Digital Signal Processing Reference
In-Depth Information
where
P
l
is
d
l
dt
l
(
t
2
1
2
l
l
!
1)
l
P
l
(
t
)
=
−
.
(10.5)
Furthermore, an important property of the Legendre polynomials is that they are
orthogonal:
Y
lm
(
ω
)
Y
lm
(
ω
−
ω
ω
=
δ
,
)
(
)
(10.6)
l
∈N
|
m
|
l
θ
,ϑ
). In this chapter, many multiscale decompositions will
be built based on the spherical harmonics and/or the HEALPix representation.
ω
=
with
ω
=
(
θ,ϑ
)et
(
10.3 ORTHOGONAL HAAR WAVELETS ON THE SPHERE
The Haar wavelet transform on the sphere (Schr oder and Sweldens 1995) at each
resolution
j
and pixel
k
=
(
k
x
,
k
y
) on the sphere is based on a scaling function
=
φ
2
−
j
(
x
k
)
, where
x
is the vector of Cartesian coordinates on the
φ
j
,
k
(
φ
j
,
k
(
x
)
−
d
j
sphere and
φ
is the Haar scaling function) and three Haar wavelet functions
ψ
,
k
(see Section 2.3.3) with
d
. It uses the idea that a given pixel on the sphere
at a given resolution
j
in the HEALPix representation is directly related to four
pixels at the next resolution
j
∈{
1
,
2
,
3
}
−
1.
,
,
,
Denoting
k
0
k
1
k
2
k
3
the four pixels at scale
j
, hierarchically related to the
pixel
k
at scale
j
+
1, scaling coefficients
c
j
+
1
,
k
at scale
j
+
1 are derived from those
at scale
j
by
3
1
4
c
j
+
1
[
k
]
=
c
j
[
k
d
]
,
(10.7)
d
=
0
and wavelet coefficients at scale
j
+
1 are derived from coefficients at scale
j
by
1
4
(
c
j
[
k
0
]
1
j
w
1
[
k
]
=
+
c
j
[
k
2
]
−
c
j
[
k
1
]
−
c
j
[
k
3
])
,
+
1
4
(
c
j
[
k
0
]
2
w
j
+
1
[
k
]
=
+
c
j
[
k
1
]
−
c
j
[
k
2
]
−
c
j
[
k
3
])
,
1
4
(
c
j
[
k
0
]
3
j
w
1
[
k
]
=
+
c
j
[
k
3
]
−
c
j
[
k
1
]
−
c
j
[
k
2
])
.
(10.8)
+
The Haar wavelet transform on the sphere is orthogonal, and its reconstruction
is exact. The inverse transformation is obtained by
J
3
j
[
k
]
j
(
x
)
c
0
[
x
]
=
c
J
[
k
]
φ
J
,
k
(
x
)
+
w
ψ
.
(10.9)
k
j
=
1
d
=
1
k
This transform is very fast, but its interest is relatively limited. Indeed, it is not rota-
tion invariant, and more important, the Haar wavelet shape is not well adapted for
most applications because of the nonregular shape of the wavelet function.
