Page 211 - Digital Analysis of Remotely Sensed Imagery

Image Geometric Rectification 175

Just as with bilinear interpolation, the final interpolated result
of cubic convolution is not affected by the sequence of interpola-
tion. Namely, the results are the same regardless of whether the
interpolation is carried out horizontally first or vertically first. The
interpolation for the pixel in Fig. 5.13c based on cubic convolution
is illustrated below:

DN : 0.3{0.3[0.3(56 − 53 + 46 − 38) + (53 − 56 − 2 × 46 + 2 × 38)]
1
+ (53 − 38)} + 46 = 48.727
DN : 0.3{0.3[0.3(55 − 51 + 41 − 36) + (51 − 55.2 × 41 + 2 × 36)]
2
+ (51 − 36)} + 41 = 44.483
DN : 0.3{0.3[0.3(48 − 42 + 34 − 32) + (42 − 48 − 2 × 34 + 2 × 32)]
3
+ (42 − 32)} + 34 = 36.316
DN : 0.3{0.3[0.3(40 − 36 + 30 − 28) + (36 − 40 − 2 × 30 + 2 × 28)]
4
+ (36 − 28)} + 30 = 31.842
DN = 0.2{0.2[0.2(31.842 − 36.316 + 44.483 − 48.727)
+ (36.316 − 31.842 − 2 × 44.483 + 2 × 48.727)]
+ (36.316 − 48.727)} + 44.483 = 42.449536 = 42

Coincidentally, the cubic convoluted pixel value is identical to
that obtained using bilinear interpolation after the pixel value is
rounded down to the nearest integer. Compared to the other two
methods, cubic convolution is much more complex and computation-
ally intensive because the output value is estimated from more neigh-
boring pixels. This method may not necessarily lead to more accurate
interpolation, as demonstrated in the above example. It should be
used with caution.
The actual implementation of image rectification in an image
analysis system requires specification of a number of image projec-
tion parameters, such as geometric correction model, order of trans-
formation, map unit, spheroid, datum, and zone number. Not all of
these options are applicable to a particular projection, depending
upon the transform model selected and whether the transformation
is 2D or 3D. All the entered projection information is stored in a
single file, together with the rectified image data. The output image
may have a larger physical dimension (e.g., more rows and columns
than the original image) after rotation (Fig. 5.14), even though the
ground area covered remains unchanged. Pixels outside the initially
covered area are considered background. They are usually allocated
a value of zero (black) that may be ignored in all subsequent pro-
cessing steps.