Jiang-Hong Ma

A general framework for error analysis in measurement-based GIS Part 1: The basic measurement-error model and related concepts

Abstract.This is the first of a four-part series of papers which proposes a general framework for error analysis in measurement-based geographical information systems (MBGIS). The purpose of the series is to investigate the fundamental issues involved in measurement error (ME) analysis in MBGIS, and to provide a unified and effective treatment of errors and their propagation in various interrelated GIS and spatial operations. Part 1 deals with the formulation of the basic ME model together with the law of error propagation. Part 2 investigates the classic point-in-polygon problem under ME. Continuing to Part 3 is the analysis of ME in intersections and polygon overlays. In Part 4, error analyses in length and area measurements are made. In this present part, a simple but general model for ME in MBGIS is introduced. An approximate law of error propagation is then formulated. A simple, unified, and effective treatment of error bands for a line segment is made under the name of “covariance-based error band”. A new concept, called “maximal allowable limit”, which guarantees invariance in topology or geometric-property of a polygon under ME is also advanced. To handle errors in indirect measurements, a geodetic model for MBGIS is proposed and its error propagation problem is studied on the basis of the basic ME model as well as the approximate law of error propagation. Simulation experiments all substantiate the effectiveness of the proposed theoretical construct.

An integrated information fusion approach based on the theory of evidence and group decision-making

Dempster-Shafer theory of evidence has been employed as a major method for reasoning with multiple evidence. The Dempsters rule of combination is however incapable of managing highly conflicting evidence coming from different information sources at the normalization step. Extending current rules, we incorporate the ideas of group decision-making into the theory of evidence and propose an integrated approach to automatically identify and discount unreliable evidence. An adaptive robust combination rule that incorporates the information contained in the consistent focal elements is then constructed to combine such evidence. This rule adjusts the weights of the conjunctive and disjunctive rules according to a function of the consistency of focal elements. The theoretical arguments are supported by numerical experiments. Compared to existing combination rules, the proposed approach can obtain a reasonable and reliable decision, as well as the level of uncertainty about it.

A general framework for error analysis in measurement-based GIS. Part 4: Error analysis in length and area measurements

Abstract.This is the final of a series of four papers on the development of a general framework for error analysis in measurement-based geographic information systems (MBGIS). In this paper, we discuss the error analysis problems in length and area measurements under measurement error (ME) of the defining points. In line with the basic ME model constructed in Part 1 of this series, we formulate the ME models for length and area measurements. For length measurement and perimeter measurement, the approximate laws of error propagation are derived. For area measurement, the exact laws of error propagation are obtained under various conditions. An important result is that area measurement is distributed as a linear combination of independent non-central chi-square variables when the joint ME vectors of vertices coordinates are normal. In addition, we also give a necessary and sufficient condition under which the area measurement estimator is unbiased. As a comparison, the approximate law of error propagation in area measurement is also considered and its approximation is substantiated by numerical experiments.

A general framework for error analysis in measurement-based GIS Part 3: Error analysis in intersections and overlays

Abstract.This is the third of a four-part series on the development of a general framework for error analysis in measurement-based geographic information systems (MBGIS). In this paper, we study the characteristics of error structures in intersections and polygon overlays. When locations of the endpoints of two line segments are in error, we analyze errors of the intersection point and obtain its error covariance matrix through the propagation of the error covariance matrices of the endpoints. An approximate law of error propagation for the intersection point is formulated within the MBGIS framework. From simulation experiments, it appears that both the relative positioning of two line segments and the error characteristics of the endpoints can affect the error characteristics of the intersection. Nevertheless, the approximate law of error propagation captures nicely the error characteristics under various situations. Based on the derived results, error analysis in polygon-on-polygon overlay operation is also performed. The relationship between the error covariance matrices of the original polygons and the overlaid polygons is approximately established.

A general framework for error analysis in measurement-based GIS. Part 2: The algebra-based probability model for point-in-polygon analysis

Abstract.This is the second paper of a four-part series of papers on the development of a general framework for error analysis in measurement-based geographic information systems (MBGIS). In this paper, we discuss the problem of point-in-polygon analysis under randomness, i.e., with random measurement error (ME). It is well known that overlay is one of the most important operations in GIS, and point-in-polygon analysis is a basic class of overlay and query problems. Though it is a classic problem, it has, however, not been addressed appropriately. With ME in the location of the vertices of a polygon, the resulting random polygons may undergo complex changes, so that the point-in-polygon problem may become theoretically and practically ill-defined. That is, there is a possibility that we cannot answer whether a random point is inside a random polygon if the polygon is not simple and cannot form a region. For the point-in-triangle problem, however, such a case need not be considered since any triangle always forms an interior or region. To formulate the general point-in-polygon problem in a suitable way, a conditional probability mechanism is first introduced in order to accurately characterize the nature of the problem and establish the basis for further analysis. For the point-in-triangle problem, four quadratic forms in the joint coordinate vectors of a point and the vertices of the triangle are constructed. The probability model for the point-in-triangle problem is then established by the identification of signs of these quadratic form variables. Our basic idea for solving a general point-in-polygon (concave or convex) problem is to convert it into several point-in-triangle problems under a certain condition. By solving each point-in-triangle problem and summing the solutions, the probability model for a general point-in-polygon analysis is constructed. The simplicity of the algebra-based approach is that from using these quadratic forms, we can circumvent the complex geometrical relations between a random point and a random polygon (convex or concave) that one has to deal with in any geometric method when probability is computed. The theoretical arguments are substantiated by simulation experiments.

A WTLS-Based Method for Remote Sensing Imagery Registration

This paper introduces a weighted total least squares (WTLS)-based estimator into image registration to deal with the coordinates of control points (CPs) that are of unequal accuracy. The performance of the estimator is investigated by means of simulation experiments using different coordinate errors. Comparisons with ordinary least squares (LS), total LS (TLS), scaled TLS, and weighted LS estimators are made. A novel adaptive weight determination scheme is applied to experiments with remotely sensed images. These illustrate the practicability and effectiveness of the proposed registration method by collecting CPs with different-sized errors from multiple reference images with different spatial resolutions. This paper concludes that the WTLS-based iteratively reweighted TLS method achieves a more robust estimation of model parameters and higher registration accuracy if heteroscedastic errors occur in both the coordinates of reference CPs and target CPs.

High-order Taylor series expansion methods for error propagation in geographic information systems

AbstractThe quality of modeling results in GIS operations depends on how well we can track error propagating from inputs to outputs. Monte Carlo simulation, moment design and Taylor series expansion have been employed to study error propagation over the years. Among them, first-order Taylor series expansion is popular because error propagation can be analytically studied. Because most operations in GIS are nonlinear, first-order Taylor series expansion generally cannot meet practical needs, and higher-order approximation is thus necessary. In this paper, we employ Taylor series expansion methods of different orders to investigate error propagation when the random error vectors are normally and independently or dependently distributed. We also extend these methods to situations involving multi-dimensional output vectors. We employ these methods to examine length measurement of linear segments, perimeter of polygons and intersections of two line segments basic in GIS operations. Simulation experiments indicate that the fifth-order Taylor series expansion method is most accurate compared with the first-order and third-order method. Compared with the third-order expansion; however, it can only slightly improve the accuracy, but on the expense of substantially increasing the number of partial derivatives that need to be calculated. Striking a balance between accuracy and complexity, the third-order Taylor series expansion method appears to be a more appropriate choice for practical applications.

An elliptical basis function network for classification of remote-sensing images

An elliptical basis function (EBF) network is proposed in this study for the classification of remotely sensed images. Though similar in structure, the EBF network differs from the well-known radial basis function (RBF) network by incorporating full covariance matrices and uses the expectation-maximization (EM) algorithm to estimate the basis functions. Since remotely sensed data often take on mixture-density distributions in the feature space, the proposed network not only possesses the advantage of the RBF mechanism but also utilizes the EM algorithm to compute the maximum likelihood estimates of the mean vectors and covariance matrices of a Gaussian mixture distribution in the training phase. Experimental results show that the EM-based EBF network is faster in training, more accurate, and simpler in structure.

Scaled total-least-squares-based registration for optical remote sensing imagery

In optical image registration, the reference control points (RCPs) used as explanatory variables in the polynomial regression model are generally assumed to be error free. However, this most frequently used assumption is often invalid in practice because RCPs always contain errors. In this situation, the extensively applied estimator, the ordinary least squares (LS) estimator, is biased and incapable of handling the errors in RCPs. Therefore, it is necessary to develop new feasible methods to address such a problem. This paper discusses the scaled total least squares (STLS) estimator, which is a generalization of the LS estimator in optical remote sensing image registration. The basic principle and the computational method of the STLS estimator and the relationship among the LS, total least squares (TLS) and STLS estimators are presented. Simulation experiments and real remotely sensed image experiments are carried out to compare LS and STLS approaches and systematically analyze the effect of the number and accuracy of RCPs on the performances in registration. The results show that the STLS estimator is more effective in estimating the model parameters than the LS estimator. Using this estimator based on the error-in-variables model, more accurate registration results can be obtained. Furthermore, the STLS estimator has superior overall performance in the estimation and correction of measurement errors in RCPs, which is beneficial to the study of error propagation in remote sensing data. The larger the RCP number and error, the more obvious are these advantages of the presented estimator.

A highly robust estimator for regression models

It is well known that classical robust estimators tolerate only less than fifty percent of outliers. However, situations with more than fifty percent of outliers often occur in practice. The efficient identification of objects from a noisier background is thus a difficult problem. In this paper, a highly robust estimator is formulated to tackle such a difficulty. The proposed estimator is called the regression density decomposition (RDD) estimator. The computational analysis of the estimator and its properties are discussed and a simulated annealing algorithm is proposed for its implementation. It is demonstrated that the RDD estimator can resist a very large proportion of noisy data, even more than fifty percent. It is successfully applied to some simulated and real-life noisy data sets. It appears that the estimator can solve efficiently and effectively general regression problems, pattern recognition, computer vision and data mining problems.