Image Processing Reference
In-Depth Information
Let us consider that each pixel in the image
I
x, y
is corrupted by additive Gaussian noise.
The noise has a mean value of zero and the (unknown) standard deviation is σ . Thus the
probability that a point in the template placed at coordinates (
i, j
) matches the corresponding
pixel at position (
x
,
y
) ∈
W
is given by the normal distribution
2
I
-
T
-
1
2
xiy j
+, +
xy
,
1
2
pxy
(,
) =
e
(5.1)
ij
,
π σ
Since noise affecting each pixel is independent, then the probability that the template is at
the position (
i
,
j
) is the combined probability of each pixel that the template covers. That is,
Π
∈
W
L
=
p
(
x y
, )
(5.2)
ij
,
ij
,
(,)
xy
By substitution of Equation 5.1, we have that
2
I
-
T
n
xiy+j
+,
x,
y
-
1
2
1
2
(,)W
xy
(5.3)
L
=
e
ij
,
where
n
is the number of pixels in the template. This function is called the
likelihood
function. Generally, it is expressed in logarithmic form to simplify the analysis. Notice that
the logarithm scales the function, but it does not change the position of the maximum.
Thus, by taking the logarithm the likelihood function is redefined as
2
I
-
T
1
2
-
1
2
xiy j
+, +
xy
,
(5.4)
ln (
L
) = ln
n
ij
,
(,)
xy
W
In
maximum likelihood estimation
, we have to choose the parameter that maximises the
likelihood function. That is, the positions that minimise the rate of change of the objective
function
ln(
L
i
)
= 0 and
ln (
L
j
)
= 0
ij
,
ij
,
(5.5)
That is,
I
xiy j
i
+, +
W
I
(
-
T
)
= 0
xiy j
+, +
xy
,
(
xy
,)
(5.6)
I
xiy j
j
+, +
W
I
(
-
T
)
= 0
xiy j
+, +
xy
,
∂
We can observe that these equations are also the solution of the minimisation problem
given by
(
xy
,)
Σ
∈
W
2
min =
e
(
I
-
T
)
(5.7)
xiy j
+, +
xy
,
(,)
xy
That is, maximum likelihood estimation is equivalent to choosing the template position
that minimises the squared error (the squared values of the differences between the template
points and the corresponding image points). The position where the template best matches
the image is the estimated position of the template within the image. Thus, if you measure
