4 KRAWTCHOUK POLYNOMIAL

AND MOMENTS

4.1 Krawtchouk Polynomial

(cid:18)

(cid:16)
ak,n,pxk(cid:17)

The n-th order of Krawtchouk polynomial is based on
the hypergeometric function and is deﬁned as:

N

Kn(x; p,N) =

∑

1
p
(2)
where x,n = 0,1,2, ...,N et N > 0, p ∈ (0,1) and

−n,−x;−N;

= 2F1

k=0

the hypergeometric function deﬁned as:

(cid:18) (a)k(b)kzk

(c)k

(cid:19)

zk
k!

∞
∑

k=0

2F1(a,b;c;z) =

(cid:19)

(3)

(4)

and satisﬁes the orthogonality condition :

N

∑

k=0

w(x; p,N)Kn(x; p,N)Km(x; p,N) = ρ(n; p,N)δnm
(6)
and δnm

where ρ(n; p,N) = (−1)n(cid:16) 1−p

(cid:17)n

p

n!
(−N) n

is the Kronecher function.
In order to eliminate the large variability in the
dynamic range, a normalization process is applied.
Then, the set of normalized (weighted) Krawtchouk
polynomials is deﬁned by (Yap et al., 2003) as:

(cid:115)

¯Kn(x; p,N) = Kn(x; p,N)

w(x; p,N)
ρ(n; p,N)

(7)

4.2 Krawtchouk Moment
VISAPP_HumanPoseEstimation