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Image Geometric Rectification 165

rectifying highly distorted images when a large contingent of GCPs
is available. For this reason it should not be the first choice if other
geometric models are applicable (Leica Geosystems, 2006) because
the output image may suffer discontinuity at the transit from the fau-
cet of one triangle to the next.
It is impossible to generalize which model is the best to use. The
answer to this question relies on many factors, the most important
being the image to be rectified. Special satellite images, such as Quick-
Bird, Landsat, and SPOT data, are best rectified with sensor-specific
models or using the polynomial coefficients provided by the data
supplier. If ancillary information about the image (e.g., RPC file) is
available, then specific models should be attempted first, supple-
mented with additional GCPs for higher accuracy. However, if the
residual of the rectification is very large, other models such as the
generic ones may be used with the addition of more GCPs if their
number is not sufficiently large yet.

5.5 Polynomial-Based Image Rectification
Of the various image transform models, the polynomial method is the
most flexible and versatile. It is the only generic model suitable for
rectifying all sorts of satellite imagery, and hence warrants an in-depth
discussion. Polynomial-based image rectification is implemented in
several steps and requires a varying number of GCPs at different accu-
racy levels. All of these issues are discussed in this section.

5.5.1 Transform Equations
In polynomial-based image rectification, Eqs. (5.9) and (5.10) are
rewritten specifically in the following polynomial form:
2 …
E = f (r, c) = a + a r + a c + a r + a rc + a c + (5.13)
2
1 0 1 2 3 4 5
N = f (r, c) = b + b r + b c + b r + b rc + b c + … (5.14)
2
2
2 0 1 2 3 4 5
where a and b (i = 0, 1, 2, ...) are the transformation coefficients. Poly-
i i
nomial equations as shown above do not recognize the internal rela-
tionship between (r, c) of a pixel and its (E, N). Instead, the two sets of
coordinates are linked up arbitrarily. The highest power of r, c, or their
combination in the above equations is known as the order of trans-
formation. It exerts a profound impact on the nature of image recti-
fication. In case of no order (e.g., r = 0 and c = 0), the rectification
degenerates into a simple shift in the origin of the image coordinate
system by a and b (see Fig. 5.10b). In the absence of shift (e.g., both a
0 0 0
and b are 0), the modifications involved are only scaling and rotation
0
in both the easting and northing directions in a first-order transfor-
mation (Fig. 5.10b, c, d). This order permits a linear transformation