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Golden section transform

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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 (Type L) or (Type H),

where

DCT-golden section transform (DCT-GST)[edit]

It is well known that the Fibonacci sequence is:

(sequence A000045 in the OEIS)
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 of 8 elements for example, we get the form of 8 is , 1-level DCT-LGST of the 8 elements is:

the "low" part of the result is:

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

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

Let be the non-normalized DCT-LGST matrix, where

when , 1-level non-normalized DCT-LGST matrix is:

when , 2-level non-normalized DCT-LGST matrix and the inverse matrix is:

;

when , 3-level non-normalized DCT-LGST matrix and the inverse matrix is:

;

when , 4-level non-normalized DCT-LGST matrix and the inverse matrix is:

;

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

;

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

;

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 is:

DCT-HGST can be also defined by a matrix , where , it is known that the non-normalized DCT matrix is:

when , 2-level non-normalized DCT-HGST matrix and the inverse matrix is:

DFT-golden section transform (DFT-GST)[edit]

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

The lifting scheme of reverse-order DCT-LGST[edit]

Let let be the original sequence, let be a sequence as the "Low" part of reverse-order DCT-LGST, let be a sequence as the "High" part of reverse-order DCT-LGST, we have

where

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

and the reconstruction algorithm is:

where

(sequence A003622 in the OEIS)
(sequence A022342 in the OEIS)
(sequence A035336 in the OEIS)

where

See also[edit]

References[edit]

  1. 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]


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