John Dolloff

The Sequential Generation of Gaussian Random Fields for Applications in the Geospatial Sciences

This paper presents practical methods for the sequential generation or simulation of a Gaussian two-dimensional random field. The specific realizations typically correspond to geospatial errors or perturbations over a horizontal plane or grid. The errors are either scalar, such as vertical errors, or multivariate, such as , , and errors. These realizations enable simulation-based performance assessment and tuning of various geospatial applications. Both homogeneous and non-homogeneous random fields are addressed. The sequential generation is very fast and compared to methods based on Cholesky decomposition of an a priori covariance matrix and Sequential Gaussian Simulation. The multi-grid point covariance matrix is also developed for all the above random fields, essential for the optimal performance of many geospatial applications ingesting data with these types of errors.

The full multi-state vector error covariance matrix: why needed and its practical representation

Whether statistically representing the errors in the estimates of sensor metadata associated with a set of images, or statistically representing the errors in the estimates of 3D location associated with a set of ground points, the corresponding “full” multi-state vector error covariance matrix is critical to exploitation of the data. For sensor metadata, the individual state vectors typically correspond to sensor position and attitude of an image. These state vectors, along with their corresponding full error covariance matrix, are required for optimal down-stream exploitation of the image(s), such as for the stereo extraction of a 3D target location and its corresponding predicted accuracy. In this example, the full error covariance matrix statistically represents the sensor errors for each of the two images as well as the correlation (similarity) of errors between the two images. For ground locations, the individual state vectors typically correspond to 3D location. The corresponding full error covariance matrix statistically represents the location errors in each of the ground points as well as the correlation (similarity) of errors between any pair of the ground points. It is required in order to compute reliable estimates of relative accuracy between arbitrary ground point pairs, and for the proper weighting of the ground points when used as control, in for example, a fusion process. This paper details the above, and presents practical methods for the representation of the full error covariance matrix, ranging from direct representation with large bandwidth requirements, to high-fidelity approximation methods with small bandwidth requirements.

Error estimation for gridded bathymetry

Estimating the uncertainty or predicted accuracy of gridded products that are generated from historical bathymetric survey data is of high interest to the maritime navigation community. Surface interpolation methods used for gridding survey data in practice are well established. This paper investigates error estimation methods for gridded bathymetry in terms of their practical utility. Of particular interest are: 1) assessing the quality of a prior uncertainty of random error in survey data; 2) the significance of autocorrelated random errors; 3) the relationship between survey point density and propagated or product uncertainty; 4) the computational feasibility of Monte Carlo (MC) methods over large regions; and 5) the value of cross-validation to estimate error in the absence of controlled truth. K-fold cross-validation is used as the basis for performance evaluation of our approach to propagate a priori random errors via MC perturbation with spline-in-tension surface interpolation. Experiments are conducted with test areas in the Norwegian archipelago of Svalbard.

Evaluating conflation methods using uncertainty modeling

The classic problem of computer-assisted conflation involves the matching of individual features (e.g., point, polyline, or polygon vectors) as stored in a geographic information system (GIS), between two different sets (layers) of features. The classical goal of conflation is the transfer of feature metadata (attributes) from one layer to another. The age of free public and open source geospatial feature data has significantly increased the opportunity to conflate such data to create enhanced products. There are currently several spatial conflation tools in the marketplace with varying degrees of automation. An ability to evaluate conflation tool performance quantitatively is of operational value, although manual truthing of matched features is laborious and costly. In this paper, we present a novel methodology that uses spatial uncertainty modeling to simulate realistic feature layers to streamline evaluation of feature matching performance for conflation methods. Performance results are compiled for DCGIS street centerline features.

Kalman filter outputs for inclusion in video-stream metadata: accounting for the temporal correlation of errors for optimal target extraction

A video-stream associated with an Unmanned System or Full Motion Video can support the extraction of ground coordinates of a target of interest. The sensor metadata associated with the video-stream includes a time series of estimates of sensor position and attitude, required for down-stream single frame or multi-frame ground point extraction, such as stereo extraction using two frames in the video-stream that are separated in both time and imaging geometry. The sensor metadata may also include a corresponding time history of sensor position and attitude estimate accuracy (error covariance). This is required for optimal down-stream target extraction as well as corresponding reliable predictions of extraction accuracy. However, for multi-frame extraction, this is only a necessary condition. The temporal correlation of estimate errors (error cross-covariance) between an arbitrary pair of video frames is also required. When the estimates of sensor position and attitude are from a Kalman filter, as typically the case, the corresponding error covariances are automatically computed and available. However, the cross-covariances are not. This paper presents an efficient method for their exact representation in the metadata using additional, easily computed, data from the Kalman filter. The paper also presents an optimal weighted least squares extraction algorithm that correctly accounts for the temporal correlation, given the additional metadata. Simulation-based examples are presented that show the importance of correctly accounting for temporal correlation in multi-frame extraction algorithms.

Geolocation system estimators: processes for their quality assurance and quality control

This paper presents recommended principles and processes for the Quality Assurance (QA) and Quality Control (QC) of estimators and their outputs in Geolocation Systems. Relevant estimators include both batch estimators, such as Weighted Least Squares (WLS) estimators, and (near) real-time sequential estimators, such as Kalman filters. The estimators typically solve for (estimate) the value of a state vector X_true containing 3d geolocations and/or corrections to the sensor metadata corresponding to the measurements supplied to the estimator. Along with a best estimate X of X_true, the estimator outputs predicted accuracy, typically an error covariance matrix CovX corresponding to the error in the solution X. It is essential that the estimator output a reliable and near-optimal estimate X as well as a reliable error covariance matrix CovX. This paper presents various procedures, including detailed algorithms, to help ensure that this is the case, and if not, flag the problem along with supporting metrics. The majority of the QA/QC procedures involve data internal to the estimator, such as measurement residuals, and can be built-in to the estimator. Examples include measurement editing, solution convergence detection, and confidence interval tests based on the reference variance.

Geopositioning with a quadcopter: Extracted feature locations and predicted accuracy without a priori sensor attitude information

This paper presents an overview of the Full Motion Video-Geopositioning Test Bed (FMV-GTB) developed to investigate algorithm performance and issues related to the registration of motion imagery and subsequent extraction of feature locations along with predicted accuracy. A case study is included corresponding to a video taken from a quadcopter. Registration of the corresponding video frames is performed without the benefit of a priori sensor attitude (pointing) information. In particular, tie points are automatically measured between adjacent frames using standard optical flow matching techniques from computer vision, an a priori estimate of sensor attitude is then computed based on supplied GPS sensor positions contained in the video metadata and a photogrammetric/search-based structure from motion algorithm, and then a Weighted Least Squares adjustment of all a priori metadata across the frames is performed. Extraction of absolute 3D feature locations, including their predicted accuracy based on the principles of rigorous error propagation, is then performed using a subset of the registered frames. Results are compared to known locations (check points) over a test site. Throughout this entire process, no external control information (e.g. surveyed points) is used other than for evaluation of solution errors and corresponding accuracy.

Methods for the specification and validation of geolocation accuracy and predicted accuracy

The specification of geolocation accuracy requirements and their validation is essential for the proper performance of a Geolocation System and for trust in resultant three dimensional (3d) geolocations. This is also true for predicted accuracy requirements and their validation for a Geolocation System, which assumes that each geolocation produced (extracted) by the system is accompanied by an error covariance matrix that characterizes its specific predicted accuracy. The extracted geolocation and its error covariance matrix are standard outputs of (near) optimal estimators, either associated (internally) with the Geolocation System itself, or with a “downstream” application that inputs a subset of Geolocation System output, such as sensor data/metadata: for example, a set of images and corresponding metadata of the imaging sensor’s pose and its predicted accuracy. This output allows for subsequent (near) optimal extraction of geolocations and associated error covariance matrices based on the application’s measurements of pixel locations in the images corresponding to objects of interest. This paper presents recommended methods and detailed equations for the specification and validation of both accuracy and predicted accuracy requirements for a general Geolocation System. The specification/validation of accuracy requirements are independent from the specification/validation of predicted accuracy requirements. The methods presented in this paper are theoretically rigorous yet practical.

Full motion video geopositioning algorithm integrated test bed

In order to better understand the issues associated with Full Motion Video (FMV) geopositioning and to develop corresponding strategies and algorithms, an integrated test bed is required. It is used to evaluate the performance of various candidate algorithms associated with registration of the video frames and subsequent geopositioning using the registered frames. Major issues include reliable error propagation or predicted solution accuracy, optimal vs. suboptimal vs. divergent solutions, robust processing in the presence of poor or non-existent a priori estimates of sensor metadata, difficulty in the measurement of tie points between adjacent frames, poor imaging geometry including small field-of-view and little vertical relief, and no control (points). The test bed modules must be integrated with appropriate data flows between them. The test bed must also ingest/generate real and simulated data and support evaluation of corresponding performance based on module-internal metrics as well as comparisons to real or simulated “ground truth”. Selection of the appropriate modules and algorithms must be both operator specifiable and specifiable as automatic. An FMV test bed has been developed and continues to be improved with the above characteristics. The paper describes its overall design as well as key underlying algorithms, including a recent update to “A matrix” generation, which allows for the computation of arbitrary inter-frame error cross-covariance matrices associated with Kalman filter (KF) registration in the presence of dynamic state vector definition, necessary for rigorous error propagation when the contents/definition of the KF state vector changes due to added/dropped tie points. Performance of a tested scenario is also presented.

Geostatistical modeling of uncertainty, simulation, and proposed applications in GIScience

Geostatistical modeling of spatial uncertainty has its roots in the mining, water and oil reservoir exploration communities, and has great potential for broader applications as proposed in this paper. This paper describes the underlying statistical models and their use in both the estimation of quantities of interest and the Monte-Carlo simulation of their uncertainty or errors, including their variance or expected magnitude and their spatial correlations or inter-relationships. These quantities can include 2D or 3D terrain locations, feature vertex locations, or any specified attributes whose statistical properties vary spatially. The simulation of spatial uncertainty or errors is a practical and powerful tool for understanding the effects of error propagation in complex systems. This paper describes various simulation techniques and trades-off their generality with complexity and speed. One technique recently proposed by the authors, Fast Sequential Simulation, has the ability to simulate tens of millions of errors with specifiable variance and spatial correlations in a few seconds on a lap-top computer. This ability allows for the timely evaluation of resultant output errors or the performance of a “down-stream” module or application. It also allows for near-real time evaluation when such a simulation capability is built into the application itself.