New Effective Filter in the Spatial Domain for Speckle Noise Reduction
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2.1 Mathematical Representation of Chebyshev Polynomials
of the First Kind
Chebyshev polynomials of the first kind T n (x) can be determined using recurrence
relations [19, 20]:
T 0 (x) = 1
T 1 (x) = x
T n+1 (x) = 2xT n (x) − T n−1 (x)
(1)
or from the relation [19, 20]
T n (x) = cos(n arccos x).
(2)
The first kind Chebyshev polynomials are orthogonal with respect to scalar
product with weight 1/
√
1 − x 2 in the interval [−1, 1], i.e., the following relation is
valid [19–21]:
1
−1
T n (x)T m (x)
dx
√
1 − x 2
=
⎧
⎨
⎩
0, n = m
π, n = m = 0
π
2
, n = m = 0
(3)
Thus, any continuous function F(x) on the interval [− 1, 1] can be represented as
a sum of infinite elements [20]:
F(x) =
∞
n=0
f n T n (x),
(4)
where coefficients f n are determined as follows [20]:
f 0 =
1
π
1
−1
F(x)T 0 (x)
dx
√
1 − x 2
(5)
f n =0 =
2
π
1
−1
F(x)T n (x)
dx
√
1 − x 2
.
(6)
Figure 1a shows the graphical representation of the first five Chebyshev polynomials on the interval [−1,1]. Figure 1b shows the function F(x) = 2 + x + sin x(red
line) and its representation as sum of 11 elements using (4) (blue rings).
We see the quite good overlay of both curves in Fig. 1b.