Golden section transform

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. The ratio of 1 level "Low" part length to "High" part length is the approximation of the golden ratio $\varphi$ (Type L) or ${\frac {1}{\varphi }}$ (Type H),

where

$\varphi ={\frac {1+{\sqrt {5}}}{2}}=1.618\ldots ;$

${\frac {1}{\varphi }}={\frac {-1+{\sqrt {5}}}{2}}=0.618\ldots$

DCT-golden section transform (DCT-GST)

It is well known that the Fibonacci sequence is:

$0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\;233,\;377,\;610,\;987\ldots \;$ (sequence A000045 in the OEIS)

$F_{0}=0,\;F_{1}=1,F_{n}=F_{n-1}+F_{n-2},$ 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)

Take a sequence $(p_{0},p_{1},p_{2},p_{3},p_{4},p_{5},p_{6},p_{7})$ of 8 elements for example, we get the form of 8 is $12212$, 1-level DCT-LGST of the 8 elements is:

$p_{0},\;{\frac {p_{1}+p_{2}}{2}},\;{\frac {p_{3}+p_{4}}{2}},\;p_{5},\;{\frac {p_{6}+p_{7}}{2}},\;{\frac {p_{1}-p_{2}}{2}},\;{\frac {p_{3}-p_{4}}{2}},\;{\frac {p_{6}-p_{7}}{2}}$

the "low" part of the result is:

$p_{0},\;{\frac {p_{1}+p_{2}}{2}},\;{\frac {p_{3}+p_{4}}{2}},\;p_{5},\;{\frac {p_{6}+p_{7}}{2}}$

according to the form $212$ of 5, 2-level DCT-LGST of the 8 elements is:

${\frac {p_{0}+{\frac {p_{1}+p_{2}}{2}}}{2}},\;{\frac {p_{3}+p_{4}}{2}},\;{\frac {p_{5}+{\frac {p_{6}+p_{7}}{2}}}{2}},\;{\frac {p_{0}-{\frac {p_{1}+p_{2}}{2}}}{2}},\;{\frac {p_{5}-{\frac {p_{6}+p_{7}}{2}}}{2}},\;{\frac {p_{1}-p_{2}}{2}},\;{\frac {p_{3}-p_{4}}{2}},\;{\frac {p_{6}-p_{7}}{2}}$

An example of 1-Level DCT-Low Golden Section Transform(DCT-LGST).

Let $L_{N}$ be the $N\times N$ non-normalized DCT-LGST matrix, where $N=F_{k},(k\geq 3,k\in \mathbb {Z} )$

when $N=2$, 1-level non-normalized DCT-LGST matrix is:

$L_{2}={\frac {1}{2}}{\begin{bmatrix}1&1\\1&-1\\\end{bmatrix}}$

when $N=3$, 2-level non-normalized DCT-LGST matrix and the inverse matrix is:

$L_{3}={\frac {1}{4}}{\begin{bmatrix}2&1&1\\2&-1&-1\\0&2&-2\\\end{bmatrix}}$; $L_{3}^{-1}={\begin{bmatrix}1&1&0\\1&-1&1\\1&-1&-1\\\end{bmatrix}}$

when $N=5$, 3-level non-normalized DCT-LGST matrix and the inverse matrix is:

$L_{5}={\frac {1}{8}}{\begin{bmatrix}2&2&2&1&1\\2&2&-2&-1&-1\\0&0&4&-2&-2\\4&-4&0&0&0\\0&0&0&4&-4\\\end{bmatrix}}$; $L_{5}^{-1}={\begin{bmatrix}1&1&0&1&0\\1&1&0&-1&0\\1&-1&1&0&0\\1&-1&-1&0&1\\1&-1&-1&0&-1\\\end{bmatrix}}$

when $N=8$, 4-level non-normalized DCT-LGST matrix and the inverse matrix is:

$L_{8}={\frac {1}{16}}{\begin{bmatrix}4&2&2&2&2&2&1&1\\4&2&2&-2&-2&-2&-1&-1\\0&0&0&4&4&-4&-2&-2\\8&-4&-4&0&0&0&0&0\\0&0&0&0&0&8&-4&-4\\0&8&-8&0&0&0&0&0\\0&0&0&8&-8&0&0&0\\0&0&0&0&0&0&8&-8\end{bmatrix}}$;$L_{8}^{-1}={\begin{bmatrix}1&1&0&1&0&0&0&0\\1&1&0&-1&0&1&0&0\\1&1&0&-1&0&-1&0&0\\1&-1&1&0&0&0&1&0\\1&-1&1&0&0&0&-1&0\\1&-1&-1&0&1&0&0&0\\1&-1&-1&0&-1&0&0&1\\1&-1&-1&0&-1&0&0&-1\end{bmatrix}}$

when $N=8$, 4-level non-normalized reverse-order DCT-LGST matrix and the inverse matrix is:

$L_{8}={\frac {1}{16}}{\begin{bmatrix}1&1&2&2&2&2&2&4\\1&1&2&2&2&-2&-2&-4\\2&2&4&-4&-4&0&0&0\\4&4&-8&0&0&0&0&0\\0&0&0&0&0&4&4&-8\\8&-8&0&0&0&0&0&0\\0&0&0&8&-8&0&0&0\\0&0&0&0&0&8&-8&0\end{bmatrix}}$;$L_{8}^{-1}={\begin{bmatrix}1&1&1&1&0&1&0&0\\1&1&1&1&0&-1&0&0\\1&1&1&-1&0&0&0&0\\1&1&-1&0&0&0&1&0\\1&1&-1&0&0&0&-1&0\\1&-1&0&0&1&0&0&1\\1&-1&0&0&1&0&0&-1\\1&-1&0&0&-1&0&0&0\end{bmatrix}}$

also, when $N=8$, 4-level non-normalized symmetrical form DCT-LGST matrix and the inverse matrix is:

$L_{8}={\frac {1}{16}}{\begin{bmatrix}1&1&2&2&2&4&2&2\\1&1&2&2&2&-4&-2&-2\\2&2&4&-4&-4&0&0&0\\4&4&-8&0&0&0&0&0\\0&0&0&0&0&8&-4&-4\\8&-8&0&0&0&0&0&0\\0&0&0&8&-8&0&0&0\\0&0&0&0&0&0&8&-8\end{bmatrix}}$;$L_{8}^{-1}={\begin{bmatrix}1&1&1&1&0&1&0&0\\1&1&1&1&0&-1&0&0\\1&1&1&-1&0&0&0&0\\1&1&-1&0&0&0&1&0\\1&1&-1&0&0&0&-1&0\\1&-1&0&0&1&0&0&0\\1&-1&0&0&-1&0&0&1\\1&-1&0&0&-1&0&0&-1\end{bmatrix}}$

DCT-high golden section transform (DCT-HGST)

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 $323$ is:

${\frac {p_{0}+p_{1}+p_{2}}{3}},\;{\frac {p_{3}+p_{4}}{2}},\;{\frac {p_{5}+p_{6}+p_{7}}{3}},\;{\frac {\sqrt {6}}{6}}p_{0}-{\frac {\sqrt {6}}{6}}p_{2},\;{\frac {\sqrt {2}}{6}}p_{0}-{\frac {\sqrt {2}}{3}}p_{1}+{\frac {\sqrt {2}}{6}}p_{2},\;{\frac {p_{3}-p_{4}}{2}},\;$

${\frac {\sqrt {6}}{6}}p_{5}-{\frac {\sqrt {6}}{6}}p_{7},\;{\frac {\sqrt {2}}{6}}p_{5}-{\frac {\sqrt {2}}{3}}p_{6}+{\frac {\sqrt {2}}{6}}p_{7}$

DCT-HGST can be also defined by a $N\times N$ matrix $H_{N}$, where $N=F_{k},(k\geq 3,k\in \mathbb {Z} )$, it is known that the $3\times 3$ non-normalized DCT matrix is:

$W={\frac {1}{3}}{\begin{bmatrix}1&1&1\\{\frac {\sqrt {6}}{2}}&0&-{\frac {\sqrt {6}}{2}}\\{\frac {\sqrt {2}}{2}}&-{\sqrt {2}}&{\frac {\sqrt {2}}{2}}\end{bmatrix}}$

when $N=8$, 2-level non-normalized DCT-HGST matrix and the inverse matrix is:

$H_{8}={\frac {1}{18}}{\begin{bmatrix}2&2&2&3&3&2&2&2\\{\sqrt {6}}&{\sqrt {6}}&{\sqrt {6}}&0&0&-{\sqrt {6}}&-{\sqrt {6}}&-{\sqrt {6}}\\{\sqrt {2}}&{\sqrt {2}}&{\sqrt {2}}&-3{\sqrt {2}}&-3{\sqrt {2}}&{\sqrt {2}}&{\sqrt {2}}&{\sqrt {2}}\\3{\sqrt {6}}&0&-3{\sqrt {6}}&0&0&0&0&0\\3{\sqrt {2}}&-6{\sqrt {2}}&3{\sqrt {2}}&0&0&0&0&0\\0&0&0&9&-9&0&0&0\\0&0&0&0&0&3{\sqrt {6}}&0&-3{\sqrt {6}}\\0&0&0&0&0&3{\sqrt {2}}&-6{\sqrt {2}}&3{\sqrt {2}}\\\end{bmatrix}};$

$H_{8}^{-1}={\begin{bmatrix}1&{\frac {\sqrt {6}}{2}}&{\frac {\sqrt {2}}{2}}&{\frac {\sqrt {6}}{2}}&{\frac {\sqrt {2}}{2}}&0&0&0\\1&{\frac {\sqrt {6}}{2}}&{\frac {\sqrt {2}}{2}}&0&-{\sqrt {2}}&0&0&0\\1&{\frac {\sqrt {6}}{2}}&{\frac {\sqrt {2}}{2}}&-{\frac {\sqrt {6}}{2}}&{\frac {\sqrt {2}}{2}}&0&0&0\\1&0&-{\sqrt {2}}&0&0&1&0&0\\1&0&-{\sqrt {2}}&0&0&-1&0&0\\1&-{\frac {\sqrt {6}}{2}}&{\frac {\sqrt {2}}{2}}&0&0&0&{\frac {\sqrt {6}}{2}}&{\frac {\sqrt {2}}{2}}\\1&-{\frac {\sqrt {6}}{2}}&{\frac {\sqrt {2}}{2}}&0&0&0&0&-{\sqrt {2}}\\1&-{\frac {\sqrt {6}}{2}}&{\frac {\sqrt {2}}{2}}&0&0&0&-{\frac {\sqrt {6}}{2}}&{\frac {\sqrt {2}}{2}}\end{bmatrix}}$

DFT-golden section transform (DFT-GST)

Similarly, DFT-golden section transform can be defined as well, the $2\times 2$ DFT matrix is identical with $2\times 2$ DCT matrix, the $3\times 3$ non-normalized DFT matrix is:

$W={\frac {1}{3}}{\begin{bmatrix}1&1&1\\1&-{\frac {1}{2}}-{\frac {\sqrt {3}}{2}}i&-{\frac {1}{2}}+{\frac {\sqrt {3}}{2}}i\\1&-{\frac {1}{2}}+{\frac {\sqrt {3}}{2}}i&-{\frac {1}{2}}-{\frac {\sqrt {3}}{2}}i\end{bmatrix}}$

The lifting scheme of reverse-order DCT-LGST

Let $F_{0}=0,\;F_{1}=1,$ let $\{X_{u}\}\,\!$ be the original sequence, let $\{S_{v}\}\,\!$ be a sequence as the "Low" part of reverse-order DCT-LGST, let $\{D_{w}\}\,\!$ be a sequence as the "High" part of reverse-order DCT-LGST, we have

$\{X_{u}\}=X_{0},X_{1},X_{2},\ldots ,X_{u};\;$ $\{S_{v}\}=S_{0},S_{1},S_{2},\ldots ,S_{v};\;$ $\{D_{w}\}=D_{0},D_{1},D_{2},\ldots ,D_{w}\;$

where $u=F_{k}-1,\;v=F_{k-1}-1,\;w=F_{k-2}-1\;(k\geq 3,k\in \mathbb {Z} )$

The lifting scheme of normalized reverse-order DCT-LGST is:

${\begin{cases}S_{M_{n}}^{0}=X_{L_{n}-1},\;D_{n-1}^{0}=X_{L_{n}}\\D_{n-1}^{1}=S_{M_{n}}^{0}-D_{n-1}^{0},S_{M_{n}}^{1}=S_{M_{n}}^{0}-0.5D_{n-1}^{1}\\S_{M_{n}}={\sqrt {2}}S_{M_{n}}^{1},\;S_{L_{m}}=X_{H_{m}},\;D_{n-1}=D_{n-1}^{1}/{\sqrt {2}}\end{cases}}$

and the reconstruction algorithm is:

${\begin{cases}S_{M_{n}}^{1}=S_{M_{n}}/{\sqrt {2}},\;D_{n-1}^{1}={\sqrt {2}}D_{n-1}\\S_{M_{n}}^{0}=S_{M_{n}}^{1}+0.5D_{n-1}^{1},\;D_{n-1}^{0}=S_{M_{n}}^{0}-D_{n-1}^{1}\\X_{L_{n}-1}=S_{M_{n}}^{0},\;X_{H_{m}}=S_{L_{m}},\;X_{L_{n}}=D_{n-1}^{0}\end{cases}}$

where

$\{L_{n}\}=1,4,6,9,12,14,17,19,\ldots \;$ (sequence A003622 in the OEIS)

$\{M_{n}\}=0,2,3,5,7,8,10,11,\ldots \;$ (sequence A022342 in the OEIS)

$\{H_{n}\}=2,7,10,15,20,23,28,31,\ldots \;$ (sequence A035336 in the OEIS)

$L_{n}=\left\lfloor {\frac {3+{\sqrt {5}}}{2}}n\right\rfloor -1,M_{n}=\left\lfloor {\frac {1+{\sqrt {5}}}{2}}n\right\rfloor -1,H_{n}=2\left\lfloor {\frac {1+{\sqrt {5}}}{2}}n\right\rfloor +n-1$

where $m=1,2,3,4,\ldots \;F_{k-3};\;n=1,2,3,4,\ldots \;F_{k-2}\;(m,n\in \mathbb {N} ^{*})$

See also

• Fibonacci number
• Golden ratio
• Discrete cosine transform
• Discrete Fourier transform
• Wavelet transform
• List of Fourier-related transforms
• JPEG 2000

References

• 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

• "discrete cosine transform". PlanetMath.
• Sloane, N. J. A. (ed.). "Sequence A000045 (Fibonacci Numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
• Weisstein, Eric W. "Golden Ratio". MathWorld.

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