An example of 2-Level DCT-Low Golden Section Transform(DCT-LGST).
The golden section transform was proposed and used in digital image multiresolution analysis in 2007 by Jun Li.[1]
The ratio of 1 level "Low" part length to "High" part length is the approximation of the golden ratio
φ
{\displaystyle \varphi }
(Type L) or
1
φ
{\displaystyle {\frac {1}{\varphi }}}
(Type H),
where
φ
=
1
+
5
2
=
1.618
…
;
{\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}=1.618\ldots ;}
1
φ
=
−
1
+
5
2
=
0.618
…
{\displaystyle {\frac {1}{\varphi }}={\frac {-1+{\sqrt {5}}}{2}}=0.618\ldots }
DCT-golden section transform (DCT-GST) [ edit ]
It is well known that the Fibonacci sequence is:
0
,
1
,
1
,
2
,
3
,
5
,
8
,
13
,
21
,
34
,
55
,
89
,
144
,
233
,
377
,
610
,
987
…
{\displaystyle 0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\;233,\;377,\;610,\;987\ldots \;}
(sequence A000045 in the OEIS )
F
0
=
0
,
F
1
=
1
,
F
n
=
F
n
−
1
+
F
n
−
2
,
{\displaystyle F_{0}=0,\;F_{1}=1,F_{n}=F_{n-1}+F_{n-2},}
make a golden section tree of the Fibonacci number 8:
The golden section tree of 8
Fibonacci number
DCT-LGST
Reverse order
Symmetrical form
DCT-HGST
Reverse order
symmetrical form
1
1
1
1
–
–
–
2
2
2
2
2
2
2
3
12
21
21
3
3
3
5
212
212
212
23
32
32
8
12212
21221
21212
323
323
323
13
21212212
21221212
21212212
23323
32332
32323
21
1221221212212
2122121221221
2121221221212
32323323
32332323
32323323
DCT-low golden section transform (DCT-LGST) [ edit ]
Take a sequence
(
p
0
,
p
1
,
p
2
,
p
3
,
p
4
,
p
5
,
p
6
,
p
7
)
{\displaystyle (p_{0},p_{1},p_{2},p_{3},p_{4},p_{5},p_{6},p_{7})}
of 8 elements for example, we get the form of 8 is
12212
{\displaystyle 12212}
, 1-level DCT-LGST of the 8 elements is:
p
0
,
p
1
+
p
2
2
,
p
3
+
p
4
2
,
p
5
,
p
6
+
p
7
2
,
p
1
−
p
2
2
,
p
3
−
p
4
2
,
p
6
−
p
7
2
{\displaystyle p_{0},\;{\frac {p_{1}+p_{2}}{2}},\;{\frac {p_{3}+p_{4}}{2}},\;p_{5},\;{\frac {p_{6}+p_{7}}{2}},\;{\frac {p_{1}-p_{2}}{2}},\;{\frac {p_{3}-p_{4}}{2}},\;{\frac {p_{6}-p_{7}}{2}}}
the "low" part of the result is:
p
0
,
p
1
+
p
2
2
,
p
3
+
p
4
2
,
p
5
,
p
6
+
p
7
2
{\displaystyle p_{0},\;{\frac {p_{1}+p_{2}}{2}},\;{\frac {p_{3}+p_{4}}{2}},\;p_{5},\;{\frac {p_{6}+p_{7}}{2}}}
according to the form
212
{\displaystyle 212}
of 5, 2-level DCT-LGST of the 8 elements is:
p
0
+
p
1
+
p
2
2
2
,
p
3
+
p
4
2
,
p
5
+
p
6
+
p
7
2
2
,
p
0
−
p
1
+
p
2
2
2
,
p
5
−
p
6
+
p
7
2
2
,
p
1
−
p
2
2
,
p
3
−
p
4
2
,
p
6
−
p
7
2
{\displaystyle {\frac {p_{0}+{\frac {p_{1}+p_{2}}{2}}}{2}},\;{\frac {p_{3}+p_{4}}{2}},\;{\frac {p_{5}+{\frac {p_{6}+p_{7}}{2}}}{2}},\;{\frac {p_{0}-{\frac {p_{1}+p_{2}}{2}}}{2}},\;{\frac {p_{5}-{\frac {p_{6}+p_{7}}{2}}}{2}},\;{\frac {p_{1}-p_{2}}{2}},\;{\frac {p_{3}-p_{4}}{2}},\;{\frac {p_{6}-p_{7}}{2}}}
An example of 1-Level DCT-Low Golden Section Transform(DCT-LGST).
Let
L
N
{\displaystyle L_{N}}
be the
N
×
N
{\displaystyle N\times N}
non-normalized DCT-LGST matrix, where
N
=
F
k
,
(
k
≥
3
,
k
∈
Z
)
{\displaystyle N=F_{k},(k\geq 3,k\in \mathbb {Z} )}
when
N
=
2
{\displaystyle N=2}
, 1-level non-normalized DCT-LGST matrix is:
L
2
=
1
2
[
1
1
1
−
1
]
{\displaystyle L_{2}={\frac {1}{2}}{\begin{bmatrix}1&1\\1&-1\\\end{bmatrix}}}
when
N
=
3
{\displaystyle N=3}
, 2-level non-normalized DCT-LGST matrix and the inverse matrix is:
L
3
=
1
4
[
2
1
1
2
−
1
−
1
0
2
−
2
]
{\displaystyle L_{3}={\frac {1}{4}}{\begin{bmatrix}2&1&1\\2&-1&-1\\0&2&-2\\\end{bmatrix}}}
;
L
3
−
1
=
[
1
1
0
1
−
1
1
1
−
1
−
1
]
{\displaystyle L_{3}^{-1}={\begin{bmatrix}1&1&0\\1&-1&1\\1&-1&-1\\\end{bmatrix}}}
when
N
=
5
{\displaystyle N=5}
, 3-level non-normalized DCT-LGST matrix and the inverse matrix is:
L
5
=
1
8
[
2
2
2
1
1
2
2
−
2
−
1
−
1
0
0
4
−
2
−
2
4
−
4
0
0
0
0
0
0
4
−
4
]
{\displaystyle L_{5}={\frac {1}{8}}{\begin{bmatrix}2&2&2&1&1\\2&2&-2&-1&-1\\0&0&4&-2&-2\\4&-4&0&0&0\\0&0&0&4&-4\\\end{bmatrix}}}
;
L
5
−
1
=
[
1
1
0
1
0
1
1
0
−
1
0
1
−
1
1
0
0
1
−
1
−
1
0
1
1
−
1
−
1
0
−
1
]
{\displaystyle L_{5}^{-1}={\begin{bmatrix}1&1&0&1&0\\1&1&0&-1&0\\1&-1&1&0&0\\1&-1&-1&0&1\\1&-1&-1&0&-1\\\end{bmatrix}}}
when
N
=
8
{\displaystyle N=8}
, 4-level non-normalized DCT-LGST matrix and the inverse matrix is:
L
8
=
1
16
[
4
2
2
2
2
2
1
1
4
2
2
−
2
−
2
−
2
−
1
−
1
0
0
0
4
4
−
4
−
2
−
2
8
−
4
−
4
0
0
0
0
0
0
0
0
0
0
8
−
4
−
4
0
8
−
8
0
0
0
0
0
0
0
0
8
−
8
0
0
0
0
0
0
0
0
0
8
−
8
]
{\displaystyle L_{8}={\frac {1}{16}}{\begin{bmatrix}4&2&2&2&2&2&1&1\\4&2&2&-2&-2&-2&-1&-1\\0&0&0&4&4&-4&-2&-2\\8&-4&-4&0&0&0&0&0\\0&0&0&0&0&8&-4&-4\\0&8&-8&0&0&0&0&0\\0&0&0&8&-8&0&0&0\\0&0&0&0&0&0&8&-8\end{bmatrix}}}
;
L
8
−
1
=
[
1
1
0
1
0
0
0
0
1
1
0
−
1
0
1
0
0
1
1
0
−
1
0
−
1
0
0
1
−
1
1
0
0
0
1
0
1
−
1
1
0
0
0
−
1
0
1
−
1
−
1
0
1
0
0
0
1
−
1
−
1
0
−
1
0
0
1
1
−
1
−
1
0
−
1
0
0
−
1
]
{\displaystyle L_{8}^{-1}={\begin{bmatrix}1&1&0&1&0&0&0&0\\1&1&0&-1&0&1&0&0\\1&1&0&-1&0&-1&0&0\\1&-1&1&0&0&0&1&0\\1&-1&1&0&0&0&-1&0\\1&-1&-1&0&1&0&0&0\\1&-1&-1&0&-1&0&0&1\\1&-1&-1&0&-1&0&0&-1\end{bmatrix}}}
when
N
=
8
{\displaystyle N=8}
, 4-level non-normalized reverse-order DCT-LGST matrix and the inverse matrix is:
L
8
=
1
16
[
1
1
2
2
2
2
2
4
1
1
2
2
2
−
2
−
2
−
4
2
2
4
−
4
−
4
0
0
0
4
4
−
8
0
0
0
0
0
0
0
0
0
0
4
4
−
8
8
−
8
0
0
0
0
0
0
0
0
0
8
−
8
0
0
0
0
0
0
0
0
8
−
8
0
]
{\displaystyle L_{8}={\frac {1}{16}}{\begin{bmatrix}1&1&2&2&2&2&2&4\\1&1&2&2&2&-2&-2&-4\\2&2&4&-4&-4&0&0&0\\4&4&-8&0&0&0&0&0\\0&0&0&0&0&4&4&-8\\8&-8&0&0&0&0&0&0\\0&0&0&8&-8&0&0&0\\0&0&0&0&0&8&-8&0\end{bmatrix}}}
;
L
8
−
1
=
[
1
1
1
1
0
1
0
0
1
1
1
1
0
−
1
0
0
1
1
1
−
1
0
0
0
0
1
1
−
1
0
0
0
1
0
1
1
−
1
0
0
0
−
1
0
1
−
1
0
0
1
0
0
1
1
−
1
0
0
1
0
0
−
1
1
−
1
0
0
−
1
0
0
0
]
{\displaystyle L_{8}^{-1}={\begin{bmatrix}1&1&1&1&0&1&0&0\\1&1&1&1&0&-1&0&0\\1&1&1&-1&0&0&0&0\\1&1&-1&0&0&0&1&0\\1&1&-1&0&0&0&-1&0\\1&-1&0&0&1&0&0&1\\1&-1&0&0&1&0&0&-1\\1&-1&0&0&-1&0&0&0\end{bmatrix}}}
also, when
N
=
8
{\displaystyle N=8}
, 4-level non-normalized symmetrical form DCT-LGST matrix and the inverse matrix is:
L
8
=
1
16
[
1
1
2
2
2
4
2
2
1
1
2
2
2
−
4
−
2
−
2
2
2
4
−
4
−
4
0
0
0
4
4
−
8
0
0
0
0
0
0
0
0
0
0
8
−
4
−
4
8
−
8
0
0
0
0
0
0
0
0
0
8
−
8
0
0
0
0
0
0
0
0
0
8
−
8
]
{\displaystyle L_{8}={\frac {1}{16}}{\begin{bmatrix}1&1&2&2&2&4&2&2\\1&1&2&2&2&-4&-2&-2\\2&2&4&-4&-4&0&0&0\\4&4&-8&0&0&0&0&0\\0&0&0&0&0&8&-4&-4\\8&-8&0&0&0&0&0&0\\0&0&0&8&-8&0&0&0\\0&0&0&0&0&0&8&-8\end{bmatrix}}}
;
L
8
−
1
=
[
1
1
1
1
0
1
0
0
1
1
1
1
0
−
1
0
0
1
1
1
−
1
0
0
0
0
1
1
−
1
0
0
0
1
0
1
1
−
1
0
0
0
−
1
0
1
−
1
0
0
1
0
0
0
1
−
1
0
0
−
1
0
0
1
1
−
1
0
0
−
1
0
0
−
1
]
{\displaystyle L_{8}^{-1}={\begin{bmatrix}1&1&1&1&0&1&0&0\\1&1&1&1&0&-1&0&0\\1&1&1&-1&0&0&0&0\\1&1&-1&0&0&0&1&0\\1&1&-1&0&0&0&-1&0\\1&-1&0&0&1&0&0&0\\1&-1&0&0&-1&0&0&1\\1&-1&0&0&-1&0&0&-1\end{bmatrix}}}
DCT-high golden section transform (DCT-HGST) [ edit ]
An example of 1-Level DCT-High Golden Section Transform(DCT-HGST).
Similar to DCT-LGST, 1-level DCT-HGST of the 8 elements by the form
323
{\displaystyle 323}
is:
p
0
+
p
1
+
p
2
3
,
p
3
+
p
4
2
,
p
5
+
p
6
+
p
7
3
,
6
6
p
0
−
6
6
p
2
,
2
6
p
0
−
2
3
p
1
+
2
6
p
2
,
p
3
−
p
4
2
,
{\displaystyle {\frac {p_{0}+p_{1}+p_{2}}{3}},\;{\frac {p_{3}+p_{4}}{2}},\;{\frac {p_{5}+p_{6}+p_{7}}{3}},\;{\frac {\sqrt {6}}{6}}p_{0}-{\frac {\sqrt {6}}{6}}p_{2},\;{\frac {\sqrt {2}}{6}}p_{0}-{\frac {\sqrt {2}}{3}}p_{1}+{\frac {\sqrt {2}}{6}}p_{2},\;{\frac {p_{3}-p_{4}}{2}},\;}
6
6
p
5
−
6
6
p
7
,
2
6
p
5
−
2
3
p
6
+
2
6
p
7
{\displaystyle {\frac {\sqrt {6}}{6}}p_{5}-{\frac {\sqrt {6}}{6}}p_{7},\;{\frac {\sqrt {2}}{6}}p_{5}-{\frac {\sqrt {2}}{3}}p_{6}+{\frac {\sqrt {2}}{6}}p_{7}}
DCT-HGST can be also defined by a
N
×
N
{\displaystyle N\times N}
matrix
H
N
{\displaystyle H_{N}}
, where
N
=
F
k
,
(
k
≥
3
,
k
∈
Z
)
{\displaystyle N=F_{k},(k\geq 3,k\in \mathbb {Z} )}
, it is known that the
3
×
3
{\displaystyle 3\times 3}
non-normalized DCT matrix is:
W
=
1
3
[
1
1
1
6
2
0
−
6
2
2
2
−
2
2
2
]
{\displaystyle W={\frac {1}{3}}{\begin{bmatrix}1&1&1\\{\frac {\sqrt {6}}{2}}&0&-{\frac {\sqrt {6}}{2}}\\{\frac {\sqrt {2}}{2}}&-{\sqrt {2}}&{\frac {\sqrt {2}}{2}}\end{bmatrix}}}
when
N
=
8
{\displaystyle N=8}
, 2-level non-normalized DCT-HGST matrix and the inverse matrix is:
H
8
=
1
18
[
2
2
2
3
3
2
2
2
6
6
6
0
0
−
6
−
6
−
6
2
2
2
−
3
2
−
3
2
2
2
2
3
6
0
−
3
6
0
0
0
0
0
3
2
−
6
2
3
2
0
0
0
0
0
0
0
0
9
−
9
0
0
0
0
0
0
0
0
3
6
0
−
3
6
0
0
0
0
0
3
2
−
6
2
3
2
]
;
{\displaystyle H_{8}={\frac {1}{18}}{\begin{bmatrix}2&2&2&3&3&2&2&2\\{\sqrt {6}}&{\sqrt {6}}&{\sqrt {6}}&0&0&-{\sqrt {6}}&-{\sqrt {6}}&-{\sqrt {6}}\\{\sqrt {2}}&{\sqrt {2}}&{\sqrt {2}}&-3{\sqrt {2}}&-3{\sqrt {2}}&{\sqrt {2}}&{\sqrt {2}}&{\sqrt {2}}\\3{\sqrt {6}}&0&-3{\sqrt {6}}&0&0&0&0&0\\3{\sqrt {2}}&-6{\sqrt {2}}&3{\sqrt {2}}&0&0&0&0&0\\0&0&0&9&-9&0&0&0\\0&0&0&0&0&3{\sqrt {6}}&0&-3{\sqrt {6}}\\0&0&0&0&0&3{\sqrt {2}}&-6{\sqrt {2}}&3{\sqrt {2}}\\\end{bmatrix}};}
H
8
−
1
=
[
1
6
2
2
2
6
2
2
2
0
0
0
1
6
2
2
2
0
−
2
0
0
0
1
6
2
2
2
−
6
2
2
2
0
0
0
1
0
−
2
0
0
1
0
0
1
0
−
2
0
0
−
1
0
0
1
−
6
2
2
2
0
0
0
6
2
2
2
1
−
6
2
2
2
0
0
0
0
−
2
1
−
6
2
2
2
0
0
0
−
6
2
2
2
]
{\displaystyle H_{8}^{-1}={\begin{bmatrix}1&{\frac {\sqrt {6}}{2}}&{\frac {\sqrt {2}}{2}}&{\frac {\sqrt {6}}{2}}&{\frac {\sqrt {2}}{2}}&0&0&0\\1&{\frac {\sqrt {6}}{2}}&{\frac {\sqrt {2}}{2}}&0&-{\sqrt {2}}&0&0&0\\1&{\frac {\sqrt {6}}{2}}&{\frac {\sqrt {2}}{2}}&-{\frac {\sqrt {6}}{2}}&{\frac {\sqrt {2}}{2}}&0&0&0\\1&0&-{\sqrt {2}}&0&0&1&0&0\\1&0&-{\sqrt {2}}&0&0&-1&0&0\\1&-{\frac {\sqrt {6}}{2}}&{\frac {\sqrt {2}}{2}}&0&0&0&{\frac {\sqrt {6}}{2}}&{\frac {\sqrt {2}}{2}}\\1&-{\frac {\sqrt {6}}{2}}&{\frac {\sqrt {2}}{2}}&0&0&0&0&-{\sqrt {2}}\\1&-{\frac {\sqrt {6}}{2}}&{\frac {\sqrt {2}}{2}}&0&0&0&-{\frac {\sqrt {6}}{2}}&{\frac {\sqrt {2}}{2}}\end{bmatrix}}}
DFT-golden section transform (DFT-GST) [ edit ]
Similarly, DFT-golden section transform can be defined as well, the
2
×
2
{\displaystyle 2\times 2}
DFT matrix is identical with
2
×
2
{\displaystyle 2\times 2}
DCT matrix, the
3
×
3
{\displaystyle 3\times 3}
non-normalized DFT matrix is:
W
=
1
3
[
1
1
1
1
−
1
2
−
3
2
i
−
1
2
+
3
2
i
1
−
1
2
+
3
2
i
−
1
2
−
3
2
i
]
{\displaystyle W={\frac {1}{3}}{\begin{bmatrix}1&1&1\\1&-{\frac {1}{2}}-{\frac {\sqrt {3}}{2}}i&-{\frac {1}{2}}+{\frac {\sqrt {3}}{2}}i\\1&-{\frac {1}{2}}+{\frac {\sqrt {3}}{2}}i&-{\frac {1}{2}}-{\frac {\sqrt {3}}{2}}i\end{bmatrix}}}
The lifting scheme of reverse-order DCT-LGST [ edit ]
Let
F
0
=
0
,
F
1
=
1
,
{\displaystyle F_{0}=0,\;F_{1}=1,}
let
{
X
u
}
{\displaystyle \{X_{u}\}\,\!}
be the original sequence, let
{
S
v
}
{\displaystyle \{S_{v}\}\,\!}
be a sequence as the "Low" part of reverse-order DCT-LGST, let
{
D
w
}
{\displaystyle \{D_{w}\}\,\!}
be a sequence as the "High" part of reverse-order DCT-LGST, we have
{
X
u
}
=
X
0
,
X
1
,
X
2
,
…
,
X
u
;
{\displaystyle \{X_{u}\}=X_{0},X_{1},X_{2},\ldots ,X_{u};\;}
{
S
v
}
=
S
0
,
S
1
,
S
2
,
…
,
S
v
;
{\displaystyle \{S_{v}\}=S_{0},S_{1},S_{2},\ldots ,S_{v};\;}
{
D
w
}
=
D
0
,
D
1
,
D
2
,
…
,
D
w
{\displaystyle \{D_{w}\}=D_{0},D_{1},D_{2},\ldots ,D_{w}\;}
where
u
=
F
k
−
1
,
v
=
F
k
−
1
−
1
,
w
=
F
k
−
2
−
1
(
k
≥
3
,
k
∈
Z
)
{\displaystyle u=F_{k}-1,\;v=F_{k-1}-1,\;w=F_{k-2}-1\;(k\geq 3,k\in \mathbb {Z} )}
The lifting scheme of normalized reverse-order DCT-LGST is:
{
S
M
n
0
=
X
L
n
−
1
,
D
n
−
1
0
=
X
L
n
D
n
−
1
1
=
S
M
n
0
−
D
n
−
1
0
,
S
M
n
1
=
S
M
n
0
−
0.5
D
n
−
1
1
S
M
n
=
2
S
M
n
1
,
S
L
m
=
X
H
m
,
D
n
−
1
=
D
n
−
1
1
/
2
{\displaystyle {\begin{cases}S_{M_{n}}^{0}=X_{L_{n}-1},\;D_{n-1}^{0}=X_{L_{n}}\\D_{n-1}^{1}=S_{M_{n}}^{0}-D_{n-1}^{0},S_{M_{n}}^{1}=S_{M_{n}}^{0}-0.5D_{n-1}^{1}\\S_{M_{n}}={\sqrt {2}}S_{M_{n}}^{1},\;S_{L_{m}}=X_{H_{m}},\;D_{n-1}=D_{n-1}^{1}/{\sqrt {2}}\end{cases}}}
and the reconstruction algorithm is:
{
S
M
n
1
=
S
M
n
/
2
,
D
n
−
1
1
=
2
D
n
−
1
S
M
n
0
=
S
M
n
1
+
0.5
D
n
−
1
1
,
D
n
−
1
0
=
S
M
n
0
−
D
n
−
1
1
X
L
n
−
1
=
S
M
n
0
,
X
H
m
=
S
L
m
,
X
L
n
=
D
n
−
1
0
{\displaystyle {\begin{cases}S_{M_{n}}^{1}=S_{M_{n}}/{\sqrt {2}},\;D_{n-1}^{1}={\sqrt {2}}D_{n-1}\\S_{M_{n}}^{0}=S_{M_{n}}^{1}+0.5D_{n-1}^{1},\;D_{n-1}^{0}=S_{M_{n}}^{0}-D_{n-1}^{1}\\X_{L_{n}-1}=S_{M_{n}}^{0},\;X_{H_{m}}=S_{L_{m}},\;X_{L_{n}}=D_{n-1}^{0}\end{cases}}}
where
{
L
n
}
=
1
,
4
,
6
,
9
,
12
,
14
,
17
,
19
,
…
{\displaystyle \{L_{n}\}=1,4,6,9,12,14,17,19,\ldots \;}
(sequence A003622 in the OEIS )
{
M
n
}
=
0
,
2
,
3
,
5
,
7
,
8
,
10
,
11
,
…
{\displaystyle \{M_{n}\}=0,2,3,5,7,8,10,11,\ldots \;}
(sequence A022342 in the OEIS )
{
H
n
}
=
2
,
7
,
10
,
15
,
20
,
23
,
28
,
31
,
…
{\displaystyle \{H_{n}\}=2,7,10,15,20,23,28,31,\ldots \;}
(sequence A035336 in the OEIS )
L
n
=
⌊
3
+
5
2
n
⌋
−
1
,
M
n
=
⌊
1
+
5
2
n
⌋
−
1
,
H
n
=
2
⌊
1
+
5
2
n
⌋
+
n
−
1
{\displaystyle L_{n}=\left\lfloor {\frac {3+{\sqrt {5}}}{2}}n\right\rfloor -1,M_{n}=\left\lfloor {\frac {1+{\sqrt {5}}}{2}}n\right\rfloor -1,H_{n}=2\left\lfloor {\frac {1+{\sqrt {5}}}{2}}n\right\rfloor +n-1}
where
m
=
1
,
2
,
3
,
4
,
…
F
k
−
3
;
n
=
1
,
2
,
3
,
4
,
…
F
k
−
2
(
m
,
n
∈
N
∗
)
{\displaystyle m=1,2,3,4,\ldots \;F_{k-3};\;n=1,2,3,4,\ldots \;F_{k-2}\;(m,n\in \mathbb {N} ^{*})}
See also [ edit ]
References [ edit ]
↑ Jun. Li (2007). "Golden Section Method Used in Digital Image Multi-resolution Analysis". Application Research of Computers (in 中文). 24 (Suppl.): 1880–1882. ISSN 1001-3695 .
Sun, Y.K.(2005). Wavelet Analysis and Its Applications . China Machine Press. ISBN 7111158768
Jin, J.F.(2004). Visual C++ Wavelet Transform Technology and Engineering Practice . Posts & Telecommunications Press. ISBN 7115119597
He, B.;Ma, T.Y.(2002). Visual C++ Digital Image Processing . Posts & Telecommunications Press. ISBN 7115109559
Ingrid Daubechies(1992). Ten Lectures on Wavelets . Society for Industrial and Applied Mathematics. ISBN 0-89871-274-2
External links [ edit ]
This article "Golden section transform" is from Wikipedia . The list of its authors can be seen in its historical. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.