Page 201 - Digital Analysis of Remotely Sensed Imagery

166 Cha pte r F i v e

between (r, c) and (E, N). The original image can only be rotated and
shifted. There is no change in the relative distance and position
between pixels in the input image and the output image. The shape
of the output image can be rectangular or skewed, depending upon
the value of the transformation coefficients. However, the two oppo-
site sides are still parallel to each other. By comparison, a second-
order transformation enables more nonlinear geometric distortions to
be removed. In addition to rotation and shift in the origin, it is also
possible to remove local scaling (e.g., stretching and compression).
Thus, the border of the output image no longer follows a straight line,
as is the case with the first order (Fig. 5.10, e). Theoretically, Eqs. (5.13)
and (5.14) can continue indefinitely with many more terms. However,
past experience has demonstrated that after the second order as
shown above, the amount of work and computation involved
increases exponentially while the accuracy of E and N hardly
improves. Therefore, the second-order transformation appears to
have the optimal balance between accuracy and complexity.
2
2
Since the terms r, c, r , rc, and c are common to both Eqs. (5.13)
and (5.14), they are rewritten more concisely in the matrix format
below:
⎛ 1⎞
⎜ r r ⎟
⎛ E⎞ a ⎛ a a a a a ⎞ ⎜ ⎟
c
⎜ ⎟ = ⎜ 0 1 2 3 4 5 ⎟ ⋅ ⎜ ⎟ (5.15)
N⎠
⎝
2
r
b
5
1 b 2 b 3 b 4 b ⎠ ⎜ ⎟
b ⎝ 0
⎜ rc ⎟
⎜ c ⎝ ⎠
2 ⎟

T
2
In the above equation, matrices (E N) and (1 r c r rc c ) are both
2 T
known for GCPs. The only unknown term in the equation is the
a ⎛ a a a a a ⎞
coefficient matrix ⎜ 0 1 2 3 4 5 ⎟ . This can be determined
b ⎝
0 b 1 b 2 b 3 b 4 b ⎠
5
through inversion of Eq. (5.15), or
⎛ 1⎞ − 1
⎜ r r ⎟
a ⎛ a a a a a ⎞ ⎛ E⎞ ⎜ ⎟
c
5
⎜ b ⎝ 0 1 2 3 4 b ⎠ ⎟ = ⎜ ⎟ ⋅ ⎜ ⎟ (5.16)
N⎠ ⎜ ⎟
⎝
2
0 b 1 b 2 b 3 b 4 5 r
⎜ rc ⎟
2 ⎟
⎜ c ⎝ ⎠

Solution of the above equation relies on GCPs. Once these coef-
ficients are determined, Eqs. (5.13) and (5.14) are then applied to
the entire image. After image transformation, all pixels in the out-
put image are expressed in ground coordinates instead of image
coordinates.