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At least 343 records · Page 19

All sources NASA NTRS DOE OSTI

DOE OSTI2014

Objective assessment of image quality and dose reduction in CT iterative reconstruction

Purpose: Iterative reconstruction (IR) algorithms have the potential to reduce radiation dose in CT diagnostic imaging. As these algorithms become available on the market, a standardizable method of quantifying the dose reduction that a particular IR method can achieve would be valuable. Such a method would assist manufacturers in making promotional claims about dose reduction, buyers in comparing different devices, physicists in independently validating the claims, and the United States Food and Drug Administration in regulating the labeling of CT devices. However, the nonlinear nature of commercially available IR algorithms poses challenges to objectively assessing image quality, a necessary step in establishing the amount of dose reduction that a given IR algorithm can achieve without compromising that image quality. This review paper seeks to consolidate information relevant to objectively assessing the quality of CT IR images, and thereby measuring the level of dose reduction that a given IR algorithm can achieve. Methods: The authors discuss task-based methods for assessing the quality of CT IR images and evaluating dose reduction. Results: The authors explain and review recent literature on signal detection and localization tasks in CT IR image quality assessment, the design of an appropriate phantom for these tasks, possible choices of observers (including human and model observers), and methods of evaluating observer performance. Conclusions: Standardizing the measurement of dose reduction is a problem of broad interest to the CT community and to public health. A necessary step in the process is the objective assessment of CT image quality, for which various task-based methods may be suitable. This paper attempts to consolidate recent literature that is relevant to the development and implementation of task-based methods for the assessment of CT IR image quality.

60 APPLIED LIFE SCIENCES↗

DOE OSTI2014

SU-F-BRD-01: A Logistic Regression Model to Predict Objective Function Weights in Prostate Cancer IMRT

Purpose: To develop a statistical model that predicts optimization objective function weights from patient geometry for intensity-modulation radiotherapy (IMRT) of prostate cancer. Methods: A previously developed inverse optimization method (IOM) is applied retrospectively to determine optimal weights for 51 treated patients. We use an overlap volume ratio (OVR) of bladder and rectum for different PTV expansions in order to quantify patient geometry in explanatory variables. Using the optimal weights as ground truth, we develop and train a logistic regression (LR) model to predict the rectum weight and thus the bladder weight. Post hoc, we fix the weights of the left femoral head, right femoral head, and an artificial structure that encourages conformity to the population average while normalizing the bladder and rectum weights accordingly. The population average of objective function weights is used for comparison. Results: The OVR at 0.7cm was found to be the most predictive of the rectum weights. The LR model performance is statistically significant when compared to the population average over a range of clinical metrics including bladder/rectum V53Gy, bladder/rectum V70Gy, and mean voxel dose to the bladder, rectum, CTV, and PTV. On average, the LR model predicted bladder and rectum weights that are both 63% closer to the optimal weights compared to the population average. The treatment plans resulting from the LR weights have, on average, a rectum V70Gy that is 35% closer to the clinical plan and a bladder V70Gy that is 43% closer. Similar results are seen for bladder V54Gy and rectum V54Gy. Conclusion: Statistical modelling from patient anatomy can be used to determine objective function weights in IMRT for prostate cancer. Our method allows the treatment planners to begin the personalization process from an informed starting point, which may lead to more consistent clinical plans and reduce overall planning time.

60 APPLIED LIFE SCIENCES↗

DOE OSTI2015

Computer aided detection of surgical retained foreign object for prevention

Purpose: Surgical retained foreign objects (RFOs) have significant morbidity and mortality. They are associated with approximately $1.5 × 10{sup 9} annually in preventable medical costs. The detection accuracy of radiographs for RFOs is a mediocre 59%. The authors address the RFO problem with two complementary technologies: a three-dimensional (3D) gossypiboma micro tag, the μTag that improves the visibility of RFOs on radiographs, and a computer aided detection (CAD) system that detects the μTag. It is desirable for the CAD system to operate in a high specificity mode in the operating room (OR) and function as a first reader for the surgeon. This allows for fast point of care results and seamless workflow integration. The CAD system can also operate in a high sensitivity mode as a second reader for the radiologist to ensure the highest possible detection accuracy. Methods: The 3D geometry of the μTag produces a similar two dimensional (2D) depiction on radiographs regardless of its orientation in the human body and ensures accurate detection by a radiologist and the CAD. The authors created a data set of 1800 cadaver images with the 3D μTag and other common man-made surgical objects positioned randomly. A total of 1061 cadaver images contained a single μTag and the remaining 739 were without μTag. A radiologist marked the location of the μTag using an in-house developed graphical user interface. The data set was partitioned into three independent subsets: a training set, a validation set, and a test set, consisting of 540, 560, and 700 images, respectively. A CAD system with modules that included preprocessing μTag enhancement, labeling, segmentation, feature analysis, classification, and detection was developed. The CAD system was developed using the training and the validation sets. Results: On the training set, the CAD achieved 81.5% sensitivity with 0.014 false positives (FPs) per image in a high specificity mode for the surgeons in the OR and 96.1% sensitivity with 0.81 FPs per image in a high sensitivity mode for the radiologists. On the independent test set, the CAD achieved 79.5% sensitivity with 0.003 FPs per image in a high specificity mode for the surgeons and 90.2% sensitivity with 0.23 FPs per image in a high sensitivity mode for the radiologists. Conclusions: To the best of the authors’ knowledge, this is the first time a 3D μTag is used to produce a recognizable, substantially similar 2D projection on radiographs regardless of orientation in space. It is the first time a CAD system is used to search for man-made objects over anatomic background. The CAD system for the μTags achieved reasonable performance in both the high specificity and the high sensitivity modes.

60 APPLIED LIFE SCIENCES↗

DOE OSTI2015

Models for predicting objective function weights in prostate cancer IMRT

Purpose: To develop and evaluate the clinical applicability of advanced machine learning models that simultaneously predict multiple optimization objective function weights from patient geometry for intensity-modulated radiation therapy of prostate cancer. Methods: A previously developed inverse optimization method was applied retrospectively to determine optimal objective function weights for 315 treated patients. The authors used an overlap volume ratio (OV) of bladder and rectum for different PTV expansions and overlap volume histogram slopes (OVSR and OVSB for the rectum and bladder, respectively) as explanatory variables that quantify patient geometry. Using the optimal weights as ground truth, the authors trained and applied three prediction models: logistic regression (LR), multinomial logistic regression (MLR), and weighted K-nearest neighbor (KNN). The population average of the optimal objective function weights was also calculated. Results: The OV at 0.4 cm and OVSR at 0.1 cm features were found to be the most predictive of the weights. The authors observed comparable performance (i.e., no statistically significant difference) between LR, MLR, and KNN methodologies, with LR appearing to perform the best. All three machine learning models outperformed the population average by a statistically significant amount over a range of clinical metrics including bladder/rectum V53Gy, bladder/rectum V70Gy, and dose to the bladder, rectum, CTV, and PTV. When comparing the weights directly, the LR model predicted bladder and rectum weights that had, on average, a 73% and 74% relative improvement over the population average weights, respectively. The treatment plans resulting from the LR weights had, on average, a rectum V70Gy that was 35% closer to the clinical plan and a bladder V70Gy that was 29% closer, compared to the population average weights. Similar results were observed for all other clinical metrics. Conclusions: The authors demonstrated that the KNN and MLR weight prediction methodologies perform comparably to the LR model and can produce clinical quality treatment plans by simultaneously predicting multiple weights that capture trade-offs associated with sparing multiple OARs.

60 APPLIED LIFE SCIENCES↗

DOE OSTI2016

TU-FG-209-05: Demonstration of the Line Focus Principle Using the Generalized Measured-Relative Object Detectability (GM-ROD) Metric

Purpose: Demonstrate and quantify the augmented resolution due to focalspot size decrease in images acquired on the anode side of the field, for both small and medium (0.3 and 0.6mm) focal-spot sizes using the experimental task-based GM-ROD metric. Theoretical calculations have shown that a medium focal-spot can achieve the resolution of a small focal-spot if acquired with a tilted anode, effectively providing a higher-output small focal-spot. Methods: The MAF-CMOS (micro-angiographic fluoroscopic complementary-metal-oxide semiconductor) detector (75µm pixel pitch) imaged two copper wire segments of different diameter and a pipeline stent at the central axis and on the anode side of the beam, achieved by tilting the x-ray C-arm (Toshiba Infinix) to 6° and realigning the detector with the perpendicular ray to correct for x-ray obliquity. The relative gain in resolution was determined using the GM-ROD metric, which compares images on the basis of the Fourier transform of the image and the measured NNPS. To emphasize the geometric unsharpness, images were acquired at a magnification of two. Results: Images acquired on the anode side were compared to those acquired on the central axis with the same target-area focal-spot to consider the effect of an angled tube, and for all three objects the advantage of the smaller effective focal-spot was clear, showing a maximum improvement of 36% in GM-ROD. The images obtained with the small focal-spot at the central axis were compared to those of the medium focal-spot at the anode side and, for all objects, the relative performance was comparable. Conclusion: For three objects, the GM-ROD demonstrated the advantage of the anode side focal-spot. The comparable performance of the medium focal-spot on the anode side will allow for a high-output small focal-spot; a necessity in endovascular image-guided interventions. Partial support from an NIH grant R01EB002873 and an equipment grant from Toshiba Medical Systems Corp.

60 APPLIED LIFE SCIENCES↗

DOE OSTI2016

A review of projection radiography of plasma and biological objects in X-Pinch radiation

A review of studies on the X-pinch as a radiation source for X-ray projection shadow radiography (XPSR) and X-ray absorption spectroscopy (XAS) is presented. The ultimate capabilities of the techniques and ways of their achievement are considered. XPSR has been successfully used to study high-energy-density plasma objects, in particular, exploding wires and wire arrays. Using XPSR, the internal structure and dynamics of a wire explosion and wire array implosion have been investigated for the first time, which has made it possible to develop an adequate consistent theory of the processes occurring in the wire loads of generators with currents from several units of kiloamperes to a few tens of megamperes. The use of XAS for diagnostics of wire loads has allowed one for the first time to measure the parameters of matter in the wire core and plasma corona of the load. X-ray images of various biological objects have obtained, including those with the use of the phase contrast method. This review is a logical continuation of the review ''X-Pinch'' [Plasma Phys. Rep. 41, 319, 493 (2015)], in which the X-pinch as a physical object was considered.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2023

Flexible and Effective Object Tiering for Heterogeneous Memory Systems

Computing platforms that package multiple types of memory, each with their own performance characteristics, are quickly becoming mainstream. To operate efficiently, heterogeneous memory architectures require new data management solutions that are able to match the needs of each application with an appropriate type of memory. As the primary generators of memory usage, applications create a great deal of information that can be useful for guiding memory tiering, but the community still lacks tools to collect, organize, and leverage this information effectively. To address this gap, this work introduces a novel software framework that collects and analyzes object-level information to guide memory tiering. Using this framework, this study evaluates and compares the impact of a variety of data tiering choices, including how the system prioritizes objects for faster memory as well as the frequency and timing of migration events. The results, collected on a modern Intel platform with conventional DRAM as well as non-volatile RAM, show that guiding data tiering with object-level information can enable significant performance and efficiency benefits compared to standard hardware- and software-directed data tiering strategies.

Kammerdiener, Brandon↗

DOE OSTI2022

Incorporating the effects of objects in an approximate model of light transport in scattering media

A computationally efficient radiative transport model is presented that predicts a camera measurement and accounts for the light reflected and blocked by an object in a scattering medium. The model is in good agreement with experimental data acquired at the Sandia National Laboratory Fog Chamber Facility (SNLFC). The model is applicable in computational imaging to detect, localize, and image objects hidden in scattering media. Here, a statistical approach was implemented to study object detection limits in fog.

42 ENGINEERING↗

DOE OSTI2025

Robust COTS objective for diffraction-limited, high-NA, long front working distance imaging

We present a robust objective lens optimized for applications requiring both high numerical aperture (NA) and long front working distance imaging, comprised of all commercial-off-the-shelf (COTS) spherical singlet lenses. Unlike traditional designs that require separate collimation and refocusing stages, our approach directly converges imaged light to the back focal plane using a single lens group. Our configuration corrects spherical aberrations and efficiently collects light to achieve diffraction-limited performance across a wide range of wavelengths while simplifying alignment and assembly. Using this approach, we design and construct an example objective lens that features a long front working distance of 61 mm and a clipped NA of 0.30 (limited by an aperture in our experimental setup). We experimentally verify that it achieves monochromatic diffraction-limited resolution at wavelengths from 375 nm to 866 nm without requiring replacement of the lenses or changing the inter-lens spacings, and its performance remains robust across a 46 mm range variation in total length (by adjusting mainly the back working distance). Additionally, we develop a quantitative method to measure the field of view (FOV) using an experimentally calibrated pinhole target. Under 397 nm illumination (i.e., from 40 Ca + ion fluorescence), the objective achieves a resolution of 0.87 μm with a 540 μm FOV. This robust, all-COTS, and versatile design is well-suited for a broad range of experiments, supporting high-precision measurements and exploring quantum phenomena.

Cui, Jiafeng [Oak Ridge National Laboratory (ORNL)↗

DOE OSTI2013

Imaging-based optical caliper for objects in hot manufacturing processes

OG Technologies, Inc. (OGT), in conjunction with its industrial and academic partners, proposes to develop an Imaging-Based Optical Caliper (hereafter referred to as OC) for Objects in Hot Manufacturing Processes. The goal is to develop and demonstrate the OC with the synergy of OGT's current technological pool and other innovations to provide a light weight, robust, safe and accurate portable dimensional measurement device for hot objects with integrated wireless communication capacity to enable real time process control. The technical areas of interest in this project are the combination of advanced imaging, Sensor Fusion, and process control. OGT believes that the synergistic interactions between its current set of technologies and other innovations could deliver products that are viable and have high impact in the hot manufacture processes, such as steel making, steel rolling, open die forging, and glass industries, resulting in a new energy efficient control paradigm in the operations through improved yield, prolonged tool life and improved quality. In-line dimension measurement and control is of interest to the steel makers, yet current industry focus is on the final product dimension only instead of whole process due to the limit of man power, system cost and operator safety concerns. As sensor technologies advances, the industry started to see the need to enforce better dimensional control throughout the process, but lack the proper tools to do so. OGT along with its industrial partners represent the indigenous effort of technological development to serve the US steel industry. The immediate market that can use and get benefited from the proposed OC is the Steel Industry. The deployment of the OC has the potential to provide benefits in reduction of energy waste, CO2 emission, waste water amount, toxic waste, and so forth. The potential market after further expended function includes Hot Forging and Freight Industries. The OC prototypes were fabricated, and were progressively tested on-site in several steel mill and hot forging facilities for evaluation. Software refinements and new calibration procedures were also carried out to overcome the discovered glitches. Progress was presented to the hot manufacture facilities worldwide. Evidence showed a great interest and practical need for this product. OGT is in the pilot commercialization mode for this new development. The R&D team also successfully developed a 3D measurement function with no additional investment of hardware or equipment to measure low or room temperature object dimensions. Several tests were conducted in the reality environment to evaluate the measurement results. This new application will require additional development in product design.

47 OTHER INSTRUMENTATION↗

DOE OSTI2014

Complementarity of LSST and WFIRST: Regarding Object Blending

Many aspects of the complementarity between LSST and WFIRST have been discussed in the WFIRST-AFTA Final Report (Spergel et al. 2013) and accepted within the community. For example, LSST’s six optical filters (u,g,r,i,z,y) and WFIRST’s four infrared filters (Y,J,H,F184)1, which will achieve effectively the same depth, will be highly complementary for photometric redshift estimation. However, the important subject of object blending has been absent from many of these discussions. In this letter we wish to focus on the complementarity of LSST and WFIRST in specific regards to object blending issues. Both how blending will affect the naively assumed complementarity of the two surveys as well as how the complementarity of the two surveys can mitigate the detrimental effects of object blending.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2017

Statistical Hypothesis Testing using CNN Features for Synthesis of Adversarial Counterexamples to Human and Object Detection Vision Systems

Validating the correctness of human detection vision systems is crucial for safety applications such as pedestrian collision avoidance in autonomous vehicles. The enormous space of possible inputs to such an intelligent system makes it difficult to design test cases for such systems. In this report, we present our tool MAYA that uses an error model derived from a convolutional neural network (CNN) to explore the space of images similar to a given input image, and then tests the correctness of a given human or object detection system on such perturbed images. We demonstrate the capability of our tool on the pre-trained Histogram-of-Oriented-Gradients (HOG) human detection algorithm implemented in the popular OpenCV toolset and the Caffe object detection system pre-trained on the ImageNet benchmark. Our tool may serve as a testing resource for the designers of intelligent human and object detection systems.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Evaluating Space Object Conjunction Probabilities Using Characteristic Function Inversion

This report discusses an approach to computing the probability of a conjunction between two space objects in the short-term encounter scenario. A conjunction is defined here as an event where the miss distance between the objects is less than some specified value. The scenario assumptions are that the motion of the objects is linear, their positions are Gaussian distributed, and their velocities are known and constant. Under these assumptions, the squared-miss distance is shown to have the generalized chi-square distribution. An established statistical technique called characteristic function inversion is employed to evaluate the distribution and obtain conjunction probabilities. The method is closely related to a recent approach based on moment generating function inversion, and a qualitative comparison of the approaches is provided. Last, the method is tested on several benchmark test cases where it agrees with numerical integration on the cases with conjunction probabilities above 10 –12 . However, the exact probability in these cases is usually not needed and this probability can be bounded above using an independent Gaussian approximation. Overall, the report shows how to compute conjunction probabilities using a standard statistical method, though numerical integration seems to perform equally well.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2025

Explainable and Differentiable Reinforcement Learning for Multi-objective Optimization in Particle Accelerators

Operating particle accelerators involves optimizing multiple goals simultaneously, which can be challenging due to trade-offs among objectives. While evolutionary algorithms like the genetic algorithm (GA) have been used for various Multi-Objective Optimization (MOO) tasks, they are not inherently suited for complex control problems. This talk highlights two variations of Reinforcement Learning (RL) for concurrently optimizing heat load and trip rates at the Continuous Electron Beam Accelerator Facility (CEBAF). The problem involves strict constraints on individual states, actions, and overall energy requirements of the beam. First, this talk highlights how differentiability can be harnessed through a Deep Differentiable Reinforcement Learning (DDRL) approach to address MOO issues within particle accelerators. We examine the DDRL method alongside Model Free Reinforcement Learning (MFRL), GA, and Bayesian Optimization (BO). The performance of these methods is assessed by generating a Pareto-front for two objectives. Our findings indicate that DDRL excels in handling high-dimensional problems more effectively than MFRL, BO, and GA. Next, we will show integration of explainable physics-based constraints into RL algorithms to enhance trans- parency and trust in decision-making processes by enabling users to verify that agents adhere to established physical principles. This surrogate function can be modeled using neural networks or sparse dictionary mod- els. By examining the mathematical form of the learned constraint function, we are able to confirm the agent has learned to use the established physics of each environment provided but the surrogate model. In addi- tion, we find that the introduction of a mathematical functional dictionary based surrogate model enables our reinforcement learning algorithms to reliably converge for difficult high-dimensional accelerator controls environments.

Rajput, Kishansingh [Thomas Jefferson National Acc↗

DOE OSTI2022

Selecting Post-Processing Schemes for Accurate Detection of Small Objects in Low-Resolution Wide-Area Aerial Imagery

In low-resolution wide-area aerial imagery, object detection algorithms are categorized as feature extraction and machine learning approaches, where the former often requires a post-processing scheme to reduce false detections and the latter demands multi-stage learning followed by post-processing. In this paper, we present an approach on how to select post-processing schemes for aerial object detection. We evaluated combinations of each of ten vehicle detection algorithms with any of seven post-processing schemes, where the best three schemes for each algorithm were determined using average F-score metric. The performance improvement is quantified using basic information retrieval metrics as well as the classification of events, activities and relationships (CLEAR) metrics. We also implemented a two-stage learning algorithm using a hundred-layer densely connected convolutional neural network for small object detection and evaluated its degree of improvement when combined with the various post-processing schemes. The highest average F-scores after post-processing are 0.902, 0.704 and 0.891 for the Tucson, Phoenix and online VEDAI datasets, respectively. The combined results prove that our enhanced three-stage post-processing scheme achieves a mean average precision (mAP) of 63.9% for feature extraction methods and 82.8% for the machine learning approach.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2024

AEPF: Attention-Enabled Point Fusion for 3D Object Detection

Current state-of-the-art (SOTA) LiDAR-only detectors perform well for 3D object detection tasks, but point cloud data are typically sparse and lacks semantic information. Detailed semantic information obtained from camera images can be added with existing LiDAR-based detectors to create a robust 3D detection pipeline. With two different data types, a major challenge in developing multi-modal sensor fusion networks is to achieve effective data fusion while managing computational resources. With separate 2D and 3D feature extraction backbones, feature fusion can become more challenging as these modes generate different gradients, leading to gradient conflicts and suboptimal convergence during network optimization. To this end, we propose a 3D object detection method, Attention-Enabled Point Fusion (AEPF). AEPF uses images and voxelized point cloud data as inputs and estimates the 3D bounding boxes of object locations as outputs. An attention mechanism is introduced to an existing feature fusion strategy to improve 3D detection accuracy and two variants are proposed. These two variants, AEPF-Small and AEPF-Large, address different needs. AEPF-Small, with a lightweight attention module and fewer parameters, offers fast inference. AEPF-Large, with a more complex attention module and increased parameters, provides higher accuracy than baseline models. Experimental results on the KITTI validation set show that AEPF-Small maintains SOTA 3D detection accuracy while inferencing at higher speeds. AEPF-Large achieves mean average precision scores of 91.13, 79.06, and 76.15 for the car class’s easy, medium, and hard targets, respectively, in the KITTI validation set. Results from ablation experiments are also presented to support the choice of model architecture.

Chemistry↗

DOE OSTI2016

LARGE SIZE AND SLOW ROTATION OF THE TRANS-NEPTUNIAN OBJECT (225088) 2007 OR{sub 10} DISCOVERED FROM HERSCHEL AND K2 OBSERVATIONS

We present the first comprehensive thermal and rotational analysis of the second most distant trans-Neptunian object (TNOs) (225088) 2007 OR{sub 10}. We combined optical light curves provided by the Kepler Space Telescope–K2 extended mission and thermal infrared data provided by the Herschel Space Observatory. We found that (225088) 2007 OR{sub 10} is likely to be larger and darker than derived by earlier studies: we obtained a diameter of d=1535{sub −225}{sup +75} km which places (225088) 2007 OR{sub 10} in the biggest top three TNOs. The corresponding visual geometric albedo is p{sub V}=0.089{sub −0.009}{sup +0.031}. The light-curve analysis revealed a slow rotation rate of P{sub rot} = 44.81 ± 0.37 hr, superseded by very few objects. The most likely light-curve solution is double-peaked with a slight asymmetry; however, we cannot safely rule out the possibility of having a rotation period of P{sub rot} = 22.40 ± 0.18 hr, which corresponds to a single-peaked solution. Due to the size and slow rotation, the shape of the object should be a MacLaurin ellipsoid, so the light variation should be caused by surface inhomogeneities. Its newly derived larger diameter also implies larger surface gravity and a more likely retention of volatiles—CH{sub 4}, CO, and N{sub 2}—on the surface.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2016

ZONAL TOROIDAL HARMONIC EXPANSIONS OF EXTERNAL GRAVITATIONAL FIELDS FOR RING-LIKE OBJECTS

We present an expression of the external gravitational field of a general ring-like object with axial and plane symmetries such as oval toroids or annular disks with an arbitrary density distribution. The main term is the gravitational field of a uniform, infinitely thin ring representing the limit of zero radial width and zero vertical height of the object. The additional term is derived from a zonal toroidal harmonic expansion of a general solution of Laplace’s equation outside the Brillouin toroid of the object. The special functions required are the point value and the first-order derivative of the zonal toroidal harmonics of the first kind, namely, the Legendre function of the first kind of half integer degree and an argument that is not less than unity. We developed a recursive method to compute them from two pairs of seed values explicitly expressed by some complete elliptic integrals. Numerical experiments show that appropriately truncated expansions converge rapidly outside the Brillouin toroid. The truncated expansion can be evaluated so efficiently that, for an oval toroid with an exponentially damping density profile, it is 3000–10,000 times faster than the two-dimensional numerical quadrature. A group of the Fortran 90 programs required in the new method and their sample outputs are available electronically.

79 ASTRONOMY AND ASTROPHYSICS↗