# Geometric Rectification of Satellite Imagery with Minimal Ground Control Using Space Oblique Mercator Projection Theory

By Ren, Liucheng; Clarke, Keith C. et al. | Cartography and Geographic Information Science, October 2010 | Go to article overview

# Geometric Rectification of Satellite Imagery with Minimal Ground Control Using Space Oblique Mercator Projection Theory

Ren, Liucheng, Clarke, Keith C., Zhou, Chenghu, Ding, Lin, Li, Gongquan, Cartography and Geographic Information Science

Introduction

There are four principal motions of concern to satellite imaging: the earth's daily rotation about its axis, the satellite's orbital trajectory, the satellite scanner's imaging sweep across the earth surface, and the earth's orbital precession. The relationship between pixels in the raw image and the geometric position of the corresponding ground points is a function of time t due to all four of these motions. Geometric positional errors in remote sensing images are also a dynamic function of time t. To prevent these motions from distorting satellite images, Colvocoresses (1974) designed a dynamic model called the Space Oblique Mercator projection (SOM), a new map projection designed for use in Landsat mapping by the U.S. Geological Survey. The SOM assumes a cylindrical surface for the projection that oscillates along its axis at a compensatory rate that varies with latitude. Geometric error in the image caused by the four movements can then be rectified by this dynamic method. The SOM projection precisely simulates the geometric relationships within the satellite viewing geometry, and so establishes the formal mathematical transformations in the image and the projection as well as pixel and ground points directly, using GCPs in the region of interest.

Colvocoresses (1974) described the geometry of SOM projection by assuming a cylinder defined by a circular orbit, with the projection surface tangent to the ellipsoid of the earth. Several years later, Snyder published his derivation of the equations for the SOM (Snyder 1978; 1981). While the 1981 Bulletin defined the projection, a subsequent paper (Snyder 1982) introduced errors that were corrected in Snyder (1993), and the projection was included in the USGS "working manual" (Snyder 1987). Junkins and colleagues published an alternative mathematical formulation for SOM (Junkins, et al. 1977). During the next two decades, Chinese scholars continued research on the SOM projection. Yang improved the speed of computation for the algorithm of the SOM projection (Yang 1996). Cheng (1996) described the Conformal Space Projection (CSP). Ren and Zhu (2001) explored the application of the SOM projection in processing Landsat TM imagery. In later work, Ren conducted a review of the theory and methods of space projection (2003). During the 1980s, American cartographers applied the SOM projection to the conversion of imagery for cartographic purposes from the Landsat MSS and TM satellite instruments. A software utility called the General Cartographic Transformation Package (GCTP) was made available for the SOM projection by the USGS (Hessler 2004); it remains in use to this day.

[FIGURE 1 OMITTED]

Georectification methods for satellite imagery were reviewed by Novak (1992), and a framework comparing the different types of conversion was provided by Jensen (2005). Traditional techniques for precise geometric rectification of remote sensing images are the mapping polynomial method and the collinearity equations method. The polynomial method works without reference to the real geometric status of remote sensor instruments and is independent of the platform used for image data acquisition. It rectifies the distortion of images based on a set of GCPs. In order to ensure the precision of geometric rectification, a large number of GCPs should be selected, and they should be dispersed across the area of interest. While there are few guidelines as to how many GCPs are typically necessary (Campbell 2002), at a minimum, three times the six parameters to be estimated by affine fitting is normally assumed to be a minimum (18), and many more are desirable as higher-order polynomials are used. The collinearity equations method depends upon establishing mathematical relationships between the row and sample coordinates of pixels in an image, and the corresponding coordinates of those points on the ground. The attitude parameters of the satellite must be provided for collinearity equations, which treat the problem as a photogrammetric model. …

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