Results for “object reconstruction” — Page 9

DOE OSTI2026

ELECTRON SHOWER RECONSTRUCTION IN THE ICARUS EXPERIMENT

This dissertation presents a study in the field of neutrino physics. Neutrinos are fundamental particles that are electrically neutral and have extremely small mass, allowing them to traverse matter with very little interaction. Because of this property, neutrinos are exceptionally difficult to detect. Nevertheless, understanding their behavior is essential for addressing fundamental questions about the origin of the Universe and the properties of matter. This work focuses on the reconstruction of electron showers produced by interactions of electron neutrinos (𝜈𝑒) in the ICARUS experiment, located at the Fermi National Accelerator Laboratory in the United States. ICARUS employs a liquid argon time projection chamber detector, which is capable of recording with high precision the tracks left by particles produced in neutrino interactions. The main objective of this research is to improve the reconstruction algorithms and techniques used to identify and characterize these electron showers, enhancing metrics such as completeness, defined as the fraction of correctly reconstructed signals, and purity, which quantifies how much of the reconstructed signal truly belongs to the candidate event. These improvements are crucial for reducing false positives and increasing the accuracy of electron photon discrimination. Consequently, this work directly contributes to improved neutrino oscillation analyses and to a deeper understanding of neutrino properties.

Salmoria, Gabrieli [Parana Tech. Fed. U., Toledo]↗

DOE OSTI2026

Search for exotic Higgs boson decays H → 𝒜𝒜 with 𝒜 → γγ in events with a semi-merged topology in proton-proton collisions at $\sqrt{s}=13$ TeV

A search for exotic Higgs boson decays H → 𝒜𝒜, with 𝒜 → γγ is presented, using events with a semi-merged topology. One of the hypothetical particles, 𝒜, is assumed to decay promptly into a semi-merged diphoton system reconstructed as a single photon-like object, while the other 𝒜 decays into two resolved photons. The search is performed using proton-proton collision data collected by the CMS experiment at $\sqrt{s}=13$ TeV, corresponding to an integrated luminosity of 138 fb −1 . The data agree with the standard model background expectation. Upper limits are set on the product of the Higgs boson production cross section and the branching fraction, σ(pp → H) ℬ(H → 𝒜𝒜 → 4γ), which range from 0.264 to 0.005 pb at 95% confidence level, for 𝒜 masses in the range 1 < m 𝒜 < 15 GeV. These limits are the most stringent to date in the 1–5 GeV m 𝒜 range.

Beyond Standard Model↗

DOE OSTI2021

Photon-Sparse, Poisson Light-Sheet Microscopy

Light-sheet microscopy has revolutionized bioimaging by enabling approximately an order of magnitude reduction in specimen irradiance compared to confocal imaging. Here, we introduce a light-sheet imaging system that enables an additional order of magnitude reduction in specimen irradiance by operating at the Poisson limit. To operate at this limit, we integrated classical illumination with single-photon detection and wavelet-based image reconstruction. This integration enabled brightness quantification and object recognition from fewer than one detected photon per image pixel, corresponding to more than 10-fold lower irradiance levels than modern systems. We demonstrate how such photon-sparse imaging can eradicate photobleaching and enable both dim and bright object imaging, thus, further enhancing the related gains of light-sheet microscopy.

60 APPLIED LIFE SCIENCES↗

DOE OSTI2020

Multi-beam X-ray ptychography for high-throughput coherent diffraction imaging

X-ray ptychography is a rapidly developing coherent diffraction imaging technique that provides nanoscale resolution on extended field-of-view. However, the requirement of coherence and the scanning mechanism limit the throughput of ptychographic imaging. Herein, we propose X-ray ptychography using multiple illuminations instead of single illumination in conventional ptychography. Multiple locations of the sample are simultaneously imaged by spatially separated X-ray beams, therefore, the obtained field-of-view in one scan can be enlarged by a factor equal to the number of illuminations. We have demonstrated this technique experimentally using two X-ray beams focused by a house-made Fresnel zone plate array. Two areas of the object and corresponding double illuminations were successfully reconstructed from diffraction patterns acquired in one scan, with image quality similar with those obtained by conventional single-beam ptychography in sequence. Multi-beam ptychography approach increases the imaging speed, providing an efficient way for high-resolution imaging of large extended specimens.

47 OTHER INSTRUMENTATION↗

NASA NTRS1977

Medical ultrasonic tomographic system

An electro-mechanical scanning assembly was designed and fabricated for the purpose of generating an ultrasound tomogram. A low cost modality was demonstrated in which analog instrumentation methods formed a tomogram on photographic film. Successful tomogram reconstructions were obtained on in vitro test objects by using the attenuation of the fist path ultrasound signal as it passed through the test object. The nearly half century tomographic methods of X-ray analysis were verified as being useful for ultrasound imaging.

Heyser, R. C.↗

NASA NTRS1991

Surface reconstruction from scattered data through pruning of unstructured grids

This paper describes an algorithm for reconstructing a surface from a randomly digitized object. Scan data (treated as a cloud of points) is first tesselated out to its convex hull using Delaunay triangulation. The line-of-sight between each surface point and the scanning device is traversed, and any tetrahedra which are pierced by it are removed. The remaining tetrahedra form an approximate solid model of the scanned object. Due to the inherently limited resolution of any scan, this algorithm requires two additional procedures to produce a smooth, polyhedral surface: one process removes long, narrow tetrahedra which span indentations in the surface between digitized points; the other smooths sharp edges. The results for a moderately resolved sample body and a highly resolved aircraft are displayed.

Maksymiuk, C. M.↗

NASA NTRS1993

Al-26 production profile and model comparisons in Canyon Diablo

The large preatmospheric size of the Canyon Diablo meteorite, a radius of about 15 m, makes it especially suitable for systematic studies of cosmogenic nuclide production rates of iron objects in a 2 pi geometry. To reconstruct the exposure history of the meteoroid, Heymann et al. investigated several fragments recovered from known geographic locations around the crater for their shock features and cosmogenic nobel gases. They applied the Signer-Nier noble gas production rate model to establish the preatmospheric depth of the specimens in the meteoroid. Cosmic ray exposure ages suggested a multi-episodic irradiation, with 170 or 540 Ma being inferred for most of the samples studied while two anomalous specimens indicated a possible third exposure age at 940 Ma. Be-10 and Cl-36 have been measured in a number of these same samples by accelerator mass spectrometry (AMS), with use being made of the preatmospheric depths determined in Heymann et al. to construct production profiles. The present study extends the cosmogenic radionuclide data to Al-26 and compares the results with both the production rate model of Reedy and Arnold and production rates determined from the cross sections used by the Reedy-Arnold model (for the major nuclear reactions making Al-26) in combination with differential fluxes calculated using the Los Alamos High Energy Transport (LAHET) Code System. Model calculations for Be-10 and Cl-36 have also been obtained.

Michlovich, E.↗

NASA NTRSDate not supplied

Recent Advances in Soft Matter Characterization Capabilities Developed at NASA GRC for Lunar Exploration: Differential Dynamic Microscopy to Spectroscopy to Computer Vision

In 1991, famous French scientist Pierre-Gilles de Genes was awarded Nobel prize for his impactful research in soft matter, more specifically polymers. He is defined as the founding father of soft matter. In his Nobel lecture (https://www.nobelprize.org/uploads/2018/06/gennes-lecture.pdf ) he described soft matter aka complex fluids as materials with two primary features – (a) complexity and (b) flexibility. The sub-categories of soft matter (e.g.- granular materials, polymers, foams, colloids etc.) are defined on the basis of Pierre-Gilles de Gennes’ definition. At NASA GRC, we are pushing the boundaries for fundamental study of soft matter on Lunar Surface. With regard to Lunar surface science, we are focusing on developing capabilities pertaining to granular materials and bio-soft/active matter to facilitate future efforts in ISRU and bio-ISRU capabilities. In order to achieve fundamental goals of soft matter research within the limitations of Lunar environment, the scientific capabilities need to be small, flexible, modular, off the shelf and the focus needs to be more on developing an interdisciplinary capability that leverages the recent growth in AI/ML and Computer Vision to augment our understanding of fundamental science. This strategy would allow us to reduce our resource requirement during launch, installation, and occupied real estate footprint on Lunar surface In this talk, we will go over 3 different capabilities that we have developed in house and in close collaboration – (a) Differential Dynamic Microscopy (DDM), (b) Portable In-situ Chemical Spectroscopy (PICS) and (c) Computer Vision Enabled Observation. At very high level, Differential Dynamic Microscopy (DDM) allows us to study the structure-property-process relation (microrheology) of bio-soft/active matter using optical microscope and improved image analysis capabilities. PICS uses AI/ML-based advanced signal deconvolution and analysis technique that can work with existing portable spectroscopy tools to perform materials analysis (e.g.- granular materials and bio-soft/active matter) inspection on the go. Finally, computer vision enabled analysis allows us to use simple camera images for 3D reconstruction of experimental process and tracking of objects of interest in an experiment. We expect that this detailed process will allow us reach a thorough understanding of soft matter in Lunar environment. The capabilities developed by us will help to validate and establish fundamental understanding in Lunar environment. This will, in turn, allow us to guide future space exploration missions and expand the knowledge base of the scientific and engineering communities.

Suman Sinha Ray↗

NASA NTRSDate not supplied

Recent Advances in Soft Matter Characterization Capabilities Developed at NASA GRC for Lunar Exploration: Differential Dynamic Microscopy to Spectroscopy to Computer Vision

In 1991, famous French scientist Pierre-Gilles de Genes was awarded Nobel prize for his impactful research in soft matter, more specifically polymers. He is defined as the founding father of soft matter. In his Nobel lecture (https://www.nobelprize.org/uploads/2018/06/gennes-lecture.pdf ) he described soft matter aka complex fluids as materials with two primary features – (a) complexity and (b) flexibility. The sub-categories of soft matter (e.g.- granular materials, polymers, foams, colloids etc.) are defined on the basis of Pierre-Gilles de Gennes’ definition. At NASA GRC, we are pushing the boundaries for fundamental study of soft matter on Lunar Surface. With regard to Lunar surface science, we are focusing on developing capabilities pertaining to granular materials and bio-soft/active matter to facilitate future efforts in ISRU and bio-ISRU capabilities. In order to achieve fundamental goals of soft matter research within the limitations of Lunar environment, the scientific capabilities need to be small, flexible, modular, off the shelf and the focus needs to be more on developing an interdisciplinary capability that leverages the recent growth in AI/ML and Computer Vision to augment our understanding of fundamental science. This strategy would allow us to reduce our resource requirement during launch, installation, and occupied real estate footprint on Lunar surface. In this talk, we will go over 3 different capabilities that we have developed in house and in close collaboration – (a) Differential Dynamic Microscopy (DDM), (b) Portable In-situ Chemical Spectroscopy (PICS) and (c) Computer Vision Enabled Observation. At very high level, Differential Dynamic Microscopy (DDM) allows us to study the structure-property-process relation (microrheology) of bio-soft/active matter using optical microscope and improved image analysis capabilities. PICS uses AI/ML-based advanced signal deconvolution and analysis technique that can work with existing portable spectroscopy tools to perform materials analysis (e.g.- granular materials and bio-soft/active matter) inspection on the go. Finally, computer vision enabled analysis allows us to use simple camera images for 3D reconstruction of experimental process and tracking of objects of interest in an experiment. We expect that this detailed process will allow us reach a thorough understanding of soft matter in Lunar environment. The capabilities developed by us will help to validate and establish fundamental understanding in Lunar environment. This will, in turn, allow us to guide future space exploration missions and expand the knowledge base of the scientific and engineering communities.

Suman Sinha-Ray↗

NASA NTRSDate not supplied

Recent Advances in Soft Matter Characterization Capabilities Developed at NASA GRC for Lunar Exploration: Differential Dynamic Microscopy to Spectroscopy to Computer Vision

In 1991, famous French scientist Pierre-Gilles de Genes was awarded Nobel prize for his impactful research in soft matter, more specifically polymers. He is defined as the founding father of soft matter. In his Nobel lecture (https://www.nobelprize.org/uploads/2018/06/gennes-lecture.pdf ) he described soft matter aka complex fluids as materials with two primary features – (a) complexity and (b) flexibility. The sub-categories of soft matter (e.g.- granular materials, polymers, foams, colloids etc.) are defined on the basis of Pierre-Gilles de Gennes’ definition. At NASA GRC, we are pushing the boundaries for fundamental study of soft matter on Lunar Surface. With regard to Lunar surface science, we are focusing on developing capabilities pertaining to granular materials and bio-soft/active matter to facilitate future efforts in ISRU and bio-ISRU capabilities. In order to achieve fundamental goals of soft matter research within the limitations of Lunar environment, the scientific capabilities need to be small, flexible, modular, off the shelf and the focus needs to be more on developing an interdisciplinary capability that leverages the recent growth in AI/ML and Computer Vision to augment our understanding of fundamental science. This strategy would allow us to reduce our resource requirement during launch, installation, and occupied real estate footprint on Lunar surface In this talk, we will go over 3 different capabilities that we have developed in house and in close collaboration – (a) Differential Dynamic Microscopy (DDM), (b) Portable In-situ Chemical Spectroscopy (PICS) and (c) Computer Vision Enabled Observation. At very high level, Differential Dynamic Microscopy (DDM) allows us to study the structure-property-process relation (microrheology) of bio-soft/active matter using optical microscope and improved image analysis capabilities. PICS uses AI/ML-based advanced signal deconvolution and analysis technique that can work with existing portable spectroscopy tools to perform materials analysis (e.g.- granular materials and bio-soft/active matter) inspection on the go. Finally, computer vision enabled analysis allows us to use simple camera images for 3D reconstruction of experimental process and tracking of objects of interest in an experiment. We expect that this detailed process will allow us reach a thorough understanding of soft matter in Lunar environment. The capabilities developed by us will help to validate and establish fundamental understanding in Lunar environment. This will, in turn, allow us to guide future space exploration missions and expand the knowledge base of the scientific and engineering communities.

Suman Sinha Ray↗

NASA NTRSDate not supplied

Recent Advances in Soft Matter Characterization Capabilities Developed at NASA GRC

In 1991, famous French scientist Pierre-Gilles de Genes was awarded Nobel prize for his impactful research in soft matter, more specifically polymers. He is defined as the founding father of soft matter. In his Nobel lecture (https://www.nobelprize.org/uploads/2018/06/gennes-lecture.pdf ) he described soft matter aka complex fluids as materials with two primary features – (a) complexity and (b) flexibility. The sub-categories of soft matter (e.g.- granular materials, polymers, foams, colloids etc.) are defined on the basis of Pierre-Gilles de Gennes’ definition. At NASA GRC, we are pushing the boundaries for fundamental study of soft matter on Lunar Surface. With regard to Lunar surface science, we are focusing on developing capabilities pertaining to granular materials and bio-soft/active matter to facilitate future efforts in ISRU and bio-ISRU capabilities. In order to achieve fundamental goals of soft matter research within the limitations of Lunar environment, the scientific capabilities need to be small, flexible, modular, off the shelf and the focus needs to be more on developing an interdisciplinary capability that leverages the recent growth in AI/ML and Computer Vision to augment our understanding of fundamental science. This strategy would allow us to reduce our resource requirement during launch, installation, and occupied real estate footprint on Lunar surface In this talk, we will go over 3 different capabilities that we have developed in house and in close collaboration – (a) Differential Dynamic Microscopy (DDM), (b) Portable In-situ Chemical Spectroscopy (PICS) and (c) Computer Vision Enabled Observation. At very high level, Differential Dynamic Microscopy (DDM) allows us to study the structure-property-process relation (microrheology) of bio-soft/active matter using optical microscope and improved image analysis capabilities. PICS uses AI/ML-based advanced signal deconvolution and analysis technique that can work with existing portable spectroscopy tools to perform materials analysis (e.g.- granular materials and bio-soft/active matter) inspection on the go. Finally, computer vision enabled analysis allows us to use simple camera images for 3D reconstruction of experimental process and tracking of objects of interest in an experiment. We expect that this detailed process will allow us reach a thorough understanding of soft matter in Lunar environment. The capabilities developed by us will help to validate and establish fundamental understanding in Lunar environment. This will, in turn, allow us to guide future space exploration missions and expand the knowledge base of the scientific and engineering communities.

Suman Sinha Ray↗

DOE OSTI2022

CodEx: A Modular Framework for Joint Temporal De-Blurring and Tomographic Reconstruction

In many computed tomography (CT) imaging applications, it is important to rapidly collect data from an object that is moving or changing with time. Tomographic acquisition is generally assumed to be step-and-shoot, where the object is rotated to each desired angle, and a view is taken. However, step-and-shoot acquisition is slow and can waste photons, so in practice fly-scanning is done where the object is continuously rotated while collecting data. However, this can result in motion-blurred views and consequently reconstructions with severe motion artifacts. In this paper, we introduce CodEx, a modular framework for joint de-blurring and tomographic reconstruction that can effectively invert the motion blur introduced in sparse view fly-scanning. The method is a synergistic combination of a novel acquisition method with a novel non-convex Bayesian reconstruction algorithm. CodEx works by encoding the acquisition with a known binary code that the reconstruction algorithm then inverts. Using a well chosen binary code to encode the measurements can improve the accuracy of the inversion process. The CodEx reconstruction method uses the alternating direction method of multipliers (ADMM) to split the inverse problem into iterative deblurring and reconstruction sub-problems, making reconstruction practical to implement. Here we present reconstruction results on both simulated and binned experimental data to demonstrate the effectiveness of our method.

42 ENGINEERING↗

DOE OSTI2026

Wasserstein Normalized Autoencoder for Anomaly Detection in ProtoDUNE Vertical-Drift Detector

ProtoDUNE Vertical Drift needs a selective triggering algorithm. The detector sits on Earth's surface, so cosmic activity dominates its data. Our goal in this paper is to trigger on neutrino events more robustly than the current deployed Analog-to-Digital Converter Simple Window (ADCSW) model and, eventually, search for signals of Beyond Standard Model (BSM) physics at DUNE as our ultimate North Star objective. As a step towards this goal, we evaluate a Wasserstein Normalized Autoencoder (WNAE) on simulated collection-plane only windows of shape $1\times10\times10$ where Neutrinos act as our BSM-proxy and Cosmic-ray Muons serve as our learned background. The network parameters are fitted using only cosmic-ray muon events as background in order to maintain an unsupervised pipeline. Training uses finite-step Langevin $x^-$ samples, positive-sample reconstruction energy, and an empirical sliced $2$-Wasserstein objective to learn a normalized Boltzmann energy model. We then calibrate on a nominal $5\,\mathrm{Hz}$ operating threshold calculated from cosmic validation data. Both WNAE and ADCSW accept 311 of 194,083 held-out cosmic background events at this $5\,\mathrm{Hz}$ threshold. We found that WNAE accepts 9,677 of 34,634 neutrino-proxy events $(27.9\pm0.24)\%$, compared with 10,076 $(29.1\pm0.24)\%$ for ADCSW, an observed WNAE-minus-ADCSW difference of $-1.15\%$. At another nominal $2\,\mathrm{Hz}$ target threshold, the corresponding efficiencies are $(20.5\pm0.22)\%$ and $(22.6\pm0.22)\%$, respectively. Of the WNAE-selected neutrino proxies at $5\,\mathrm{Hz}$, $(20.8\pm0.4)\%$ of the classified neutrino-proxy events are unique to WNAE, where the uncertainty is an absolute binomial standard error of $0.4\%$.

Zheng, Jake [U. Chicago (main)] (ORCID:00090002189↗

DOE OSTI2026

Wasserstein Normalized Autoencoder for Anomaly Detection in ProtoDUNE Vertical-Drift Detector

ProtoDUNE Vertical Drift needs a selective triggering algorithm. The detector sits on Earth's surface, so cosmic activity dominates its data. Our goal in this paper is to trigger on neutrino events more robustly than the current deployed Analog-to-Digital Converter Simple Window (ADCSW) model and, eventually, search for signals of Beyond Standard Model (BSM) physics at DUNE as our ultimate North Star objective. As a step towards this goal, we evaluate a Wasserstein Normalized Autoencoder (WNAE) on simulated collection-plane only windows of shape $1\times10\times10$ where Neutrinos act as our BSM-proxy and Cosmic-ray Muons serve as our learned background. The network parameters are fitted using only cosmic-ray muon events as background in order to maintain an unsupervised pipeline. Training uses finite-step Langevin $x^-$ samples, positive-sample reconstruction energy, and an empirical sliced $2$-Wasserstein objective to learn a normalized Boltzmann energy model. We then calibrate on a nominal $5\,\mathrm{Hz}$ operating threshold calculated from cosmic validation data. Both WNAE and ADCSW accept 311 of 194,083 held-out cosmic background events at this $5\,\mathrm{Hz}$ threshold. We found that WNAE accepts 9,677 of 34,634 neutrino-proxy events $(27.9\pm0.24)\%$, compared with 10,076 $(29.1\pm0.24)\%$ for ADCSW, an observed WNAE-minus-ADCSW difference of $-1.15\%$. At another nominal $2\,\mathrm{Hz}$ target threshold, the corresponding efficiencies are $(20.5\pm0.22)\%$ and $(22.6\pm0.22)\%$, respectively. Of the WNAE-selected neutrino proxies at $5\,\mathrm{Hz}$, $(20.8\pm0.4)\%$ of the classified neutrino-proxy events are unique to WNAE, where the uncertainty is an absolute binomial standard error of $0.4\%$.

Zheng, Jake [Chicago U.] (ORCID:0009000218901379)↗

NASA NTRS1989

Image understanding and interpretation in microwave diversity imaging

The authors investigate microwave imaging of metallic objects using a diversity method and interpret and predict the reconstructed image from an approach based on analysis of the scattering mechanism and a back-projection algorithm used in image retrieval. The connection between the various scattering mechanisms and the reconstructed images is discussed, what the images represent is interpreted, and a prediction is made as to what the image will look like over given spectral and angular windows. A brief description is given of the microwave diversity imaging system and the formulation of the microwave diversity imaging based on the physical optics approximation. The scattering mechanism of a complex shaped metallic object is then briefly reviewed and an alternate approach to interpreting the reconstructed image based on the understanding of the scattering mechanism and the reconstruction algorithm is given. Several numerical and experimental examples are included to support this interpretation approach.

Li, Hsueh-Jyh↗

DOE OSTI2020

Multiple monoenergetic gamma radiography (MMGR) with a compact superconducting cyclotron

Smuggling of special nuclear materials and nuclear devices through borders and ports of entry constitutes a major risk to global security. Technologies are needed to reliably screen the flow of commerce for the presence of high-Z materials such as uranium and plutonium. Here, we present an experimental proof-of-concept of a technique that uses inelastic (p,p') nuclear reactions to generate monoenergetic photons, which provide means to measure the areal density and the effective-Z (Z eff ) of an object with an accuracy surpassing that achieved by current methods. We use an ION-12 SC superconducting 12 MeV proton cyclotron to produce 4.4, 6.1, 6.9, and 7.1 MeV photons from a variety of nuclear reactions. Using these photons in a transmission mode, we show that we are able to accurately reconstruct the areal densities and Zeff of a test object. This methodology could enable mobile applications to screen commercial cargoes with high material specificity, providing a means of distinguishing common cargo materials from high-Z materials that include uranium and plutonium.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2025

X-ray and Neutron Radiography for Quantitative Material Reconstructions

Radiography is a powerful tool to determine the interior structure of objects. X-ray radiography is widely used and provides high-resolution images, though X-rays have limited transmission through materials of high atomic number (Z) and density. In contrast, neutrons can penetrate many materials that are heavily attenuating to X-rays, such as metals, providing contrast in the inner layers of highly attenuating items. Past work has shown the value in using both X-ray and neutron radiography for estimating material thicknesses, though that work was limited to simulated data. Here, we demonstrate quantitative material reconstructions using experimental X-ray and neutron radiography data from lab-based systems, accurately modeling radiography system responses to within a few percent to enable quantitative measures of material thickness. We demonstrate the utility of neutron radiography and X-ray radiography for these quantitative reconstructions and introduce methods for using their complementarity to improve image quality and optimize experimental design.

Gilbert, Andrew J.↗

NASA NTRS1985

Intersection of three-dimensional geometric surfaces

Calculating the line of intersection between two three-dimensional objects and using the information to generate a third object is a key element in a geometry development system. Techniques are presented for the generation of three-dimensional objects, the calculation of a line of intersection between two objects, and the construction of a resultant third object. The objects are closed surfaces consisting of adjacent bicubic parametric patches using Bezier basis functions. The intersection determination involves subdividing the patches that make up the objects until they are approximately planar and then calculating the intersection between planes. The resulting straight-line segments are connected to form the curve of intersection. The polygons in the neighborhood of the intersection are reconstructed and put back into the Bezier representation. A third object can be generated using various combinations of the original two. Several examples are presented. Special cases and problems were encountered, and the method for handling them is discussed. The special cases and problems included intersection of patch edges, gaps between adjacent patches because of unequal subdivision, holes, or islands within patches, and computer round-off error.

Crisp, V. K.↗