMAT (3) the most rapid dose falloff rang is between 95%–80% IDL for both modalities. Conclusion: Tomo has more rapid dose falloff outside of PTV. In some areas, the gradient is double for Tomo helical than that for LINAC VMAT at same points. Future work will examine the differences between optimization of doses and inherent delivery limitations.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2016

SU-D-206-06: Task-Specific Optimization of Scintillator Thickness for CMOS-Detector Based Cone-Beam Breast CT

Purpose: To optimize the cesium iodide (CsI:Tl) scintillator thickness in a complimentary metal-oxide semiconductor (CMOS)-based detector for use in dedicated cone-beam breast CT. Methods: The imaging task considered was the detection of a microcalcification cluster comprising six 220µm diameter calcium carbonate spheres, arranged in the form of a regular pentagon with 2 mm spacing on its sides and a central calcification, similar to that in ACR-recommended mammography accreditation phantom, at a mean glandular dose of 4.5 mGy. Generalized parallel-cascades based linear systems analysis was used to determine Fourier-domain image quality metrics in reconstructed object space, from which the detectability index inclusive of anatomical noise was determined for a non-prewhitening numerical observer. For 300 projections over 2π, magnification-associated focal-spot blur, Monte Carlo derived x-ray scatter, K-fluorescent emission and reabsorption within CsI:Tl, CsI:Tl quantum efficiency and optical blur, fiberoptic plate transmission efficiency and blur, CMOS quantum efficiency, pixel aperture function and additive noise, and filtered back-projection to isotropic 105µm voxel pitch with bilinear interpolation were modeled. Imaging geometry of a clinical prototype breast CT system, a 60 kV Cu/Al filtered x-ray spectrum from 0.3 mm focal spot incident on a 14 cm diameter semi-ellipsoidal breast were used to determine the detectability index for 300–600 µm thick (75µm increments) CsI:Tl. The CsI:Tl thickness that maximized the detectability index was considered optimal. Results: The limiting resolution (10% modulation transfer function, MTF) progressively decreased with increasing CsI:Tl thickness. The zero-frequency detective quantum efficiency, DQE(0), in projection space increased with increasing CsI:Tl thickness. The maximum detectability index was achieved with 525µm thick CsI:Tl scintillator. Reduced MTF at mid-to-high frequencies for 600µm thick CsI:Tl lowered the detectability index than 525µm CsI:Tl. Conclusion: For the x-ray spectrum and imaging conditions considered, a 525µm thick CsI:Tl scintillator integrated with the CMOS detector is optimal for detecting microcalcification cluster. Funding support: Supported in part by NIH R01 CA195512. The contents are solely the responsibility of the authors and do not reflect the official views of the NIH or the NCI. Disclosures: SV, GV and AK - Research collaboration, Koning Corp., West Henrietta, NY.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2016

SU-F-R-35: Repeatability of Texture Features in T1- and T2-Weighted MR Images

Purpose: To evaluate repeatability of lung tumor texture features from inspiration/expiration MR image pairs for potential use in patient specific care models and applications. Repeatability is a desirable and necessary characteristic of features included in such models. Methods: T1-weighted Volumetric Interpolation Breath-Hold Examination (VIBE) and/or T2-weighted MRI scans were acquired for 15 patients with non-small cell lung cancer before and during radiotherapy for a total of 32 and 34 same session inspiration-expiration breath-hold image pairs respectively. Bias correction was applied to the VIBE (VIBE-BC) and T2-weighted (T2-BC) images. Fifty-nine texture features at five wavelet decomposition ratios were extracted from the delineated primary tumor including: histogram(HIST), gray level co-occurrence matrix(GLCM), gray level run length matrix(GLRLM), gray level size zone matrix(GLSZM), and neighborhood gray tone different matrix (NGTDM) based features. Repeatability of the texture features for VIBE, VIBE-BC, T2-weighted, and T2-BC image pairs was evaluated by the concordance correlation coefficient (CCC) between corresponding image pairs, with a value greater than 0.90 indicating repeatability. Results: For the VIBE image pairs, the percentage of repeatable texture features by wavelet ratio was between 20% and 24% of the 59 extracted features; the T2-weighted image pairs exhibited repeatability in the range of 44–49%. The percentage dropped to 10–20% for the VIBE-BC images, and 12–14% for the T2-BC images. In addition, five texture features were found to be repeatable in all four image sets including two GLRLM, two GLZSM, and one NGTDN features. No single texture feature category was repeatable among all three image types; however, certain categories performed more consistently on a per image type basis. Conclusion: We identified repeatable texture features on T1- and T2-weighted MRI scans. These texture features should be further investigated for use in specific applications such as tissue classification and changes during radiation therapy utilizing a standard imaging protocol. Authors have the following disclosures: a research agreement with Philips Medical systems (Hugo, Weiss), a license agreement with Varian Medical Systems (Hugo, Weiss), research grants from the National Institute of Health (Hugo, Weiss), UpToDate royalties (Weiss), and none(Mahon, Ford, Karki). Authors have no potential conflicts of interest to disclose.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2017

New Possibilities of Probe Detection of Anisotropic Charged-Particle Distribution Functions in an Arbitrary-Symmetry Plasma

We have analyzed the precision and systematic errors of the familiar method of a plane single-ended probe for measuring the anisotropic distribution functions for electrons and ions in plasma. Analytic relations that connect the plasma parameters and the required number of terms in a series are obtained under the conditions when anisotropy is due to the presence of an electric field in the plasma. It has been shown that ten terms of the series are usually sufficient to adequately describe the ion distribution function, except for the case of strong fields. For strongly anisotropic charged particle distribution functions, the spline interpolation technique for experimental data is proposed and tested, which substantially reduces the systematic error for a preset number of terms in the series. It has been shown for abundant actual plasma objects with the mirror symmetry that the number of required orientations of a plane probe for the experimental distribution function can be reduced by half compared to the most general case of the absence of any symmetry for a fixed number of terms in the Legendre series, while for the same number of orientations, the number of terms in the series, which are determined from these measurements, doubled accordingly.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2016

Hyperspherical Sparse Approximation Techniques for High-Dimensional Discontinuity Detection

This work proposes a hyperspherical sparse approximation framework for detecting jump discontinuities in functions in high-dimensional spaces. The need for a novel approach results from the theoretical and computational inefficiencies of well-known approaches, such as adaptive sparse grids, for discontinuity detection. Our approach constructs the hyperspherical coordinate representation of the discontinuity surface of a function. Then sparse approximations of the transformed function are built in the hyperspherical coordinate system, with values at each point estimated by solving a one-dimensional discontinuity detection problem. Due to the smoothness of the hypersurface, the new technique can identify jump discontinuities with significantly reduced computational cost, compared to existing methods. Several approaches are used to approximate the transformed discontinuity surface in the hyperspherical system, including adaptive sparse grid and radial basis function interpolation, discrete least squares projection, and compressed sensing approximation. Moreover, hierarchical acceleration techniques are also incorporated to further reduce the overall complexity. In conclusion, rigorous complexity analyses of the new methods are provided, as are several numerical examples that illustrate the effectiveness of our approach.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2019

Bayesian Assimilation of Multiscale Precipitation Data and Sparse Ground Gauge Observations in Mountainous Areas

Estimating the spatial distribution of precipitation is important for understanding ecohydrological processes at catchment scales. However, this estimation is difficult in mountainous areas because ground-based observation stations are often sparsely located and do not represent the spatial variability of precipitation. In this study, we develop a Bayesian assimilation method based on data collected on the Tibetan Plateau from 1980 to 2014 to estimate monthly and daily precipitation. To accomplish this, point-scale ground meteorological observations are combined with large-scale precipitation data such as satellite observations or reanalysis data. First, we remove the terrain effects from ground observations by fitting the precipitation data as functions of elevation, and then we spatially interpolate the residuals to 5-km-resolution grids to obtain monthly and daily precipitation. Additionally, we use Tropical Rainfall Measuring Mission (TRMM) satellite observations and ERA-Interim reanalysis data. Cross-validation methods are used to evaluate our method; the results show that our method not only captures the change in precipitation with terrain but also significantly reduces the associated uncertainty. The improvements are more evident in the main river source areas on the edge of the Tibetan Plateau, where elevation changes dramatically, and in high-altitude areas, where the ground gauges are sparse compared with those in low-altitude areas. Our assimilation method is applicable to other regions and is particularly useful for mountainous watersheds where ground meteorological stations are sparse and precipitation is considerably influenced by terrain.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2017

Zonal wavefront reconstruction in quadrilateral geometry for phase measuring deflectometry

There are wide applications for zonal reconstruction methods in slope-based metrology due to its good capability of reconstructing the local details on surface profile. It was noticed in the literature that large reconstruction errors occur when using zonal reconstruction methods designed for rectangular geometry to process slopes in a quadrilateral geometry, which is a more general geometry with phase measuring deflectometry. In this paper, we present a new idea for the zonal methods for quadrilateral geometry. Instead of employing the intermediate slopes to set up height-slope equations, we consider the height increment as a more general connector to establish the height-slope relations for least-squares regression. The classical zonal methods and interpolation-assisted zonal methods are compared with our proposal. Results of both simulation and experiment demonstrate the effectiveness of the proposed idea. In implementation, the modification on the classical zonal methods is addressed. Finally, the new methods preserve many good aspects of the classical ones, such as the ability to handle a large incomplete slope dataset in an arbitrary aperture, and the low computational complexity comparable with the classical zonal method. Of course, the accuracy of the new methods is much higher when integrating the slopes in quadrilateral geometry.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗