IEEE Geoscience and Remote Sensing Magazine - December 2020 - 30
platform-induced distortions. Hence, a generic polynomial
model is applied to register images to one another. We rely
on a polynomial distortion model because these models
are widely used as approximating functions in data analysis
[18]. A bivariate polynomial model relates the coordinates
in the sensed image to those in the reference image. Let x
and y be the scan and pixel coordinates of the sensed image, respectively. Also, let x ref and y ref be the scan and pixel
coordinates of the reference image, respectively. The generic
polynomial model is given by [19]
M M-i
x= /
/ a ij x iref y refj ,
y= /
/ b ij x iref y refj ,
i=0 j=0
M M-i
(5)
i=0 j=0
where a ij and b ij are the coefficients for scan lines and pixels, respectively. The level of details in the data that can be
approximated depends directly on the degree of polynomials, i.e., M. In this article, we consider the convergence in the
mean-square error sense, and therefore we measured rootmean-square error (RMSE) [16] and compared up to the
fourth degree of polynomials. It is found that, from the GEO
platform, the terrain variations of the Earth do not appear to
be significant, and a second degree of polynomial approximation is sufficient to model the image data. Hence, a bivariate
quadratic polynomial is used in this implementation.
The quadratic polynomial equation is (6) below, where
the coefficients of the quadratic polynomial contribute to
total warp, i.e., shift, scale, shear, dependent scale, nonlinear
scale, and nonlinear shear in x and y. As the distortions in
image data from satellites are random, we need to use the
least-squares fit method [20] to fit the surface coordinates
to warp. The matching pairs of GCPs have been established.
Using these GCPs, the least-squares surface fit for (6) determines the unknown coefficients. This will lead to six pairs
of equations. Now, the problem of least squares is to find
the linear solution for the given scan and pixel by fitting the
match pairs to (6). The method of least squares says that
the best representative surface is that for which the sum
of the squares of the residuals is a minimum. Let m be the
numbers of match pairs established. The sum of squares of
the residuals is shown in (7) below, where f(x) and f(y) are
the minimum with respect to a and b, respectively.
The solution to the set of linear equations is obtained by the
Gauss elimination method [21]. This estimated transformation
x = a 00 + a 10 x ref + a 01 y ref +
7 7
7
shift
scale
shear
y = b 00 + b 10 x ref + b 01 y ref +
7 7
7
shift
shear
scale
a 11
7
function is now used to define a mapping from the reference
image coordinates to the sensed image coordinates:
The mapping functions constructed are used to transform
the sensed image and thus to register the image. The function
is implemented by stepping through the integer coordinates
of the reference image one by one and calculating the transformed coordinate values. Since the transformed coordinates
are, in general, not integers, a new pixel must be estimated between existing pixels by an appropriate interpolation process
known as resampling. Basically, this is the interpolation technique of generating an image on a system of coordinates and
taking the input image from a different set of coordinates. The
corrected output image is then created by filling each location
of a new, initially empty array with the pixel calculated by the
transformed coordinates in the sensed image.
There are various resampling (interpolation) methods
available for digital imagery [22]. Selection of an appropriate resampling technique can be treated as an applicationdependent criterion. Nearest neighbor (NN) resampling is
used for this application software. It is found that the other
techniques put more computational load on the program
and consequently slow down the registration process in addition to tampering with actual brightness values in the sensed
data. However, NN resampling introduces blocking artifacts
in the processed image because of its round-off property, i.e.,
geometrical discontinuities on the order of ! 1 2 pixels.
Nevertheless, NN interpolation replaces a pixel value with
its nearest (pixel) value within the coordinate neighborhood
of an image. Therefore, the NN interpolation does not introduce any newer pixel values in the resultant resampled image. Hence, the statistical distribution of the image remains
unaltered, so the processed image can be conveniently used
for generating data products like OLR, QPE, UTH, AMV, and
NDVI, for further analysis and predictions where radiometric
accuracy is also of importance. This is required in the present
case with meteorological applications.
A FLOWCHART OF SOFTWARE IMPLEMENTATION
Finally, we present a simplified flowchart of the complete
software implementation in Figure 6. It shows the data flow
diagram for this operational software. As shown in the flowchart, the sensed images are first read into 2D arrays. Match
x ref y ref +
y - dependent scale
b 11
7
x - dependent scale
^ x, y h = f ^ x ref , y ref h , 6x ref , y ref .(8)
a 20
7
x 2ref +
nonlinear scale
x ref y ref +
b 20
7
nonlinear shear
a 02
7
y 2ref ,
nonlinear shear
x 2ref +
b 02
7
y 2ref
(6)
nonlinear scale
f (x) = / 6x i - ^a 00 + a 10 x ref + a 01 y ref + a 11 x ref y ref + a 20 x 2ref + a 02 y 2ref h@ ,
2
30
m
2
2
h@
f (y) = / 6y j - ^ b 00 + b 10 x ref + b 01 y ref + b 11 x ref y ref + b 20 x ref
+ b 02 y ref
2
(7)
m
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE
DECEMBER 2020
IEEE Geoscience and Remote Sensing Magazine - December 2020
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