matching model. For the target ,
y x
g and search
, s
l q
images, the line and sample coordinates pixel are expressed as
y a
x a
a l
2 1
and
y b
x b
b s
2 1
, with the
, ,
1
a a
1 2
, ,
b b
a
and
2
b
symbols expressing affinity. Differencing pixel values may lead to a gray-level function as
. ,
,
2 1
2 1
1
y x
g y
b x
b b
y a
x a
a q
h h
v
i i
i
Inde x i varies for n pixels in a window. Symbol
i
v
denotes a zero- mean residual error having the Gaussian distribution, or
,
2 i
N
with the
i
symbol meaning the standard deviation;
h
and
1
h
linearly modify pixel values. Linear expansion at approximate unknowns results in a system
of error equations, defined as
l Ax
v
with
Q
2
.
By referring to Mikhail 1976, one obtains the least-squares
solution of unknown parameters as
l Q
A Q
x
1
T x
2 where apart from scaling, the covariance matrix,
1 1
A
Q A
Q
T x
, results from error propagation. The solution for measurement residuals is given by the
Ax l
term, resulting in
l Q
Q v
1
v
, where
Q
Q
v T
x
A AQ
is the corresponding covariance matrix.
A variance-component estimator is eventually formed by Wu et al. 2009 as
tr
1 1
1 1
2
Q
C Q
Q v
Q C
Q v
i v
i T
i
for
} ,
, 2
, 1
{ m
i
3
The simplified BIQUE Best Invariant Quadratic Unbiased Estimator by Crocetto et al. 2000 is an alternative method.
Intuitively, adaptive weights in terms of the inverse of
i i
m i
C
2 1
, should be better than weights in
1
Q
that remain unchanged during iterative processing.
2.3 Reliability of the Matched Points
As implied, RANSAC Random Sample Consensus distinguishes inliers from outliers by requiring that any n-
member subgroup of inlying data points leads to only one set of model parameters. When the w symbol stands for the
probability of a data point consistent with an affine transformation model and the z symbol for the probability of
selecting an error-free n-member set, Fischler and Bolles 1981 have shown that the number k of attempts for the removal of
outliers can be computed by equating the
k n
w 1
term with the
z
1
term. The solution for k results in
1 log
1 log
n
w z
k
4 According to Schwarz and Kok 1993, the
i
v
residual normalized by the standard deviation of
i
v
has a tau distribution, which is related to Student’s t distribution by
2 1
1
1
u n
u n
u n
t u
n t
u n
5
where
u n
means the degree of freedom. The normalized residuals are treated in hypothesis testing as a test statistic,
gauged by a threshold at a 5 percent significance level. The blunder is removed one at a time so that the algorithm is termed
IDS Iterated Data Snooping. As calculation of the covariance matrix for a large number of
data residuals can grow burdensome, the RANSAC algorithm is usually conducted first. For the remaining data points, IDS
could be invoked to ensure that indeed they are regular samples of an experiment at hand. Because of the difference in theory,
RANSAC and IDS are expected to be complementary. 2.4
Thin-plate Spline Interpolation
TPS Thin-Plate Splines stands for a flexible function in that it emulates the minimized bending energy of a metal plate on
multiple tie-point constraints. A trend surface stems from a global, affine transformation between two overlapping images.
If the x and
y
symbols denote the transformed line and sample coordinates pixel, coordinate discrepancies at m
conjugate points can be expressed as
,
1 1
x x
T
x
, ,
2 2
m m
x x
x x
and
T
y
, ,
2 2
1 1
y y
y y
,
m m
y y
, respectively. For an interpolation point, TPS actually involves both trend and
discrepancy values, conforming to the concept of a remove-and- restore operation Darbeheshti and Featherstone, 2009.
In determining weights, a special matrix is defined as
2 1
2 21
1 12
m m
m m
r K
r K
r K
r K
r K
r K
K
6 where with
jk
r
meaning a Euclidean distance between points j and k, both
} ,
2, {1,
m
,
r K
is the so-called fundamental solution of the biharmonic equation and takes on the form
2 2
log r
r r
K
. An interim vector consisting of weight coefficients is computed to be
x K
w
1
x
and y
K w
1
y
, in the respective line and sample directions.
At a place i other than the locations of the m tie points, a vector exists,
, ,
,
2 1
im i
i T
i
r K
r K
r K
k
. In association with the
x
w
and
y
w
weights, the
i
k
vector is employed for an interpolation in a field of coordinate discrepancies, as follows
Du et al., 2008
i T
i T
x i
x k
K x
k w
1
i T
i T
y i
y
k K
y k
w
1
7 The corresponding values from the trend function have to be
added to those of Equation 7, or
i i
x x
and
i i
y y
, in order to achieve the TPS-based warping and resampling for the
purpose of image registration. Later, the difference in performance between an affine transform and the thin-plate
spline warping methods will be assessed.
3. MULTISOURCE IMAGE TESTS
