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NASA NTRSDate not supplied

Markov Decision Process based Trajectory Planning for UAVs under Uncertain Wind Conditions

In this paper we propose a Markov Decision Process (MDP) algorithm for path-planning of Unmanned Aviation Vehicles (UAVs) under varying wind conditions. Solutions to path-planning for UAVs are becoming increasingly necessary as autonomous UAVs continue to enter commercial and government spaces. Path-planning is inherently challenging, as UAVs needs to account for dynamically changing flying conditions such as weather, obstacle or no-fly zones, degraded vehicle health and off-nominal battery power consumption. Machine learning methods such as Markov Decision Process (MDPs) have the potential to revolutionize how vehicles navigate in such uncertain environments. Previous papers have demonstrated the use of MDPs to optimize UAV path-planning for energy consumption under time-varying wind distribution. In this study, UAV trajectories from a pre-determined waypoint to target cell, will be computed on a 7X7 grid environment by optimizing parameters for mission assurance and safety limits in addition to the energy consumption, and operation time. The UAV navigates the grid by taking actions to move in either of the eight cardinal and intercardinal directions, under constant thrust profile. The next state of the UAV is calculated by considering its action, transition probability, obstacle cells and the wind speed magnitude and direction. Both constant and stochastic wind will be considered in this paper, the parameters being extracted from real wind measurements in proximity to an experimental UAV flight. One of the studies to be demonstrated in this paper is that as the unmanned airspace gets more complex with multiple vehicles and environmental uncertainties, trade-offs between energy consumption, operation time, risk tolerance, and mission assurance needs to be made. Further, MDPs are capable of fast computation of UAV trajectories under varying wind, hence making them suitable for in-flight path planners.

decision-making↗

NASA NTRSDate not supplied

Tailoring Early Concept Maturity Levels for Small Satellite Missions

The Concept Maturity Level (CML) framework has gained widespread acceptance as a process to assess mission concept maturity, particularly for early mission concepts in the formulation phase. Popularized by the Jet Propulsion Laboratory almost 15 years ago to evaluate large NASA science mission concepts, the CML method offers wide-ranging programmatic and technical evaluation criteria to assess and measure the progress of mission concepts at all stages of maturity. It also provides a useful tool to facilitate stakeholder relationships through effective engagement, interaction, and communication. The CML approach is modeled largely after the widespread Technology Readiness Level (TRL) scale, a measure of how new technology progresses from early concept through to being operational at the mission flight system level. Tailoring the CML framework for small satellites allows for more relevant technical and programmatic evaluations for these missions. Given the diversity of small satellite mission objectives, cost constraints, complexity, and risk tolerance, this modified CML framework enables study teams and stakeholders to determine the appropriate CML fidelity to implement based upon the opportunity and strategic investment approach. A decision process flow method is proposed to guide teams through the logic of determining what CML fidelity to implement in evaluating small satellite mission concepts. The goal is to provide enhanced stakeholder engagement such that CML assessments are clearly articulated and understood early in the design study. This would allow for more appropriate application of CML methodology by study teams developing University led missions, technology demonstrations, commercial constellations, and more complex planetary science missions.

Concept Maturity Level↗

NASA NTRSDate not supplied

Evacuation and Shelter Plans for Asteroid Impacts

Lessons from Analogs - Hurricanes - Forecasting in time to take action - Large scale evacuations - Risk tolerance - Nuclear Explosions - Effects of large blast waves - Shelters - Fatality rates Application to Asteroid Impacts - PDC 2021: small, little warning, poorly constrained - PDC 2023: large, well characterized

ATAP↗

NASA NTRSDate not supplied

Safety Expertise and the Perils of Novelty

Emerging aviation markets such as urban air mobility are giving rise to new technologies and means of operation. However, novelty may hide ‘unknown unknowns,’ raising new hazards. This paper examines how expertise and safety techniques enable transformative technologies such as reduced crew operations, hybrid wing-borne and rotor-born flight, federated air traffic services, and urban operations. We explore how analysts use expertise to address common-cause failures, collect and interpret safety data, and perform exacting tradeoffs between dissimilarity, redundancy, independence, and diversity (human, process lifecycle, or otherwise) to ensure safety. When novelty is present, analysts might not possess the expertise needed to fully understand the implications of design decisions and tradeoffs being made, especially in early lifecycle phases, on emergent properties such as safety. Safety expertise must be carefully cultivated. The conflicting views of safety experts must be unpacked to identify the divergence in fundamental assumptions, models, means, and methods that may be causing them. Once systems venture beyond the basis of what safety expertise can reliably guarantee, projects take on risk that must be managed. The paper contains key takeaways and actionable recommendations for novel OEMs and regulators touching on topics such as robust monitoring; clear and transparent reporting; incremental approaches to fielding novel systems in hazard-rich, risk-tolerant environments; the cultivation of safety culture and expertise in an organization; and the use of scientific study to reduce epistemic uncertainty in novel operations with new technologies. Since excessive novelty in aviation can undermine the current foundation of safety, humility and incrementalism are necessary to enable emerging aviation markets safely.

safety expertise↗

NASA NTRSDate not supplied

Safety Expertise and the Perils of Novelty

safety expertise↗

NASA NTRS2024

Computing Proximity to Threat Along Uncertain Trajectory to Support Urban Air Mobility

Many airspace threats affect the selection of a flight route, such as terrain, physical obstacles, adverse weather, and special-use airspace, among others. Threat avoidance during Urban Air Mobility (UAM) and Low Altitude Mobility (LAM) flights is especially challenging due to their lower cruising altitudes. These operations may be exposed to buildings, towers, trees, terrain undulations, etc., along much of their flight route. Moreover, these low-altitude flights in urban environments are expected to encounter very busy airspace, increasing the workload associated with threat avoidance. The Proximity to Threat (PtT) function aims to support onboard or remote pilots by computing the risk from geospatial threats in the airspace along an aircraft’s flight path. The threats are both static entities and dynamic airspace restrictions that have been modeled and stored in geo-referenced mapping databases. The flight path may be specified by waypoints, airways, or similar discrete route elements or by a time series of closely spaced position and altitude points. Flight path uncertainty can also be taken into account by assuming both the position and altitude to be sampled from two normal distributions, each being specified by a mean and variance. PtT uses this additional information to verify that the flight path remains clear of threats along a wider path or if a position is reached earlier or later than anticipated, assuring the pilot or operator that there will be an available safety margin even if the aircraft deviates due to one or more occurrences of unexpected wind gusts, mechanical failures, airspeed changes, collision avoidance maneuvers, etc. The client can specify an uncertainty confidence level to bound the set of trajectories evaluated for threats so that the confidence level corresponds to the client’s risk tolerance under the expected conditions. PtT can be used as a pre-flight planning tool to determine a safe route through known threats and in-flight to avoid emerging or changing threats. In this report, we present the PtT function, discuss how the trajectory uncertainty is incorporated into the PtT function, and describe the use of this PtT function in a representative scenario.

urban air mobility↗

NASA NTRSDate not supplied

Energy-Optimized Path Planning for Uas in Varying Winds Via Reinforcement Learning

In this paper we propose a reinforcement learning (RL) algorithm for path planning of Unmanned Aviation Vehicles (UAVs) under varying wind conditions. Solutions to UAV path planning problems are becoming increasingly necessary as autonomous UAVs continue to enter commercial and government spaces. Path-planning is inherently challenging, as UAVs need to account for dynamically changing flying conditions such as weather, obstacle or no-fly zones, degraded vehicle health, and off-nominal battery power consumption. Machine learning methods such as reinforcement learning (RL) have the potential to revolutionize how vehicles navigate in such uncertain environments. In this study, we compute UAV trajectories from a pre-determined starting position to a target cell within a 7X7 grid environment by optimizing parameters for mission assurance and safety limits in addition to the energy consumption and operation time. The UAV navigates the grid by taking actions to move in any of the eight cardinal and inter-cardinal directions, under constant thrust profile. The resultant UAV state is sampled from a probability distribution which accounts for the UAV’s action, local wind velocity, and the presence of obstacles or boundaries. As the unmanned airspace gets more complex due to multiple vehicles and environmental uncertainties, trade-offs between energy consumption, operation time, risk tolerance, and mission assurance need to be made. Our Markov Decision Process (MDP) environment model can capture any combination of these in the optimization objective, making it novel compared to other work in the field.

trajectory planning↗

NASA NTRSDate not supplied

Full Lunar Surface Visualization and Simulation Platform

This paper reports the results of the innovative work completed to facilitate the creation of a lunar surface visualization and simulation platform for the entire surface of the moon as a design, test, training, verification, and mission support environment. The mathematical approach to the visualization of the data sets utilizes fractal mathematics to generate not only the highest possible detail that the Lunar Reconnaissance Orbiter (LRO) data can provide, but it also additionally supplies an estimation of surface reflectivity and lighting based upon location on the lunar surface. This level of detail and accuracy is required to conduct accurate integrated crew / vehicle risk assessment throughout the design-to-disposal life cycle of vehicles and their supporting procedures and mission priorities. This process starts with safety assessments of integrated hardware / software / crew/ and support environments. Unlike the Apollo Program, the Artemis Program is not risk tolerant.

Simulation↗

NASA NTRS2024

The NASA Radiation Hardness Assurance (RHA) Process Standard

This paper presents status on the upcoming NASA Agency-level RHA Process Standard. Based on recommendations provided by the NASA Engineering and Safety Center (NESC), the NASA Electronic Parts and Packaging (NEPP) Program has led development of an agency-level standard for space flight avionics and electronics RHA. The standard introduces a novel RHA taxonomy and prescribes the process required by NASA programs and projects to baseline their RHA programs consistent with the MEAL (Mission, Environment, Application, and Lifetime) criteria including the risk tolerance posture. The standard also provides requirements for the RHA schedule integration in the program or project lifecycle and requirements for data deliverables content. Additional non-prescriptive technical content is provided in the standard appendices to guide development of RHA programs for NASA missions.

Avionics↗

NASA NTRS2024

NASA Risk Management Handbook: Version 2.0, Part 1

The purpose of this handbook is to provide an in-depth reference for the practice of risk management in NASA, updating the guidance offered in its original version, NASA/SP-2011-3422 (November 2011), and closely aligning the updated guidance with the current NASA Procedural Requirements for Agency Risk Management, NPR 8000.4, and the parent NASA Policy Directive for NASA Governance and Strategic Management, NPD 1000.0. NPD 1000.0 introduces with emphasis the concept of “Risk Leadership,” making it a fundamental tenet and pillar of the risk management culture that it advocates for the Agency. NPR 8000.4 applies this concept and establishes Risk Management (RM) requirements for the Agency as an integrated enterprise, as well as the RM requirements for portfolio elements within the enterprise. Such elements include the various programs and projects that contribute to the Agency’s objectives and the various institutional activities carried out by entities that contribute to mission support. The present version of the handbook also emphasizes the integration of risk management processes across activity and project life cycles and their coordination and interaction with day-to-day programmatic and organizational functions. Areas of application of risk assessment and management that were not covered with specific guidance in the preceding version are addressed in this version with in-depth examples. The handbook is structured into two parts, whose chapters are in turn organized in a sequential order intended to facilitate a gradual and progressive introduction of the reader to risk management principles and practices. Part 1 of the handbook is dedicated to the introduction of the basic foundations of the NASA integrated risk management framework, the related fundamental risk concepts, the description of the risk management and decision processes that are to be implemented within the framework, the discussion of the risk assessment techniques that should be utilized in support of such processes, and the management and organizational interactions and interfaces that should be enabled to implement an effective integration of risk management activities within the Agency. Part 2 provides self-contained, end-to-end examples of application of the processes and techniques introduced in Part 1, in the context of both programmatic (i.e., project and/or mission related) and institutional activities.

Uncertainty↗

NASA NTRS2002

An Approach to Risk-Based Design Incorporating Damage Tolerance Analyses

Incorporating risk-based design as an integral part of spacecraft development is becoming more and more common. Assessment of uncertainties associated with design parameters and environmental aspects such as loading provides increased knowledge of the design and its performance. Results of such studies can contribute to mitigating risk through a system-level assessment. Understanding the risk of an event occurring, the probability of its occurrence, and the consequences of its occurrence can lead to robust, reliable designs. This paper describes an approach to risk-based structural design incorporating damage-tolerance analysis. The application of this approach to a candidate Earth-entry vehicle is described. The emphasis of the paper is on describing an approach for establishing damage-tolerant structural response inputs to a system-level probabilistic risk assessment.

Knight, Norman F., Jr.↗

NASA NTRS2012

Response Surface Modeling Tolerance and Inference Error Risk Specifications: Proposed Industry Standards

This paper reviews the derivation of an equation for scaling response surface modeling experiments. The equation represents the smallest number of data points required to fit a linear regression polynomial so as to achieve certain specified model adequacy criteria. Specific criteria are proposed which simplify an otherwise rather complex equation, generating a practical rule of thumb for the minimum volume of data required to adequately fit a polynomial with a specified number of terms in the model. This equation and the simplified rule of thumb it produces can be applied to minimize the cost of wind tunnel testing.

DeLoach, Richard↗

NASA NTRS2011

Shooting the Moon

This story is about an unlikely NASA mission to the Moon. It was unlikely because it was started with far too little time and too-little money to complete. It was unlikely because it was able to take chances to accept risk of failure. It was unlikely because it was searching for the unthinkable: water-ice on the moon... Figure 1-1: LCROSS Mission. The mission of the Lunar CRater Observation and Sensing Satellite (LCROSS) was to investigate the possibility of water ice in craters on the Moon s poles. This is certainly an interesting scientific topic in itself, but I intend to focus on the compelling experience of managing the LCROSS Project in the context of this storied Agency. Perhaps most interesting are the implications this story has for managing any development effort, lunar or not, and working a balance to achieve success. NASA is by design a risk-taking agency within the US Government. It could be argued that NASA s purpose in the aerospace community is to take on the really big challenges that either the corporate world can t afford, are not yet profitable endeavors, or are just too risky for private corporations to entertain. However, expectations of the Agency have evolved. A combination of grim human tragedies and some very public cost and schedule overruns have challenged the public s and Congress s tolerance for risk-taking within the Agency. NASA, which is supposed to be in the business of taking risks to do bold, difficult things, has become less and less able to do so within its cost framework. Yet effectively replacing prudent risk management with attempts to "risk-eliminate" is completely unaffordable. So where does risk-taking fit within the Agency, or within private/corporate organizations for that matter? Where astronauts play there is clearly concern about risk. When an organization puts humans in harm s way, it is understandably going to take extra effort to assure nobody gets hurt. Doing so, of course, costs money - a lot of money to pay for labor and hardware which is attempting to assure nothing will go wrong. Sophisticated designs, with doubly- or triply-redundant systems, extensive testing to verify those systems, and numerous engineering test units built to learn and evolve a hardware design, all drive the cost and time required to implement. Human spaceflight is an expensive business because of the exceptional system complexity and levels of assurance required for human space travel. What about missions that do not involve human spaceflight? What about missions whose potential failure will not take a human life, whose costs are small and whose urgency and importance are limited by design? A portfolio consisting of this type of mission can be designed to be risk tolerant, not requiring large expenditures to guarantee against failure. With the money saved, the number of missions that can be executed within the portfolio grows, or the total cost of the portfolio can be reduced. The NASA LCROSS mission is a pathfinder example of a low-cost, quick turn-around mission which struck a balance on mission risk, while accomplishing big objectives, like defining how we understand the Moon.

Andrews, Daniel R.↗

NASA NTRS1999

The Use of the Satellite Breakup Risk Assessment Model (SBRAM) to Characterize Collision Risk to Manned Spacecraft

NASA uses environment models such as ORDEM96 to characterize the long-term orbital debris collision hazard for spacecraft in LEO. Occasionally, however, there are breakups of satellites or rocket bodies that create enhanced collision hazard for a period of time. This enhanced collision hazard can pose increased risks to space operations - especially those involving manned missions where the tolerance for risk is very low. NASA has developed SBRAM to simulate the enhanced debris environment in the days and weeks that follow such a breakup. This simulation provides the kind of risk probabilities that can be used by mission planners to consider if changes are warranted for the mission. Announcements of breakups come to NASA from US Space Command as soon as they are identified. The pre-breakup orbit and time of breakup are used to determine the initial conditions of the explosion. SBRAM uses the latest explosion models developed at NASA to simulate a debris cloud for the breakup. The model uses a Monte Carlo technique to create a random debris cloud from the probability distributions in the breakup model. Each piece of debris randomly created in the cloud is propagated in a deterministic manner to include the effects of drag and other orbital perturbations. The detailed geometry of each simulated close approach to the target spacecraft is noted and logged and the collision probability is computed using an estimated probability density in down-range and cross-range positions of both the target spacecraft and debris object. The collision probability is computed from the overlap of these probability densities for each close-approach geometry and summed over all computed conjunctions. Cloud propagation runs over the desired time interval are then repeated until the scale of the collision risk can be estimated to a desired precision. This paper presents an overview of the SBRAM model and a number of examples, both real and hypothetical, to demonstrate its use. In addition, a number of different examples are shown how the data can be used by decision makers on issues such as spacecraft orientation and timing of EVAs.

Matney, Mark J.↗

NASA NTRS2024

NASA Risk Management Handbook: Version 2.0, Part 2

The purpose of this handbook is to provide an in-depth reference for the practice of risk management in NASA, updating the guidance offered in its original version, NASA/SP-2011-3422 (November 2011), and closely aligning the updated guidance with the current NASA Procedural Requirements for Agency Risk Management, NPR 8000.4, and the parent NASA Policy Directive for NASA Governance and Strategic Management, NPD 1000.0 (January 2020). NPD 1000.0 introduces with emphasis the concept of “Risk Leadership,” making it a fundamental tenet and pillar of the risk management culture that it advocates for the Agency. NPR 8000.4 applies this concept and establishes Risk Management (RM) requirements for the Agency as an integrated enterprise, as well as the RM requirements for portfolio elements within the enterprise. Such elements include the various programs and projects that contribute to the Agency’s objectives and the various institutional activities carried out by entities that contribute to mission support. The present version of the handbook also emphasizes the integration of risk management processes across activity and project life cycles and their coordination and interaction with day-to-day programmatic and organizational functions. Areas of application of risk assessment and management that were not covered with specific guidance in the preceding version are addressed in this version with in-depth examples. The handbook is structured into two parts, whose chapters are in turn organized in a sequential order intended to facilitate a gradual and progressive introduction of the reader to risk management principles and practices. Part 1 of the handbook is dedicated to the introduction of the basic foundations of the NASA integrated risk management framework, the related fundamental risk concepts, the description of the risk management and decision processes that are to be implemented within the framework, the discussion of the risk assessment techniques that should be utilized in support of such processes, and the management and organizational interactions and interfaces that should be enabled to implement an effective integration of risk management activities within the Agency. Part 2 provides self-contained, end-to-end examples of application of the processes and techniques introduced in Part 1, in the context of both programmatic (i.e., project and/or mission related) and institutional activities.

Risk Leadership↗

NASA NTRS2001

Biomarkers of exposure and dose: state of the art

Biomarkers provide methods to measure changes in biological systems and to relate them to environmental insults and disease processes. Biomarkers can be classified as markers of exposure and dose, markers of sensitivity, and markers of disease. It is important that the differences and applications of the various types of biomarkers be clearly understood. The military is primarily interested in early biomarkers of exposure and dose that do not require high levels of sensitivity but can be used to rapidly triage war fighters under combat or terrorist conditions and determine which, if any, require medical attention. Biomarkers of long-term radiation risk represent the second area of interest for the military. Biomarkers of risk require high sensitivity and specificity for the disease and insult but do not require rapid data turnaround. Biomarkers will help provide information for quick command decisions in the field, characterise long-term troop risks and identify early stages of radiation-induced diseases. This information provides major positive reassurances about individual exposures and risk that will minimise the physical and psychological impact of wartime radiation exposures.

Review, Tutorial↗

NASA NTRS2000

Overcoming Dynamic Disturbances in Imaging Systems

We develop and discuss a methodology with the potential to yield a significant reduction in complexity, cost, and risk of space-borne optical systems in the presence of dynamic disturbances. More robust systems almost certainly will be a result as well. Many future space-based and ground-based optical systems will employ optical control systems to enhance imaging performance. The goal of the optical control subsystem is to determine the wavefront aberrations and remove them. Ideally reducing an aberrated image of the object under investigation to a sufficiently clear (usually diffraction-limited) image. Control will likely be distributed over several elements. These elements may include telescope primary segments, telescope secondary, telescope tertiary, deformable mirror(s), fine steering mirror(s), etc. The last two elements, in particular, may have to provide dynamic control. These control subsystems may become elaborate indeed. But robust system performance will require evaluation of the image quality over a substantial range and in a dynamic environment. Candidate systems for improvement in the Earth Sciences Enterprise could include next generation Landsat systems or atmospheric sensors for dynamic imaging of individual, severe storms. The technology developed here could have a substantial impact on the development of new systems in the Space Science Enterprise; such as the Next Generation Space Telescope(NGST) and its follow-on the Next NGST. Large Interferometric Systems of non-zero field, such as Planet Finder and Submillimeter Probe of the Evolution of Cosmic Structure, could benefit. These systems most likely will contain large, flexible optomechanical structures subject to dynamic disturbance. Furthermore, large systems for high resolution imaging of planets or the sun from space may also benefit. Tactical and Strategic Defense systems will need to image very small targets as well and could benefit from the technology developed here. We discuss a novel speckle imaging technique with the potential to separate dynamic aberrations from static aberrations. Post-processing of a set of image data, using an algorithm based on this technique, should work for all but the lowest light levels and highest frequency dynamic environments. This technique may serve to reduce the complexity of the control system and provide for robust, fault-tolerant, reduced risk operation. For a given object, a short exposure image is "frozen" on the focal plane in the presence of the environmental disturbance (turbulence, jitter, etc.). A key factor is that this imaging data exhibits frame-to-frame linear shift invariance. Therefore, although the Point Spread Function is varying from frame to frame, the source is fixed; and each short exposure contains object spectrum data out to the diffraction limit of the imaging system. This novel speckle imaging technique uses the Knox-Thompson method. The magnitude of the complex object spectrum is straightforward to determine by well-established approaches. The phase of the complex object spectrum is decomposed into two parts. One is a single-valued function determined by the divergence of the optical phase gradient. The other is a multi-valued function determined by the circulation of the optical phase gradient-"hidden phase." Finite difference equations are developed for the phase. The novelty of this approach is captured in the inclusion of this "hidden phase." This technique allows the diffraction-limited reconstruction of the object from the ensemble of short exposure frames while simultaneously estimating the phase as a function of time from a set of exposures.

Young, Eric W.↗

NASA NTRS2000

Overcoming Dynamic Disturbances in Imaging Systems

We develop and discuss a methodology with the potential to yield a significant reduction in complexity, cost, and risk of space-borne optical systems in the presence of dynamic disturbances. More robust systems almost certainly will be a result as well. Many future space-based and ground-based optical systems will employ optical control systems to enhance imaging performance. The goal of the optical control subsystem is to determine the wavefront aberrations and remove them. Ideally reducing an aberrated image of the object under investigation to a sufficiently clear (usually diffraction-limited) image. Control will likely be distributed over several elements. These elements may include telescope primary segments, telescope secondary, telescope tertiary, deformable mirror(s), fine steering mirror(s), etc. The last two elements, in particular, may have to provide dynamic control. These control subsystems may become elaborate indeed. But robust system performance will require evaluation of the image quality over a substantial range and in a dynamic environment. Candidate systems for improvement in the Earth Sciences Enterprise could include next generation Landsat systems or atmospheric sensors for dynamic imaging of individual, severe storms. The technology developed here could have a substantial impact on the development of new systems in the Space Science Enterprise; such as the Next Generation Space Telescope(NGST) and its follow-on the Next NGST. Large Interferometric Systems of non-zero field, such as Planet Finder and Submillimeter Probe of the Evolution of Cosmic Structure, could benefit. These systems most likely will contain large, flexible optormechanical structures subject to dynamic disturbance. Furthermore, large systems for high resolution imaging of planets or the sun from space may also benefit. Tactical and Strategic Defense systems will need to image very small targets as well and could benefit from the technology developed here. We discuss a novel speckle imaging technique with the potential to separate dynamic aberrations from static aberrations. Post-processing of a set of image data, using an algorithm based on this technique, should work for all but the lowest light levels and highest frequency dynamic environments. This technique may serve to reduce the complexity of the control system and provide for robust, fault-tolerant, reduced risk operation. For a given object, a short exposure image is "frozen" on the focal plane in the presence of the environmental disturbance (turbulence, jitter, etc.). A key factor is that this imaging data exhibits frame-to-frame linear shift invariance. Therefore, although the Point Spread Function is varying from frame to frame, the source is fixed; and each short exposure contains object spectrum data out to the diffraction limit of the imaging system. This novel speckle imaging technique uses the Knox-Thompson method. The magnitude of the complex object spectrum is straightforward to determine by well-established approaches. The phase of the complex object spectrum is decomposed into two parts. One is a single-valued function determined by the divergence of the optical phase gradient. The other is a multi-valued function determined by, the circulation of the optical phase gradient-"hidden phase." Finite difference equations are developed for the phase. The novelty of this approach is captured in the inclusion of this "hidden phase." This technique allows the diffraction-limited reconstruction of the object from the ensemble of short exposure frames while simultaneously estimating the phase as a function of time from a set of exposures.

Young, Eric W.↗