Biomedical Engineering Reference
In-Depth Information
2
⎞
⎠
dx dy
M
−
1
2
M
−
1
(
x
)
0
,
k
,
l
(
x
,
y
)
(
x
)
0
,
k
,
l
(
x
,
y
)
q
k
,
l
φ
q
k
,
l
φ
+
+
k
,
l
=
0
k
,
l
=
0
⎛
M
−
1
2
M
−
1
(
x
)
0
,
k
,
l
(
x
,
y
)
−
⎝
Z
k
,
l
φ
p
k
,
l
φ
0
,
k
,
l
(
x
,
y
)
+
k
,
l
=
0
k
,
l
=
0
2
⎞
⎠
dx dy
M
−
1
M
−
1
(
y
)
0
,
k
,
l
(
x
,
y
)
−
(
x
)
0
,
k
,
l
(
x
,
y
)
Z
k
,
l
φ
q
k
,
l
φ
+
k
,
l
=
0
k
,
l
=
0
⎛
⎝
M
−
1
2
M
−
1
(
xx
)
0
,
k
,
l
(
x
,
y
)
−
(
x
)
0
,
k
,
l
(
x
,
y
)
Z
k
,
l
φ
p
k
,
l
φ
k
,
l
=
0
k
,
l
=
0
2
⎞
⎠
dx dy
.
M
−
1
M
−
1
(
yy
)
0
,
k
,
l
(
x
,
y
)
−
(
y
)
0
,
k
,
l
(
x
,
y
)
+
Z
k
,
l
φ
q
k
,
l
φ
k
,
l
=
0
k
,
l
=
0
There are total of 3
M
2
unknown variables (they are the function samples of
Z
,
p
, and
q
):
p
k
,
l
,
q
k
,
l
,
and
Z
k
,
l
,
where the indices run on
M
×
M
grid (see (5.69)).
It is remarkable that the interpolating property (5.69) simplified the compu-
tation significantly. The integrals we need to compute in energy function are
only involved with the integrals which are the inner product of the scaling func-
tion
φ
(
x
,
y
):
=
φ
0
,
0
,
0
(
x
,
y
), its shifting
φ
k
,
l
(
x
,
y
):
=
φ
0
,
k
,
l
(
x
,
y
), and their partial
derivatives of first and second orders. Note that we have dropped the scale (or
the resolution) index 0 for simplicity
,
since the discussion here does not relate
to other scales. Now we assume that the scaling function
φ
is the Daubechies
scaling function with 2
N
+
1 filter coefficients
c
i
(see (5.63)). These definite
integrals are called connection coefficients [5]:
φ
(
xx
)
k
,
l
(4)
(
xx
)
(
x
,
y
)
φ
(4)
(
k
)
D
(
l
)
,
x
(
k
,
l
)
=
(
x
,
y
)
dx dy
=
φ
(
yy
)
k
,
l
(4)
(
yy
)
(
x
,
y
)
φ
(4)
(
k
)
,
y
(
k
,
l
)
=
(
x
,
y
)
dx dy
=
D
(
k
)
φ
(
xy
)
k
,
l
(4)
(
xy
)
(
x
,
y
)
φ
(2)
(
k
)
(2)
(
l
)
,
xy
(
k
,
l
)
=
(
x
,
y
)
dx dy
=
