International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
MIF x,y)1=( W'S TFG + | WEI) (5)
MIfQ xy) > T2} S
Which is the modulus of the wavelet transform at the scale
2' The maximum of the differences calculated at each pixel in
the x and y directions are squared and summed and their square
root is taken, which will be the modulus maxima (5). A
threshold procedure is applied on the wavelet transform
modulus image in order to eliminate non-significant feature
points. Then, a point (x ,y) is recorded only if (6) is valid.
T»? 7 a(o7 * uz), a is a constant whose initial value is defined
by the user and o» and p ? are the standard deviation and mean
of the wavelet transform modulus image at level 2
respectively. The parameter a controls the number of feature
points selected . Since the number of feature points increases in
the finer resolutions the parameter a is also increased in the
higher levels in order to select the most significant feature
points in the images. The matching pairs of control points are
identified using correlation. Reliable matches can be identified
through consistenéy check and RMSE verification, which are
used to determine a warping model that gives the best
registration of the LL sub bands to the precision available at
that level. Solving the warping model and determining the
unknown coefficients create new pixel locations. Bilinear
interpolation can assign gray values to these new locations. The
point matching and image warping steps can be performed at
progressively higher resolutions in a similar fashion to that
described. 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. The
implementation issues are discussed in the next section.
3.2 FFT TECHNIQUE
The FFT- based automatic registration algorithm relies on the
Fourier shift theorem (De Castro, 1987), if two images I: and b
differ only by a shift, (xo, yo), i.e. b (x, y)» hi [x- xo, y-yo], then
their Fourier transforms are related by
F (En) = e 1565 +m) F1(&;m) (7)
The ratio of two images Ll: and b is defined as
F1 ($3) * conj ( F($ ))
Abs (F1 ($m) * abs (F ($9)
Where conj is the complex conjugate, and abs is absolute value.
By taking the Inverse Fourier Transform of R, we see that the
resulting function is approximately zero every-where except for
a small neighborhood around a single point (B.S.Reddy,
1996) . This single point is where the absolute value of the
Inverse Fourier Transform of R attains its maximum value. It
can be shown that the location of this point is exactly the
displacement (xo, yo), needed to optimally register the images.
Then converting these images from rectangular coordinates (x,
y) to log-polar coordinates (log (r,9)) makes it possible to
represent both rotation and scaling as shifts. To transfer the
image from rectangular to log-polar coordinates (Young. D.
2000) the steps of the angle (Dtheta) and the logarithmic base
(b) are calculated. In order to attain high accuracy, we must
require that the polar plane have the same number of rows as
the rectangular plane. The implementation issues are discussed
in the next section.
3.3 MORPHOLOGICAL PYRAMID APPROACH
Mathematical morphology is a set-theoretic approach to image
analysis (Zhongxiu Hu , 2000). The morphological filters, such
as open and close, can be designed to preserve edges or shapes
of objects, while eliminating noise and details in an image.
Successive application of morphological filtering and sub
sampling (A. Morales, 1995) can construct the Morphological
Pyramid (MP) of an image:
L=[(l.oK) eK]|d L-0,2,...n (9)
Where K is a structuring element, d is a down sampling factor, o
and e are open and close filters. Thus the image at any level L
can be created. The spatial-mapping function and parameters in
MPIR are described by a global affine transformation. The
global affine transformation (Allieny .S 1986, L.G. Brown
1982), includes translation (tx, ty), rotation (0), scaling (sx, sy),
and shearing (shx, shy).
g1(X) = alx+a2y+a5S (10)
gl(y) = a3x+ady+a6 (11)
In addition to these errors this approach compensate for
brightness and contrast (P.Thevenaz, 1998) variation between
the images. The intensity-mapping function is defined as
@ = à7 g1+ ag (12)
g: and g»are the gray scale images. Levenberg - Marquardt
(LM) algorithm used to estimate the transformation parameters
iteratively. The LM nonlinear optimization algorithm is well
suited for performing registration based on least-squares
criterion. Combining the spatial mapping and the intensity
mapping functions, we achieve the complete relationship
between the two input images:
g 2 (r,c)7| a; gi(p.q) *as] * n(r,c) (13)
where n(r,c) is due to noise existing in both images, and the
eight transformation parameters ax , k = 1, 2, ... , 8 are
estimated using the intensity-based method for matching, since
registration methods based on initial intensity values can make
effective use of all data available. The parameters (al to a8) are
estimated by the procedure similar to (Y.Keller, 2002). The
implementation issues are discussed in the next section.
3.4 REGISTRATION USING GENETIC ALGORITHM
Highest similarity between the input and the reference images
indicates the proper registration of images. This similarity can
be achieved by properly identifying the correct transformation
procedure. Unlike traditional linear search, the GAs adaptively
explore the search solution space in a hyper - dimension
fashion (J.H. Holland, 1975, D.E. Goldburg, 1989), so that they
can improve computational efficiency.
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