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Article Excerpt
Introduction
Coordinate transformation is among the most frequently encountered operations in mapping, geodesy, surveying, engineering, photogrammetry, computer vision, and geographical information science (GIS). There are many examples of coordinate transformation applications. Sprinsky (1987) utilized an affine transformation to convert digitizer coordinates to map or world coordinates; Morad et al. (1996) transformed aerial photographs to match a digital file from the cadastre of New Zealand; Greenfeld (1997) converted geodetic data from state plane coordinates in the North American Datum (NAD) of 1927 to state plane coordinates in NAD 83; and Mikhail et al. (2001, p. 399) performed linear transformations as part of the inner orientation process in photogrammetry.
A coordinate transformation is the process of converting spatial data from a source coordinate system, such as image or map coordinates, to a target coordinate system (world or object coordinate system). A variety of transformation models have been described in the literature, including the polynomial, projective, affine, and similarity models (e.g., Wolf and Ghilani 1997, pp.335-356). These models have different properties and consist of different parameters that will, consequently, have different effects on the data being transformed. This paper focuses on linear coordinate transformations; and it describes in detail the most common methods, namely, the affine (i.e., six-parameter) and the similarity (i.e., four-parameter) transformations.
To estimate the transformation parameters, a set of control points, measured in both coordinate systems, is used. Generally, extra control points are measured and a least squares adjustment process is employed to compute the best parameters from the redundant data. The least squares (LS) adjustment procedure provides an accurate estimate of the parameters and a methodology for error estimation and analysis. This method estimates a vector of parameters [xi] from a linear model (y = A[xi] +e) that includes an observation vector y, a vector of normally distributed errors e, and a data matrix A. However, in this linear model, also known as the Gauss-Markov model, the data matrix A is considered fixed or error free. In other words, the basic assumption of the standard least squares adjustment approach is that all the errors are confined to the target coordinates (in the observation vector y).
Unfortunately, this assumption is frequently not true; various types of errors exist in almost any measured quantity. Errors due to faulty measuring instruments, human errors, modeling errors, and errors due to the sampling effect, all contribute to the fact that the source coordinates (in the data matrix A) include unknown errors as well.
In order to incorporate these errors, surveyors, cartographers, and GIS experts use the first-order Taylor series linearization followed by the Gauss-Helmert least squares method, a procedure known as the generalized least squares (Wolf and Ghilani 1997). Due to the first-order approximation of the Taylor series, however, the generalized least squares (GLS) method often converges to the standard least squares solution, and not to the true Error-In-Variables (EIV) solution. The peculiar behavior of the GLS method was investigated by Lenzmann and Lenzmann (2004a; b) and Kupferer (2005) in various examples. In this contribution, the GLS approach is not described mathematically, but the numerical example demonstrates that, indeed, the GLS converges to the standard LS solution.
The Total Least Squares (TLS) principle, as developed in Golub and Van Loan (1980), is an elegant method to treat problems where all the data are affected by random errors. The TLS idea can be traced back to the beginning of the century when the term "error-in-variables regression" came into use. Over the years, it was rediscovered many times, often independently; however, only in the last two decades did it start to be used in many practical applications (Van Huffel and Vandewalle 1991).
An additional complexity in computing the coordinate transformation parameters is the special structure of the data matrix A, where some elements appear twice. This special structure should be preserved and taken into account in the adjustment process. To circumvent this problem, Felus and Schaffrin (2005) utilized the Structured TLS (STLS) algorithm by Cadzow (1988). This iterative algorithm computes the unstructured TLS solution matrices and then enforces a structure by reconstructing these matrices using the matrix-property mapping function. The Cadzow algorithm, however, is only suboptimal with respect to the STLS criterion (cf., De Moor 1994). Schaffrin and Felus (2006) were able to avoid the structure in the case of affine transformations by expressing the mathematical model as a multivariate TLS problem. Nevertheless, their approach cannot treat similarity transformations.
In this paper, a novel application of the Structured Total Least-Norm (STLN) algorithm, as developed by Rosen et al. (1996), is applied to estimate the parameters of affine and similarity transformations while using the Error-In-Variables model and considering the special structure of the data matrix. The paper is organized as follows: The next section reviews the mathematical principles of the affine and similarity transformations. This is followed by a description of the Gauss-Markov model and the least squares solution. Next, the Error-in-Variables (EIV) model, and the Total Least Squares (TLS) and the Structured Total Least Squares (STLS) problems are defined. The Structured Total Least-Norm (STLN) problem is presented along with the STLN algorithm, and a numerical example is provided to demonstrate the advantages of the STLN approach over the LS approach for coordinate transformation applications. The paper...
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