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Physical Sciences and Mathematics Commons

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2018

Air Force Institute of Technology

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Articles 1 - 30 of 118

Full-Text Articles in Physical Sciences and Mathematics

Electrical Characterization Of Crystalline Uo2, Tho2 And U0.71Th0.29O2, Christina L. Dugan Dec 2018

Electrical Characterization Of Crystalline Uo2, Tho2 And U0.71Th0.29O2, Christina L. Dugan

Theses and Dissertations

Uranium dioxide (UO2), thorium dioxide (ThO2), and UxTh1-xO2 alloys are characterized for suitability in uranium-based neutron detectors. ThO2 was studied for an envisioned UO2/ThO2 heterojunction. A U0.71Th0.29O2 alloy was studied because of its resistance to oxidation and potential use in surface passivation. The U0.71Th0.29O2 effective Debye temperature of 217± 24 K was measured using temperature-dependent x-ray photoemission spectroscopy (XPS). The specific heat capacity for the U0.71Th0.29O2 alloy was calculated from the Debye …


Enabling Auditing And Intrusion Detection Of Proprietary Controller Area Networks, Brent C. Stone Dec 2018

Enabling Auditing And Intrusion Detection Of Proprietary Controller Area Networks, Brent C. Stone

Theses and Dissertations

The goal of this dissertation is to provide automated methods for security researchers to overcome ‘security through obscurity’ used by manufacturers of proprietary Industrial Control Systems (ICS). `White hat' security analysts waste significant time reverse engineering these systems' opaque network configurations instead of performing meaningful security auditing tasks. Automating the process of documenting proprietary protocol configurations is intended to improve independent security auditing of ICS networks. The major contributions of this dissertation are a novel approach for unsupervised lexical analysis of binary network data flows and analysis of the time series data extracted as a result. We demonstrate the utility …


Kinetics Of Higher Lying Potassium States After Excitation Of The D2 Transition In The Presence Of Helium, Austin J. Wallerstein Dec 2018

Kinetics Of Higher Lying Potassium States After Excitation Of The D2 Transition In The Presence Of Helium, Austin J. Wallerstein

Theses and Dissertations

A kinetic model for the performance of a potassium Diode Pumped Alkali Laser (DPAL), including the role of higher lying states is developed to assess the impact on device efficiency and performance. A rate package for a nine level kinetic model including recommended rate parameters is solved under steady-state conditions. Energy pooling and far wing absorption populates higher lying states, with single photon and Penning ionization leading to modest potassium (K) dimer ion concentrations. The fraction of the population removed from the basic three levels associated with the standard model is less than 10% for all reasonable laser conditions, including …


Improving Detection Of Dim Targets: Optimization Of A Moment-Based Detection Algorithm, Shannon R. Young Dec 2018

Improving Detection Of Dim Targets: Optimization Of A Moment-Based Detection Algorithm, Shannon R. Young

Theses and Dissertations

Wide area motion imagery (WAMI) sensor technology is advancing rapidly. Increases in frame rates and detector array sizes have led to a dramatic increase in the volume of data that can be acquired. Without a corresponding increase in analytical manpower, much of these data remain underutilized. This creates a need for fast, automated, and robust methods for detecting dim, moving signals of interest. Current approaches fall into two categories: detect-before-track (DBT) and track-before-detect (TBD) methods. The DBT methods use thresholding to reduce the quantity of data to be processed, making real time implementation practical but at the cost of the …


Sequence Pattern Mining With Variables, James S. Okolica, Gilbert L. Peterson, Robert F. Mills, Michael R. Grimaila Nov 2018

Sequence Pattern Mining With Variables, James S. Okolica, Gilbert L. Peterson, Robert F. Mills, Michael R. Grimaila

Faculty Publications

Sequence pattern mining (SPM) seeks to find multiple items that commonly occur together in a specific order. One common assumption is that all of the relevant differences between items are captured through creating distinct items, e.g., if color matters then the same item in two different colors would have two items created, one for each color. In some domains, that is unrealistic. This paper makes two contributions. The first extends SPM algorithms to allow item differentiation through attribute variables for domains with large numbers of items, e.g, by having one item with a variable with a color attribute rather than …


A Statistical Analysis Of Steve, Bea Gallardo‐Lacourt, Y. Nishimura, E. Donovan, G. W. Gillies, W. E. Archer, Omar A. Nava, E. L. Spanswick Nov 2018

A Statistical Analysis Of Steve, Bea Gallardo‐Lacourt, Y. Nishimura, E. Donovan, G. W. Gillies, W. E. Archer, Omar A. Nava, E. L. Spanswick

Faculty Publications

There has been an exciting recent development in auroral research associated with the discovery of a new subauroral phenomenon called STEVE (Strong Thermal Emission Velocity Enhancement). Although STEVE has been documented by amateur night sky watchers for decades, it is as yet an unidentified upper atmosphere phenomenon. Observed first by amateur auroral photographers, STEVE appears as a narrow luminous structure across the night sky over thousands of kilometers in the east‐west direction. In this paper, we present the first statistical analysis of the properties of 28 STEVE events identified using Time History of Events and Macroscale Interactions during Substorms (THEMIS) …


Optimal Policy For Sequential Stochastic Resource Allocation, Kalyanam Krishnamoorthy, Meir Pachter, David W. Casbeer Nov 2018

Optimal Policy For Sequential Stochastic Resource Allocation, Kalyanam Krishnamoorthy, Meir Pachter, David W. Casbeer

Faculty Publications

A gambler in possession of R chips/coins is allowed N(>R) pulls/trials at a slot machine. Upon pulling the arm, the slot machine realizes a random state i ɛ{1, ..., M} with probability p(i) and the corresponding positive monetary reward g(i) is presented to the gambler. The gambler can accept the reward by inserting a coin in the machine. However, the dilemma facing the gambler is whether to spend the coin or keep it in reserve hoping to pick up a greater reward in the future. We assume that the gambler has full knowledge of the reward distribution function. We …


Wireless Intrusion Detection And Device Fingerprinting Through Preamble Manipulation, Benjamin W. Ramsey, Barry E. Mullins Oct 2018

Wireless Intrusion Detection And Device Fingerprinting Through Preamble Manipulation, Benjamin W. Ramsey, Barry E. Mullins

AFIT Patents

A method of establishing a hardware identity of a coordinating device in a wireless network is provided. A standard PHY preamble is modified to a preamble that can be received by the coordinating device having an expected hardware configuration. The modified PHY preamble is transmitted with an association request by a joining device. In response to not receiving a reply containing an association response from the coordinating device by the joining device, determining the hardware configuration of the coordinating device is not the expected hardware configuration. A further method of characterizing a hardware identity of a device in a wireless …


Estimation Of Atmospheric Turbulence Using Differential Motion Of Extended Features In Time-Lapse Imagery, Santasri Bose-Pillai, Jack E. Mccrae, Christopher A. Rice, Ryan A. Wood, Conner E. Murphy, Steven T. Fiorino Oct 2018

Estimation Of Atmospheric Turbulence Using Differential Motion Of Extended Features In Time-Lapse Imagery, Santasri Bose-Pillai, Jack E. Mccrae, Christopher A. Rice, Ryan A. Wood, Conner E. Murphy, Steven T. Fiorino

Faculty Publications

We address the design, development, and testing of a pointer/tracker as a probe beam for the purpose of making high-speed, aero-optical measurements of the flow over a scaled beam director turret. The tracker uses retro-reflection of the probe beam off of a Reflexite annulus surrounding the turret. The constraints of the design required a near-total-commercial off the shelf system that could be quickly installed and removed in a rented aircraft. Baseline measurements of environmental vibrations are used to predict pointing performance; mitigation of line-of-sight jitter on the probe beam is achieved through passive isolation and the design of relay optics. …


Equiangular Tight Frames That Contain Regular Simplices, Matthew C. Fickus, John Jasper, Emily J. King, Dustin G. Mixon Oct 2018

Equiangular Tight Frames That Contain Regular Simplices, Matthew C. Fickus, John Jasper, Emily J. King, Dustin G. Mixon

Faculty Publications

An equiangular tight frame (ETF) is a type of optimal packing of lines in Euclidean space. A regular simplex is a special type of ETF in which the number of vectors is one more than the dimension of the space they span. In this paper, we consider ETFs that contain a regular simplex, that is, have the property that a subset of its vectors forms a regular simplex. As we explain, such ETFs are characterized as those that achieve equality in a certain well-known bound from the theory of compressed sensing. We then consider the so-called binder of such an …


Equiangular Tight Frames From Group Divisible Designs, Matthew C. Fickus, John Jasper Oct 2018

Equiangular Tight Frames From Group Divisible Designs, Matthew C. Fickus, John Jasper

Faculty Publications

An equiangular tight frame (ETF) is a type of optimal packing of lines in a real or complex Hilbert space. In the complex case, the existence of an ETF of a given size remains an open problem in many cases. In this paper, we observe that many of the known constructions of ETFs are of one of two types. We further provide a new method for combining a given ETF of one of these two types with an appropriate group divisible design (GDD) in order to produce a larger ETF of the same type. By applying this method to known …


Design And Optimization Of A 3-D Plasmonic Huygens Metasurface For Highly-Efficient Flat Optics, Bryan M. Adomanis Sep 2018

Design And Optimization Of A 3-D Plasmonic Huygens Metasurface For Highly-Efficient Flat Optics, Bryan M. Adomanis

Theses and Dissertations

For miniaturization of future USAF unmanned aerial and space systems to become feasible, accompanying sensor components of these systems must also be reduced in size, weight and power (SWaP). Metasurfaces can act as planar equivalents to bulk optics, and thus possess a high potential to meet these low-SWaP requirements. However, functional efficiencies of plasmonic metasurface architectures have been too low for practical application in the infrared (IR) regime. Huygens-like forward-scattering inclusions may provide a solution to this deficiency, but there is no academic consensus on an optimal plasmonic architecture for obtaining efficient phase control at high frequencies. This dissertation asks …


Reconstruction Of The 3d Temperature And Species Concentration Spatial Distribution Of A Jet Engine Exhaust Plume Using An Infrared Fourier Transform Spectrometer Hyperspectral Imager, Mason D. Paulec Sep 2018

Reconstruction Of The 3d Temperature And Species Concentration Spatial Distribution Of A Jet Engine Exhaust Plume Using An Infrared Fourier Transform Spectrometer Hyperspectral Imager, Mason D. Paulec

Theses and Dissertations

The measurement of combustion byproducts is useful for determining pollution of any fuel burning application, efficiency of combustion, and determining detectability of aircraft exhausts. Both intrusive and non-intrusive techniques have been utilized to measure these quantities. For the majority of the non-intrusive techniques, the absorption and emission spectra of the gases are utilized for measurements. For this research, the use of the Telops Infrared Fourier Transform Spectrometer (IFTS) Hyperspectral Imager (HSI) was explored within the scope of combustion diagnostic methods, as an option for remote measurements of a jet turbine to determine concentration of species and temperature of the combustion …


A Macro-Level Order Metric For Self-Organizing Adaptive Systems, David W. King, Gilbert L. Peterson Sep 2018

A Macro-Level Order Metric For Self-Organizing Adaptive Systems, David W. King, Gilbert L. Peterson

Faculty Publications

Analyzing how agent interactions affect macro-level self-organized behaviors can yield a deeper understanding of how complex adaptive systems work. The dynamic nature of complex systems makes it difficult to determine if, or when, a system has reached a state of equilibrium or is about to undergo a major transition reflecting the appearance of self-organized states. Using the notion of local neighborhood entropy, this paper presents a metric for evaluating the macro-level order of a system. The metric is tested in two dissimilar complex adaptive systems with self-organizing properties: An autonomous swarm searching for multiple dynamic targets and Conway's Game of …


The Effectiveness Of Using Diversity To Select Multiple Classifier Systems With Varying Classification Thresholds, Harris K. Butler Iv, Mark A. Friend, Kenneth W. Bauer, Trevor J. Bihl Sep 2018

The Effectiveness Of Using Diversity To Select Multiple Classifier Systems With Varying Classification Thresholds, Harris K. Butler Iv, Mark A. Friend, Kenneth W. Bauer, Trevor J. Bihl

Faculty Publications

In classification applications, the goal of fusion techniques is to exploit complementary approaches and merge the information provided by these methods to provide a solution superior than any single method. Associated with choosing a methodology to fuse pattern recognition algorithms is the choice of algorithm or algorithms to fuse. Historically, classifier ensemble accuracy has been used to select which pattern recognition algorithms are included in a multiple classifier system. More recently, research has focused on creating and evaluating diversity metrics to more effectively select ensemble members. Using a wide range of classification data sets, methodologies, and fusion techniques, current diversity …


Application Of Spectral Solution And Neural Network Techniques In Plasma Modeling For Electric Propulsion, Joseph R. Whitman Sep 2018

Application Of Spectral Solution And Neural Network Techniques In Plasma Modeling For Electric Propulsion, Joseph R. Whitman

Theses and Dissertations

A solver for Poisson's equation was developed using the Radix-2 FFT method first invented by Carl Friedrich Gauss. Its performance was characterized using simulated data and identical boundary conditions to those found in a Hall Effect Thruster. The characterization showed errors below machine-zero with noise-free data, and above 20% noise-to-signal strength, the error increased linearly with the noise. This solver can be implemented into AFRL's plasma simulator, the Thermophysics Universal Research Framework (TURF) and used to quickly and accurately compute the electric field based on charge distributions. The validity of a machine learning approach and data-based complex system modeling approach …


A Methodology For Evaluating Relational And Nosql Databases For Small-Scale Storage And Retrieval, Ryan D. Engle Sep 2018

A Methodology For Evaluating Relational And Nosql Databases For Small-Scale Storage And Retrieval, Ryan D. Engle

Theses and Dissertations

Modern systems record large quantities of electronic data capturing time-ordered events, system state information, and behavior. Subsequent analysis enables historic and current system status reporting, supports fault investigations, and may provide insight for emerging system trends. Unfortunately, the management of log data requires ever more efficient and complex storage tools to access, manipulate, and retrieve these records. Truly effective solutions also require a well-planned architecture supporting the needs of multiple stakeholders. Historically, database requirements were well-served by relational data models, however modern, non-relational databases, i.e. NoSQL, solutions, initially intended for “big data” distributed system may also provide value for smaller-scale …


Breaking Down The Barriers To Operator Workload Estimation: Advancing Algorithmic Handling Of Temporal Non-Stationarity And Cross-Participant Differences For Eeg Analysis Using Deep Learning, Ryan G. Hefron Sep 2018

Breaking Down The Barriers To Operator Workload Estimation: Advancing Algorithmic Handling Of Temporal Non-Stationarity And Cross-Participant Differences For Eeg Analysis Using Deep Learning, Ryan G. Hefron

Theses and Dissertations

This research focuses on two barriers to using EEG data for workload assessment: day-to-day variability, and cross- participant applicability. Several signal processing techniques and deep learning approaches are evaluated in multi-task environments. These methods account for temporal, spatial, and frequential data dependencies. Variance of frequency- domain power distributions for cross-day workload classification is statistically significant. Skewness and kurtosis are not significant in an environment absent workload transitions, but are salient with transitions present. LSTMs improve day- to-day feature stationarity, decreasing error by 59% compared to previous best results. A multi-path convolutional recurrent model using bi-directional, residual recurrent layers significantly increases …


Evaluation And Quantification Of Diffractive Plenoptic Camera Algorithm Performance, Jack A. Shepherd Iii Sep 2018

Evaluation And Quantification Of Diffractive Plenoptic Camera Algorithm Performance, Jack A. Shepherd Iii

Theses and Dissertations

A diffractive plenoptic camera is a novel approach to the traditional plenoptic camera which replaces the main optic with a Fresnel zone plate making the camera sensitive to wavelength instead of range. However, algorithms are necessary to reconstruct the image produced by plenoptic cameras. While many algorithms exist for traditional plenoptic cameras, their ability to create spectral images in a diffractive plenoptic camera is unknown. This paper evaluates digital refocusing, super resolution, and 3D deconvolution through a Richardson-Lucy algorithm as well as a new Gaussian smoothing algorithm. All of the algorithms worked well near the Fresnel zone plate design wavelength, …


Techniques For Improved Space Object Detection Performance From Ground-Based Telescope Systems Using Long And Short Exposure Images, David J. Becker Aug 2018

Techniques For Improved Space Object Detection Performance From Ground-Based Telescope Systems Using Long And Short Exposure Images, David J. Becker

Theses and Dissertations

Space object detection is of great importance in the highly dependent yet competitive and congested space domain. Detection algorithms employed play a crucial role in fulfilling the detection component in the space situational awareness mission to detect, track, characterize and catalog unknown space objects. Many current space detection algorithms use a matched filter or a spatial correlator on long exposure data to make a detection decision at a single pixel point of a spatial image based on the assumption that the data follows a Gaussian distribution. This research focuses on improving current space object detection algorithms and developing new algorithms …


Compressive Sampling For Phenotype Classification, Eric L. Brooks Aug 2018

Compressive Sampling For Phenotype Classification, Eric L. Brooks

Theses and Dissertations

Phenotype classification has become an increasingly important genomic research method for disease identification and treatment. Phenotype classification is the investigation into the genetic information concerned with locating biomarkers (features) in order to identify an observed effect. The primary challenge associated with phenotype classification is with analyzing the data due to the inherent high-dimensionality of DNA data. As a result, phenotype classification faces challenges with feature selection, and consequently, classification accuracy. This research developed a methodology to alleviate these challenges while improving classification performance. The methodology leverages concepts of compressive sampling, to arrive at a process that identifies features most relevant …


Numerical Simulation Of High Energy Laser Propagation, Dana F. Morrill Aug 2018

Numerical Simulation Of High Energy Laser Propagation, Dana F. Morrill

Theses and Dissertations

High energy lasers have many applications, such as in aerospace, weapons, wireless power transfer, and manufacturing. Fluid-laser interaction is important to predicting power at receiver, and other measures of laser beam quality. Typically the carrying medium of the laser is modeled statistically. This dissertation describes a novel method of coupling fluid dynamics to beam propagation in free space. The coupled laser-fluid solver captures dynamic interaction of fluid temperature and beam intensity. Ultimately, the model captures the effects of fluid convection in the laser intensity-field. Boundary conditions play an important role for fluid dynamics, more so than for beam dynamics. Simulation …


Statistical Inference To Evaluate And Compare The Performance Of Correlated Multi-State Classification Systems, Beau A. Nunnally Aug 2018

Statistical Inference To Evaluate And Compare The Performance Of Correlated Multi-State Classification Systems, Beau A. Nunnally

Theses and Dissertations

The current emphasis on including correlation when comparing diagnostic test performance is quite important, however, there are cases in which correlation effects may be negligible with respect to inference. This proposed work examines the impact of including correlation between classification systems with continuous features by comparing the optimal performance of two diagnostic tests with multiple outcomes as well as providing inference for a sequence of tests. We define the optimal point using Bayes Cost, a metric that sums the weighted misclassifications within a diagnostic test using a cost/benefit structure. Through simulation, we quantify the impact of correlation on standard errors …


Automating Mobile Device File Format Analysis, Richard A. Dill Aug 2018

Automating Mobile Device File Format Analysis, Richard A. Dill

Theses and Dissertations

Forensic tools assist examiners in extracting evidence from application files from mobile devices. If the file format for the file of interest is known, this process is straightforward, otherwise it requires the examiner to manually reverse engineer the data structures resident in the file. This research presents the Automated Data Structure Slayer (ADSS), which automates the process to reverse engineer unknown file for- mats of Android applications. After statically parsing and preparing an application, ADSS dynamically runs it, injecting hooks at selected methods to uncover the data structures used to store and process data before writing to media. The resultant …


Cyber Anomaly Detection: Using Tabulated Vectors And Embedded Analytics For Efficient Data Mining, Robert J. Gutierrez, Kenneth W. Bauer, Bradley C. Boehmke, Cade M. Saie, Trevor J. Bihl Aug 2018

Cyber Anomaly Detection: Using Tabulated Vectors And Embedded Analytics For Efficient Data Mining, Robert J. Gutierrez, Kenneth W. Bauer, Bradley C. Boehmke, Cade M. Saie, Trevor J. Bihl

Faculty Publications

Firewalls, especially at large organizations, process high velocity internet traffic and flag suspicious events and activities. Flagged events can be benign, such as misconfigured routers, or malignant, such as a hacker trying to gain access to a specific computer. Confounding this is that flagged events are not always obvious in their danger and the high velocity nature of the problem. Current work in firewall log analysis is manual intensive and involves manpower hours to find events to investigate. This is predominantly achieved by manually sorting firewall and intrusion detection/prevention system log data. This work aims to improve the ability of …


Evaluation Criteria For Selecting Nosql Databases In A Single Box Environment, Ryan D. Engle, Brent T. Langhals, Michael R. Grimaila, Douglas D. Hodson Aug 2018

Evaluation Criteria For Selecting Nosql Databases In A Single Box Environment, Ryan D. Engle, Brent T. Langhals, Michael R. Grimaila, Douglas D. Hodson

Faculty Publications

In recent years, NoSQL database systems have become increasingly popular, especially for big data, commercial applications. These systems were designed to overcome the scaling and flexibility limitations plaguing traditional relational database management systems (RDBMSs). Given NoSQL database systems have been typically implemented in large-scale distributed environments serving large numbers of simultaneous users across potentially thousands of geographically separated devices, little consideration has been given to evaluating their value within single-box environments. It is postulated some of the inherent traits of each NoSQL database type may be useful, perhaps even preferable, regardless of scale. Thus, this paper proposes criteria conceived to …


Securing Zigbee Commercial Communications Using Constellation Based Distinct Native Attribute Fingerprinting, Christopher M. Rondeau, J. Addison Betances, Michael A. Temple Jul 2018

Securing Zigbee Commercial Communications Using Constellation Based Distinct Native Attribute Fingerprinting, Christopher M. Rondeau, J. Addison Betances, Michael A. Temple

Faculty Publications

This work provides development of Constellation Based DNA (CB-DNA) Fingerprinting for use in systems employing quadrature modulations and includes network protection demonstrations for ZigBee offset quadrature phase shift keying modulation. Results are based on 120 unique networks comprised of seven authorized ZigBee RZSUBSTICK devices, with three additional like-model devices serving as unauthorized rogue devices. Authorized network device fingerprints are used to train a Multiple Discriminant Analysis (MDA) classifier and Rogue Rejection Rate (RRR) estimated for 2520 attacks involving rogue devices presenting themselves as authorized devices. With MDA training thresholds set to achieve a True Verification Rate (TVR) of TVR = …


Cybersecurity Architectural Analysis For Complex Cyber-Physical Systems, Martin Trae Span Iii, Logan O. Mailloux, Michael R. Grimaila Jul 2018

Cybersecurity Architectural Analysis For Complex Cyber-Physical Systems, Martin Trae Span Iii, Logan O. Mailloux, Michael R. Grimaila

Faculty Publications

In the modern military’s highly interconnected and technology-reliant operational environment, cybersecurity is rapidly growing in importance. Moreover, as a number of highly publicized attacks have occurred against complex cyber-physical systems such as automobiles and airplanes, cybersecurity is no longer limited to traditional computer systems and IT networks. While architectural analysis approaches are critical to improving cybersecurity, these approaches are often poorly understood and applied in ad hoc fashion. This work addresses these gaps by answering the questions: 1. “What is cybersecurity architectural analysis?” and 2. “How can architectural analysis be used to more effectively support cybersecurity decision making for complex …


Arrhenius Rate Chemistry-Informed Inter-Phase Source Terms (Arciist), Matthew J. Schwaab, Robert B. Greendyke, Bryan J. Steward Jul 2018

Arrhenius Rate Chemistry-Informed Inter-Phase Source Terms (Arciist), Matthew J. Schwaab, Robert B. Greendyke, Bryan J. Steward

Faculty Publications

Currently, in macro-scale hydrocodes designed to simulate explosive material undergoing shock-induced ignition, the state of the art is to use one of numerous reaction burn rate models. These burn models are designed to estimate the bulk chemical reaction rate. Unfortunately, these burn rate models are largely based on empirical data and must be recalibrated for every new material being simulated. We propose that the use of Arrhenius Rate Chemistry-Informed Interphase Source Terms (ARCIIST) in place of empirically derived burn models will improve the accuracy for these computational codes. A reacting chemistry model of this form was developed for the cyclic …


Efficient Phase Retrieval For Off-Axis Point Spread Functions, Salome Esteban Carrasco Jun 2018

Efficient Phase Retrieval For Off-Axis Point Spread Functions, Salome Esteban Carrasco

Theses and Dissertations

A novel pairing of phase retrieval tools allows for efficient estimation of pupil phase in optical systems from images of point spread functions (PSFs). The phase retrieval algorithm uses correlation of modeled phase in the focal plane to decouple aberrations that are difficult to identify in complex PSFs. The use of a phase kernel that departs from the Fresnel approximation for off-axis PSFs is a more accurate representation of wavefront phase in finite conjugate imaging. The combination of the approximation and phase correlation algorithm can be more efficient and accurate than generic algorithms.