Image Processing Reference
In-Depth Information
Naturally, this correct length is independent of y
1
and z
1
. Now, the con-
tinuous line segments leading to a particular tuple t are all found within the
region Domain(t). As a measure of the error for the length estimator L(t), we
use a quantity dubbed RDEV (L(t),n) from [79]. It is the root mean square
difference between the original length and the estimated length, averaged over
all strings of n elements and divided by n. The last division is performed to
render the measure scale invariant and assumes a sampling density of n
3
points
per unit cube.
RDEV (L(t),n) =
(1/n)
(L(t)−f(n,φ,θ)
2
p(φ,y
1
,θ,z
1
)dθdz
1
dφdy
1
∀t∈Domain(t)
where p(φ,y
1
,θ,z
1
) is the probability density function of the lines in
(φ,y
1
,θ,z
1
) space. We assume that y
1
and z
1
are uniformly distributed in
the range [0,1), and θ and φ are uniformly distributed in the range [0,π/4)
and [0,tan
−1
(sinφ)), respectively. The ranges are selected in accordance with
our discussion in the previous sections. Hence we get,
−1
sin(φ)−0))
p(θ,φ,y
1
,z
1
) = (1/(π/4−0))∗(1/(tan
−1
(sinφ)).
= 4/(π∗tan
For notational convenience, I(L(t);e,f;g,h;a,b;c,d) denotes the following
integration:
b
d
f
h
(L(t)−nsecφsecθ)
2
.(4/π)/tan
1
(sinφ))dz
1
dθdy
1
dφ.
a
c
e
g
3.3.3.1 (n)-characterization [39]
A simple estimator for the n-characterization has the form L(n) = αn. For
the choice of α to be 1 we have L(n) = n. Now, the RDEV for this estimator
can be found:
−1
(sinφ); 0,1; 0,π/4; 0,1).
X = I(n; 0,tan
√
RDEV (L(n),n) =
X/n = 21.77%.
Of course, this RDEV is true asymptotically.
To compute the unbiased linear estimator using this characterization, we
have to find an α that minimizes the RDEV. Thus, we have to minimize
RDEV (αn,n) with respect to α. Setting d(RDEV )/dα to 0 and solving for
α, we get optimum α = 1.1307 and the corresponding error becomes 15.26%.
