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Research Engineer

Location:
Saugus, MA
Posted:
February 17, 2017

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Resume:

EDWARD CHARLES BARILE

* **** *****

SAUGUS, MA. *1906

781-***-**** (HOME)

617-***-****(CELL)

EMAIL acyvhx@r.postjobfree.com

Areas of Experience and Education

Radar Systems Engineering, Radar detection, discrimination, tracking and signal processing. Algorithm development. Dynamics and Kinematics of Rigid Bodies for Missile Defense. System level requirements and development. System models and analysis to support radar requirements and performance analysis. Electromagnetic Theory, Acoustics, Optics, Space-Time Adaptive Processing, Adaptive Array and signal Processing, Detection, Adaptive Estimation and Filtering, Antennas and Scattering, Diffraction, Microwaves, Imaging Systems and Analysis, Pattern Recognition, Classification, Applied Mathematics and Physics, Imaging, Image Reconstruction, Image Processing and Enhancement, Communication Theory, Inverse Scattering, Synthetic Aperture Radar/Sonar Proficient in MATLAB, C/C++, Fortran, Java.

Employment History

2015-2016: HMMH Burlington, MA

2000-2011: Raytheon Company, Woburn, MA

1999-2000 Witten Technology. Washington, DC

1998-1999: Raytheon Company, Bedford, MA

1994-1998: SenCom Corporation, Bedford, MA

1986-1994: The MITRE corporation, Bedford, MA

1981-1986: ITEK Optical Systems, Lexington, MA

1978-1981: MIT Lincoln Laboratory, Lexington, MA

1972-1978: Naval Underwater Systems Center, Newport, R.I. 1968-1970: Raytheon, Wayland, MA

Education

M.S., Electrical Engineering, Northeastern University, 1971 B.S., Electrical Engineering, Northeastern University, 1968 Graduate Courses in Applied Mathematics, Brown University, 1974 Graduate Courses in Physics, University of Rhode Island, 1975 Honors, Awards:

MITRE Best Paper of the Year Award, 1992

Senior Member of I.E.E.E., 1989

Eta Kappa Nu Engineering Honor Society, 1968

Publications (open literature)

Barile, E.C., Fante, R.L., Torres, J.A.; "Some Limitations on the Effectiveness of Airborne Adaptive Radar"; I.E.E.E. Transactions on Aerospace and Electronic Systems; Vol. 28, No.4; October, 1992.

Fante, R.L., Barile, E.C., Guella, T.P., “Clutter Covariance Smoothing by Subaperture Averaging,” I.E.E.E. Transactions on Aerospace and Electronic Systems, July, 1994.

Barile, E.C., Guella, T.P., Lamensdorf, D.; “Adaptive Antenna Space-Time Processing Techniques to Suppress Platform Scattered Clutter for Airborne Radar”, I.E.E.E. Transactions on Aerospace and Electronics Systems, January 1995.

Copies of significant original work may be viewed and downloaded from https://www.researchgate.net/profile/Edward_Barile?ev=hdr_xprf Significant Original Contributions Throughout Work Experience: 6/2015-2/2016 HMMH Burlington, MA

ANALYSIS OF AIRPORT TRACKS AND THEIR CLUSTERING

Analysis of airport departure and arrival track data and its clustering into similar track bundles or clusters, the mean for each cluster and its dispersion. The tracks are used to calculate sound exposure in neighborhoods around the airport. Analysis and software (MATLAB).

2/2000-4/2011 Raytheon Company, Woburn, MA

LAUNCH COMPLEX ASSOCIATION

The problem of Launch Complex Association is addressed as one of separating tracks into groups, each group corresponding to one parent body which has separated into a set of children. Within a group, the tracked objects are children of a common ancestor or parent. We need to be able to do this whether or nor we have a track of the parent. We develop a method based on the assumed ballistic property of each child track, in particular the conservation of angular momentum. We are able to take any child track and obtain its location and time

(space-time event) of deployment from the parent in a simple and direct way. Those children with nearby space-time deployment events are thereby assigned to the same parent (or complex).

REGISTRATION AND BEAM POINTING ERRORS AND THEIR CORRECTION FOR SEQUENTIAL RADAR HANDOVER

In this study, we investigate methods for aligning two or more radars

(registration or “grid lock”) so that any relative pointing error between radars and consequent conflict of target locations in a track are resolved. We have two types of registration errors, one being an incorrect knowledge of the orientation of a base coordinate system (such as the Earth Centered Rotating – ECR or its accompanying Earth Centered Inertial – ECI) by one of the radars relative to that of another (with which it is to share its track data) and the other being due to a beam pointing error of the radar’s phase steered array when the array elements have an unknown displacement from their design locations. These two errors can be combined as a rotation of one radar’s estimate of the ECR system relative to the other radar’s estimate of it, by an unknown angle about an unknown axis.

We show that the actual rotation matrix can be calculated by using the data for each radar for no less than 3 targets In addition a method for properly associating the targets is derived as well.

ROTATION DYAD OPERATORS AND THE PROBLEM OF MULTIPLE COMPOUND ROTATIONS FOR MOVING RADARS ON MOVING PLATFORMS This is a complete analysis for calculating the trajectory of a target as seen by radars mounted on a ship or other moving platform. This has been done in the past using transformation matrices and their deriviatives, and becomes very complicated when considering moving antennas on moving platforms on a rotating earth. This is a special case of a general compound rotation problem. We provide a new method using rotation dyad operators, which has only lately become of interest in engineering practice even though it has been known for a long time. It enables a great simplification in this problem particularly in determining the order of the rotation sequences. This method, along with a convenient notation created by the author, alleviates confusion, especially in such a complicated analysis.

A target trajectory, its velocity and acceleration are calculated in either an antenna or inertial frame, given the radar antenna location on the ship and its rotational motion about its center of gravity or as given by an inertial navigation IMU device located on or near the antenna, the ship trajectory and rotational motion about its center of gravity, and the earth rotation relative to the inertial system.

This is given in terms of sequences of rotation dyad operators and corresponding angular velocity dyad operators for the ship and the antenna. DUAL TRACKERS FOR BURNOUT DETECTION

A method of improving the performance for tracking a boosting object using a both a 10 –state (boosting) and a 6-state (ballistic) Kalman Filter was devised where the measurement and prediction residual state is used to determine burnout and when to switch filters. Also, a complete analysis of burnout detection methods is included

MULTISITE RADAR ANALYSIS

Survey of all existing methods of using multiple radar and other sensor platform data to improve track on threat targets. A method of combining tracks including the effect of common model correlation of the state error covariance matrices was derived. Also the method of combining measurement or observation vectors with their known and uncorrelated covariances was derived. A comparison of various track or observation fusion methods was made to determine the simplest and most effective means of acquiring a system wide track and association was done. Also the applicability of Space-Time Adaptive Processing (STAP) was studied to improve the individual observation accuracy was studied. A complete technical report detailing all derivations, associated analysis software and powerpoint presentations was written.

HIT ASSESSMENT FOR BALLISTIC MISSILE DEFENSE RADAR SYSTEM Analysis of current Hit assessment algorithms for determining from radar data, whether an engaged target was hit and or destroyed by an interceptor missile. Analysis of Impact Physics for bodies in both translational and rotational motion was done as part of the study. Adaptive Process Noise Kalman Tracker developed for tracking the (possibly maneuvering) interceptor. Findings incorporated into system algorithm enhancements for future system development.

MULTIPLE SCATTERING BETWEEN CLOSELY SPACED RADAR TARGETS A new method of solving for the multiple scattered field between proximal bodies illuminated by a radar signal was developed. The solution is composed of Admittance dyad solutions of the individual bodies in free space. The approximate double bounce solution is derived from the general formal rigorous solution. It is believed that this is a new approach that has apparently been overlooked in the literature. The study was done to see how close the bodies had to be in terms of separation distance divided by maximum body dimension to have multiple scattering affect the echoes from the scenario. Also the effect of separation in wavelengths was studied.

STRETCH PROCESSED COMPLEX ECHOES – WIDE BAND TO MEDIUM BAND FORMS

This study provides a derivation of Stretch Processing – a method to implement pulse compression for echoes when matched filtering (crosscorrelation with a replica of the transmit pulse) is impractical. The derivation is quite general and is in terms of the impulse response of the illuminated target, rather than any simplified point scatterer model. The only restriction is that the transmit pulse is a Linear Frequency Modulated (LFM) pulse. Then a method is presented whereby one may develop a medium band version of the stretch processed echo from measured wide band stretch processed data, thus alleviating the need for a separate medium band measurement. Of course the medium band must be a sub band of the original wide band.

REQUIREMENTS ANALYSIS FOR DISCRIMINATION AND PATTERN RECOGNITION FOR GROUND BASED MISSILE DEFENSE RADAR Calculation of probability of correct classification for various scenarios pertinent to Ground Based missile defense radar. Analysis is for worst case scenario. All decision surfaces for the feature space calculated, and all derivations, equations, computer codes documented.

A separate study investigates the performance of the radar for providing an estimate of the type of target detected based on measured features. RANGE PROFILE ALIGNMENT AND RANGE –DOPPLER IMAGING This work investigates methods of removal of the center of gravity (CG) motion from a set of range profiles that are to be used to form a Range-Doppler Image. A complete theory of complex profile measurement from wide band radar returns, based on an Electromagnetic Analysis of scattering from moving rotating bodies is provided, as well as a study comparing performance of methods of estimating the Doppler due to CG motion and its removal, after range profiles are aligned so that scatterers remain in the same range bin throughout the coherent processing time.

TOMOGRAPHIC (INVERSE SAR) AND RANGE-DOPPLER IMAGING In this study, Range-Doppler imaging for arbitrarily long integration times is investigated. We find that images may be formed using a matched filter tuned to all scatterer location histories, and that the image so formed is in fact a tomographic image. As a special case, when the motion is restricted to integration times short enough so that scatterers remain in their initial range bins, the tomographic image reduces to the typical Range-Doppler image, formed by Fourier Transforming each series of I and Q voltages for each range bin, over pulses.

Also investigated were Coherent Integration degradation effects vs CG estimate errors, degradation effects of cases where the amount of target rotation is not small, the behavior of various other features used in discrimination, and features related to electromagnetic scattering properties of typical targets. 2/1999-3-2000 Witten Industries, 2121 K Street, NW Suite 650, Washington, DC DIFFRACTION TOMOGRAPHY, SAR, AND TIME REVERSAL IMAGING FOR GROUND PENETRATING MULTIMONOSTATIC AND MULTISTATIC RADAR. Application and extension of method of Time Reversal Imaging to Ground Penetrating Radar Data for Multistatic Configuration. Applied Synthetic Aperture Radar methods to multimonostatic and Multistatic data as well. 3/1998-2/1999 Raytheon Company, Bedford, MA

ANTI RADIATION MISSILE MITIGATION BY LEAST SQUARES VIRTUAL SOURCE SYNTHESIS OVER THE MISSILE ARRAY FACE

A method of creating an artificial wavefront over the face of an Anti- Radiation Missile (ARM) by using separated radars or one single radar array was developed. The method creates a wavefront over the ARM antenna that appears to emanate from some chosen virtual radar location. A computer simulation and analysis for the method was devised.

WIDE BAND RANGE PROFILE CORRELATOR FOR DETERMINNG THE PRESENCE OF WARHEADS ON MISSILE AFTER ENGAGEMENT FOR SECOND SHOT DISCRIMINATION

A method of using a set of aspect dependent range profiles of a warhead to detect the presence of it in the received profile from an engaged threat missile was developed. The output of a cross - correlation peak between the warhead profile and the data for range windows up to the range extent of the warhead was used to determine the presence of an unkilled warhead after engagement. The method was tested with simulated data from XPATCH models and worked well. A complete analysis and computer simulation was developed. 11/1994 – 3/1998: SenCom Corporation, 3 Preston Court, Bedford, MA, Senior Engineer

SPACE-TIME ADAPTIVE PROCESSING (STAP) FOR BISTATIC RADAR A complete theoretical analysis was done for STAP as applied to bistatic radar. A computer simulation with an accurate clutter and target model was developed as well to aid in the performance analysis of a typical bistatic radar scenario. The distribution of clutter in the wavenumber – Doppler domain was studied, and it is found to have characteristics that profoundly affect the performance in a manner distinct from the monostatic case. In addition, MUSIC was applied to both bistatic and monostatic STAP and impressive improved resolution was obtained even in the presence of near field scattering. ACCURACY AND RESOLUTION FOR BISTATIC RADAR

Resolution and accuracy for target location and tracking is of concern for bistatic radar, especially for the case of wide bistatic angles (150 degrees or more). An analysis was developed wherein the Cramer Rao lower bound was calculated and a simulation was developed to provide accuracy estimates for a typical bistatic scenario with a typical target model incorporated. Accuracy for the bistatic radar is compared with that of a monostatic radar residing at the location of the bistatic radar transmitter. It was found that enhanced signal to noise ratio

(SNR) in some bistatic scenarios can actually provide better accuracy than the monostatic case, even though the resolution is worse. ELECTROMAGNETIC SCATTERING FROM ROTATING HELICOPTER ROTOR BLADES AND HELICOPTER IDENTIFICATION

As part of an effort to extend present radar capability to identify (as friend or foe) ground and low flying targets, an investigation into the discrimination of helicopters by their reflected signals was undertaken. The problem of electromagnetic scattering from a moving, deforming, or rotating conducting body was formulated and formally solved. The frequency of interest enabled approximating the problem with Physical Optics. The signal received by a generic radar due to illumination of a helicopter rotor while rotating was calculated and a computer simulation was developed. The minimum dwell time and pulse repetition frequency (PRF) was derived so that the magnitude spectrum of the signal could be resolved with no aliasing, enabling estimation of the blade rotation rate (from the spectral line spacing) and the blade tip velocity

(from the spectral bandwidth). These measurements, however were shown to be insufficient to determine the number of blades and the rotor rotation rate uniquely. Subsequent study showed that if the phase of the complex spectrum is measured, the ambiguity can be resolved. Thus a prescription was developed to determine whether or not an existing radar could discriminate helicopters, and what had to be modified in order that it can do so uniquely. METHODS OF CLASSIFICATION OF TARGETS USING TRANSFORMED DATA The problem of classifying groups of targets by examining their radar data was investigated, relative to modifying Early Warning Radars to detect warheads in a ballistic missile launch. Since the data is sparse and has variable sampling rates, it must be transformed so as to remove features that are meaningless. Since the motion of the scattering bodies (translation and rotation) are not obtainable from these Radars, typically only mean, peak and variance of RCS history has been used as a three dimensional feature vector. We have employed the use of a histogram approximant to the probability density function for RCS, as a feature vector, since it includes mean, peak and variance and retains all the data without committing any preconceived notion about the unmeasurable body motion. Another similar feature vector space involves reshuffling the data sequence to a monotonically increasing (or decreasing) sequence, or “amplitude ranking”. Testing these algorithms with simulated data has shown improvement over conventional methods. Other data transformations to a feature space also have been investigated.

2/1986 - 11/1994: The MITRE Corporation, Bedford, MA, Technical Staff SPACE-TIME ADAPTIVE METHODS OF DETECTING TARGETS IN CLUTTER AND OTHER INTERFERENCE (INCLUDING NEAR FIELD SCATTERING).

An extensive electromagnetic analysis was conducted to investigate Displaced Phase Center Antenna (DPCA) and Space-Time Adaptive Processing (STAP) methods for detection of targets in interference from a moving radar . DPCA and STAP were related as filters in the wavenumber frequency (space-time spectral) domain, and DPCA is shown to be a deterministic version of the more general STAP adaptive filter. The space-time spectral domain

(Wavenumber - Doppler) is used to analyze the performance of STAP and DPCA on clutter suppression, and the important effects of clutter internal motion, and antenna - clutter-airframe interaction (near field scattering). Eigenanalysis methods are also used to study such effects as effective receive pattern as a function of Doppler, and its variation with eigenvalue equalization or truncation, the effects of clutter internal motion, near field scattering, and errors.

The work includes a new formulation of the received field on a moving array in terms of cascaded impedance and admittance dyad operators which enables exact representation of the antenna-airframe interaction, and also enables the expression of clutter multipath scattering. It was found that these two effects limit STAP performance because they transform the clutter returns into signals that are ambiguous with the target return. In addition, the effect of mutual coupling in the array both on transmit and receive is included, as well as the effect of compensating for it. A computer program for both personal computer and mainframe use has been developed which simulates the effect of actual airborne platforms that may be used on a candidate system. This work was reported in MITRE Technical Reports, MITRE Seminars, the IEEE URSI - Antennas and Propagation, MTT joint Symposium on May 10, 1990 in Dallas Texas., and a paper was published in 10-92 in IEEE Transactions on Aerospace and Electronic Systems. That paper won MITRE’s Best Paper 1992. The effect of antenna airframe interaction (“near field scattering”) was shown to be significant, and it was shown that the effect could be minimized but never completely removed.

PREADAPTIVE COLUMN NULLING FOR NEAR FIELD SCATTERING SUPPRESSION IN AIRBORNE RADAR APPLICATIONS OF STAP A method of preadaptive vertical near field suppression followed by line array STAP was developed and shown to approach the performance of planar array STAP for some cases, thus significantly reducing the amount of processing. A paper describing this work appeared in January 1995 I.E.E.E. Transactions on Aerospace and Electronic Systems, and the reviewers thought it to be a significant breakthrough in the state of the art. The effects of near field scatterer motion was also studied, accompanied by complete theoretical and computer studies.

ADAPTIVE MULTIPLE BEAM ANTENNA AND ITS USE IN

COMMUNICATIONS IN A JAMMING ENVIRONMENT

An analytical model and extensive computer simulation was developed for the purpose of evaluating the performance of a satellite borne multiple beam antenna based communications system operating with a given earth surface coverage requirement and with the adaptive function of suppressing jammers within the coverage region. The antenna, geometry and adaptive algorithm were simulated for the steady state and transient condition. Based on this work, it was found that the contractor’s proposed algorithm performed only marginally, and other more robust algorithms were simulated and suggested. SYNTHETIC APERTURE RADAR METHODS OF THREE-DIMENSIONAL SURFACE IMAGING

Interferometric Synthetic Aperture Radar (INSAR) was investigated for use in imaging or detecting perturbations in the ground surface to classify targets with vertical structure. Two SAR images from two sensors are mixed with the result that surface perturbations are manifest as a system of null lines in the image which have local curvatures, enabling estimation of the actual height of the corresponding pixels above the ground. This method was evaluated for performance as a function of signal to noise in its effect on height resolution.

A complete theoretical and mathematical investigation of SAR and INSAR was conducted for this study, and all levels of detail were used. An accompanying set of computer programs were developed and used in the analysis. Also, variants of the INSAR technique were also investigated. MOTION EMULATION OF RADAR DATA FROM A FIXED PLATFORM In order to test the various advanced airborne radar methods such as STAP, a method of emulating platform motion from a fixed radar is desired. In this way, the tremendous expense of actually building and flying a large complex radar system is avoided. A site on the top of a mountain is to be used for the radar, and the emulator is supposed to transform the received data to appear as if the platform were moving. A general optimal method of motion emulation was devised and analyzed, and is a candidate for the actual experiment. The general optimal method will emulate both the antenna-airframe interaction and in addition will correctly emulate the clutter multipath effects. RAID RECOGNITION WITH TARGET CLASSIFICATION

The effect of a target classifier on a radar's ability to detect a raid of hostile aircraft was investigated. A complete theoretical analysis and computer simulation was developed to study the effective increase in the probability of detecting a raid with fixed false alarm rate due to using a generic target classifier over simply counting the number of unidentified targets and declaring a raid when a threshold is exceeded. The generic classifier is characterized by its probability of correct classification of both a hostile target and those found in normal traffic.

MUTUAL COUPLING IN DIPOLE ARRAYS AND ITS COMPENSATION An analysis and computer simulation was created concerning mutual coupling of dipole arrays using the method of moments: given a set of applied driving point element voltages, the resultant element current distributions are solved. A method was then developed to find the driving point voltages required to realize a desired current distribution on the elements in the least square sense. The current distribution is chosen to obtain some desired radiation pattern. The work then gives a method of compensation of mutual coupling, by prescribing the correct driving point voltages. A computer simulation was also developed for this work.

LOW FREQUENCY ACTIVE RADAR CROSS SECTION SIMULATION USING CROSSED DIPOLE RERADIATED FIELDS

The ability of a pair of crossed dipole reradiators to simulate the monostatic scattered electromagnetic field from a target illuminated at wavelengths comparable to or larger than the target was investigated. A complete analysis and computer simulation was developed which calculated the receive to reradiate coupling coefficients that would simulate the target monostatic scattered field over a given sector of incidence angles in the least squares sense. The analysis and computer program includes the case where the target and simulating dipoles are located over a perfectly conducting ground plane. In addition, the case of bistatic scattering simulation was also treated.

METHOD OF IMAGES FOR SCATTERING FROM OBJECTS OVER

A GROUND PLANE

The theory of electromagnetic images is employed to analyze the scattering from perfectly conducting bodies over an infinite perfectly conducting ground plane. The resultant integral equation for the current induced on the body, including its interaction with the ground plane, is numerically solved using the method of moments. A computer simulation was developed with the analysis. As an example, scattering from an infinite elliptical cylinder over a ground plane is calculated for both horizontal and vertical polarization. In addition, the analysis is extended to the case where the body to altitude ratio is small enough so that only first order multipath contributions to the body / ground plane interaction are significant, leading to a simpler form, since now free space currents may be used to evaluate the scattering.

CONFORMAL VS. PLANAR ARRAYS FOR AXIAL PLANE SIDELOBES Conformal and Planar arrays are evaluated in their sidelobe structure for the axial plane radiation pattern. It is shown that, neglecting mutual coupling effects, the conformal array has lower axial plane sidelobes than the equivalent

(in number of elements) planar array, for narrow transaxial scans. For higher transaxial plane scans, the sidelobes of the planar array are lower. A computer simulation was developed for this work. Also the off-principle plane sidelobe structure was investigated showing mostly lower planar array sidelobes.

CONFORMAL ARRAY PATTERN SYNTHESIS

A method whereby weights for a conformal array may be chosen to realize the pattern of a planar array was investigated. One of the methods examined was known in the field, although only in an heuristic way. A mathematical theory was developed to explain why it works and it was also discovered how to generalize the method to arbitrary conformal array surfaces as well as how to steer such an array.

Also, a least squares method of calculating the weights was developed and investigated.

Computer programs were written and used for the study. ITEK CORPORATION (subsidiary of Litton Industries) OPTICAL SYSTEMS DIVISION - 10 Maguire Rd., Lexington, Mass, 2-81 - 2-86. ACTIVE OPTICS

A complete theoretical analysis and computer simulation was developed for simulation of a closed loop optical wavefront correction system using Kalman Filtering for actuator controlled deformable mirrors. A method was then developed to estimate a poorly known or completely unknown observation matrix by state perturbation element by element or mode by mode followed by supplemental Kalman Filtering. Observation error and state error covariance matrix modifications were derived to account for residual observation matrix uncertainty. Complete implementation by Singular Value Decomposition and system performance evaluation for soft (low frequency) and hard (high frequency) mirror influence functions in terms of modes. OPTICAL PROPAGATION

Analysis and software was developed for propagation of randomly or deterministically distorted wavefronts with major quadratic (doubly curved) phase by extension of previous method for wavefronts with large spherical component. The method was further extended to include wavefronts with significant third and higher order distortion. The technique enables propagation of wavefronts which would otherwise require extremely fine sampling due to the aliasing effect of undersampling wavefronts with large changes in phase between adjacent samples. OPTICAL SYSTEM ANALYSIS

Analysis and simulation was created for an optical system consisting of an array of mirrors with the purpose of redirecting incident light for illumination of specified region. Region intensity as function of mirror roughness distribution and statistics was derived. Expression for partial coherence effects given mirror orientations was derived in a unique and compact manner. Complete computer simulation was developed.

NON-PARAXIAL ANALYSIS OF FOCAL PLANE FIELDS

Field at the focal plane for a converging spherical wave with aperture comparable to the focal length (non paraxial propagation) and deterministic or random phase error was calculated by two derived methods developed for the purpose of cutting computational intensity.

OPTICAL ENCODER ANALYSIS AND SIMULATION

A highly detailed physical optics model for a double grating optical encoder was developed, including method of modifying Angular Spectrum for propagation in bounded region and from tilted planes. Supervised development of software simulation to be used as analysis and design tool for optical encoders. Method for treating multiple reflections was developed for future incorporation into model.

CCD ARRAY IMAGE ENHANCEMENT BY MASKING

Analysis of concept of image resolution improvement from CCD array by masking and reconstruction. Derived optimality criterion for masking, optimal estimator for arbitrary mask including effects of cell and photon noise as well as registration error effects. Software for analysis was also developed. DISTORTED IMAGE SEQUENCE RESTORATION

Developed algorithm and software for taking several noisy distorted images of an object and mapping each to coincide with any one of the set, so that averaging for noise reduction may be



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