
142
5
Compression
In order to minimize the mean-square error between f(ra, n) and f(m, n\
the coefficients a(k, I) must satisfy the following orthogonality conditions:
E[f(m,
ή)- ΣΣ
a
^ W(
m
~Kn- I)
\f(ij)
= 0
i and; = 0, 1,
2,...,
Ν + 1, (ij) Φ (m, n) (79)
Substituting (78) in (79), we get
p
\m-i\
p
\n-)\
_ £ £ φ
9
/)
p
l«-k-i|
p
l»-l-j|
=
0
(k,Z)*(0,0)
ι and; = 0, 1,
2,...,
Ν + 1, (U) Φ (m, n) (80)
Proceeding in a manner analogous to the one-dimensional case, one can
show that when (m, n) is one of the interior pixels (i.e., it is not on the bound-
ary),
the coefficients a(k, /) are given by
a(l,0) =
fl
(0, 1) = a(-1, 0) = α(0, -1) = α
and
a(l,
1) =