Image Processing Reference
In-Depth Information
the chain code C is not typically used to estimate the properties of the pre-
digitized CSLS. Most of the digitized image (in this case the chain code C)
is characterized by a fixed number of parameters, also called a tuple t. For
example, we discuss in Section 3.1.2 that (n,q,p,s) provides a 4-tuple charac-
terization given any chain code C of a DSLS. The length estimator is expressed
in terms of t, say g(t). Thus, the length of all lines whose chain code could
have generated a tuple t would be estimated by the same g(t). Let the length
of a line l be f(l) and let l be digitized to generate a chain code C that is char-
acterized by a tuple t. It would be important to minimize the error between
f(l) and g(t) for all lines l whose chain code after characterization becomes t.
Two types of estimators are considered here.
The best linear unbiased estimator (BLUE) [79] minimizes the mean square
error (MSE) between f(l) and g(t) over all l, which are in Domain(t), the
domain of t. This is equivalent to
g
BLUE
(t) =
f(l)p(l)dl
Domain(t)
where p(l) is the probability density of lines.
The most probable estimator of the length is the most probable original
value of length given t is obtained as
g
MPO
(t) = f(argmax{p(l)|l ∈Domain(t)})
where argmax{p(l)} indicates the value of l maximizing p(l) in that range.
The estimators that are provided in this section are found in [76, 79].
Before we discuss the estimators for different characterizations of the chain
code, let us formulate the length of a line segment and the probability density
function of random lines in the 2-D plane.
The length of a line with m slope, and c intercept between x = 0 and
x = n is given by
f(c,m,n) = n
1 + m
2
.
The probability density function of random lines in the 2-D plane is taken as
uniform in polar parameters and is given by
√
2(1 + m
2
)
−3/2
.
p(c,m,n) =
As a measure of the accuracy of the length estimator g we use RDEV (g,n),
which is defined by
RDEV (g(t),n) = (1/n)
(g(t)−f(l))
2
p(l)dl.
Domain(t)
t
