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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
were identified to cover, one from each class.
Wavelet-Modulus Maxima method (Leila M. Fonesca, 1997),
uses image pixel values, similar to that described in (Q. Zheng,
1993) except determining the feature selection. The probable
control points are detected from the local modulus maxima of
the wavelet transform, applied to the input and reference
images, after performing the wavelet decomposition up to two
levels. The correlation coefficient is used as a similarity
measure and only the best pair-wise fitting, among all pairs of
feature points, are taken as actual control points. A polynomial
transform, which can take care of translation and rotational
errors, is then used to model the deformation between the
images and their parameters are estimated in a coarse to fine
manner. The refinement matching is achieved using the warped
image and the set of feature points detected in the reference
image. After processing all levels the final parameters are
determined and used to warp the original input image.
Fast Fourier Transform (FFT) technique (B.S.Reddy, 1996,
Y.Keller, 2002) is a frequency domain approach in which it
does not use any control points, instead the FFT ratio is
computed. The displacement between two given images can be
determined by computing the ratio Fl.conj(F2)/|F1.F2|, the
inverse of this ratio results as an impulse like function. This is
approximately zero everywhere except at the displacement, this
determines the translation error between the images. Converting
these images from rectangular coordinates to log-polar
coordinates and by calculating the similar ratio, we can
represent rotation and scaling errors also as shifts. These three
parameters are used to establish the mathematical model and
the image is geometrically rectified with respect to the
reference image.
Morphological Pyramid Image Registration algorithm
(Zhongxiu HU, 2000) uses the low level shape features to
determine the global affine transformation model along with the
radiometric changes between the images. The multi resolution
images are represented by a Morphological Pyramid (MP), as
the MP’s have the capability to eliminate details and to
maintain shape features. The Levenberg Marquardt non-linear
optimization algorithm is employed to estimate the matching
parameters of translation, rotation and scaling errors up to sub
pixel accuracy. In this approach intensity mapping function is
integrated into geometric mapping function.
Image Registration using Genetic Algorithms (GAs) (J.H.
Holland, 1975) uses the comparison of identified solutions to
ensure a populations survival under changing environmental
conditions. GAs are iterative procedures that maintain a
population of candidate solutions encoded in the form of
chromosome strings. The initial population can be generated
randomly. These candidates will be selected using a selection
criterion for the reproduction in the next generation based on
their fitness values. GAs search is used to efficiently explore
the huge solution space required by the image registration to a
sub pixel accuracy.
3. METHODOLOGY
Mathematical modeling techniques are used to correct the
geometric errors like translation, rotation and scaling of the
input image to that of the reference image. Let the image to be
warped be called the input image and to which it is reduced is
called the reference image. There are two cases to consider for
the image registration algorithms:
a) The images have the same ground resolution (pixel size)
b) The images are taken from different sensors and have
different ground resolutions. as
Each of the above algorithms models the same deformation In
its own way. The input image needs to be interpolated while
warping. The simplest scheme for gray-level interpolation is
based on the nearest neighbor approach called zero-order
interpolation. But the nearest neighbor interpolation yields
undesirable artifacts such as stair-stepped effect around
diagonal lines and the curves. Bilinear interpolation produces
the output images that are smoother and without the stair
stepped effect. It’s a reasonable compromise between
smoothness and computational cost.
‘3.1 Wavelet-Modulus Maxima method
As the wavelet approach (Leila M. Fonesca, 1997), assumes
that the images have the same ground resolution, so the image
with the highest resolution is reduced to the lower resolution.
After reducing the images to the same spatial resolution,
compute the discrete multi-resolution wavelet transform (L
levels). This helps in decomposing the signal into the coarser
resolution, which consists of the low frequency approximation
information and the high frequency detail information called
sub bands. During the decomposition, the resolution decreases
exponentially at the base of 2. For generating the sub bands the
algorithm proposed in (S.G.Mallat 1989), is used for its
computational efficiency. In sub band coding, an image is
decomposed into a set of band-limited components, called sub
bands, which can be reassembled to reconstruct the original
image without error. We call LL, LH, HL, HH the four images
created at each level of decomposition. When the
decomposition level j decreases, the resolution decreases in the
spatial domain and increases in the frequency domain. The next
phase aims to identify features that are present in both images
in each level of decomposition. The modulus maxima of the
wavelet transform is used to detect sharp variation points,
which correspond to edge points in the images.Let us call a
smoothing function dx, y), the impulse response of a 2-D low-
pass filter. The first order derivative of ®(x,y) decomposed in
two components along the x and y directions , respectively, are
0 D (x, y)
WON) Tee (1)
ox
0 o (x, y)
y(x,y) — (2)
Jy
and these functions can be used as wavelets. For calculating the
partial derivatives, the difference between each pixel and its
adjacent pixel is calculated. This difference is calculated both
in the x and y directions, separately. For any function f, the
wavelet transform at scale a=2' defined with respect to these
two wavelets has two components.
WAY) -f*y5 Gy» -
f * (2!0/0x e)! (x,y))
2i0/üx(f * «3)(x,y) (3)
f * y^ (xy)
f * (2i0/0y 4j (x,y))
= 2i0/0y(f *))(x.y) (4)
Il
Il
W'j [fGy)]
ll
Therefore, these two components of the wavelet transform are
proportional to the coordinates of the gradient vector of f(x,y)
smoothened by ®:'(x,y). They characterize the singularities
along x and y directions, respectively.
