![Page 1: Image cryptosystems based on PottsNICA algorithms](https://reader036.vdocument.in/reader036/viewer/2022062314/5681320e550346895d9862e1/html5/thumbnails/1.jpg)
Image cryptosystems based on PottsNICA algorithms
Meng-Hong ChenJiann-Ming WuDepartment of Applied MathematicsNational Donghwa University
![Page 2: Image cryptosystems based on PottsNICA algorithms](https://reader036.vdocument.in/reader036/viewer/2022062314/5681320e550346895d9862e1/html5/thumbnails/2.jpg)
Blind Source Separation (BSS)
Sources
Observations
Unknown MixingStructure
![Page 3: Image cryptosystems based on PottsNICA algorithms](https://reader036.vdocument.in/reader036/viewer/2022062314/5681320e550346895d9862e1/html5/thumbnails/3.jpg)
BSS by PottsICA
Observations
Recovered sources
PottsNICA
![Page 4: Image cryptosystems based on PottsNICA algorithms](https://reader036.vdocument.in/reader036/viewer/2022062314/5681320e550346895d9862e1/html5/thumbnails/4.jpg)
The ICA problem
Unknown mixing structure:
Unkown statistical independent sources: S=
Observations:
![Page 5: Image cryptosystems based on PottsNICA algorithms](https://reader036.vdocument.in/reader036/viewer/2022062314/5681320e550346895d9862e1/html5/thumbnails/5.jpg)
The goal of ICA
The goal is to find W to recover independent sources by
The joint distribution is as close as possible to the product of the marginal distributions
such that
![Page 6: Image cryptosystems based on PottsNICA algorithms](https://reader036.vdocument.in/reader036/viewer/2022062314/5681320e550346895d9862e1/html5/thumbnails/6.jpg)
The criterion on independency of components of y can be quantified by he Kullback-Leibler divergence
The Kullback-Leibler Divergence
![Page 7: Image cryptosystems based on PottsNICA algorithms](https://reader036.vdocument.in/reader036/viewer/2022062314/5681320e550346895d9862e1/html5/thumbnails/7.jpg)
Then
The Kullback-Leibler Divergence
![Page 8: Image cryptosystems based on PottsNICA algorithms](https://reader036.vdocument.in/reader036/viewer/2022062314/5681320e550346895d9862e1/html5/thumbnails/8.jpg)
Partition the range of each output component
… …
Potts Modeling
![Page 9: Image cryptosystems based on PottsNICA algorithms](https://reader036.vdocument.in/reader036/viewer/2022062314/5681320e550346895d9862e1/html5/thumbnails/9.jpg)
Energy function for ICA
To minimize L’ is to solve a mixed integer and linear programming
![Page 10: Image cryptosystems based on PottsNICA algorithms](https://reader036.vdocument.in/reader036/viewer/2022062314/5681320e550346895d9862e1/html5/thumbnails/10.jpg)
Annealed neural dynamics
Boltzmann distribution
Use mean field equations to find the mean configuration at each
![Page 11: Image cryptosystems based on PottsNICA algorithms](https://reader036.vdocument.in/reader036/viewer/2022062314/5681320e550346895d9862e1/html5/thumbnails/11.jpg)
Derivation of mean field equations
Free energy by
![Page 12: Image cryptosystems based on PottsNICA algorithms](https://reader036.vdocument.in/reader036/viewer/2022062314/5681320e550346895d9862e1/html5/thumbnails/12.jpg)
Mean field equations
![Page 13: Image cryptosystems based on PottsNICA algorithms](https://reader036.vdocument.in/reader036/viewer/2022062314/5681320e550346895d9862e1/html5/thumbnails/13.jpg)
A hybrid of mean field annealing
MFE
( 1 )
( 2 )
![Page 14: Image cryptosystems based on PottsNICA algorithms](https://reader036.vdocument.in/reader036/viewer/2022062314/5681320e550346895d9862e1/html5/thumbnails/14.jpg)
Natural gradient descent method
W’W
W’W ( 3 )
![Page 15: Image cryptosystems based on PottsNICA algorithms](https://reader036.vdocument.in/reader036/viewer/2022062314/5681320e550346895d9862e1/html5/thumbnails/15.jpg)
The PottsNICA algorithm
![Page 16: Image cryptosystems based on PottsNICA algorithms](https://reader036.vdocument.in/reader036/viewer/2022062314/5681320e550346895d9862e1/html5/thumbnails/16.jpg)
SimulationsWe test the PottsICA method using facial images where the last one is a noise image. The parameters for the PottsICA algorithm are K=10, c₁=8, c₂=2 and η=0.001; the β parameter has an initial value of and each time it is increased to β by the scheduling process. The diagonal and last column of the mixing matrix A are lager than others. As follows,
![Page 17: Image cryptosystems based on PottsNICA algorithms](https://reader036.vdocument.in/reader036/viewer/2022062314/5681320e550346895d9862e1/html5/thumbnails/17.jpg)
Figure1
Original images
Mixtures of the sources by the mixing matrix A(4x4)
Recovered images by PossNICA
N = 4
![Page 18: Image cryptosystems based on PottsNICA algorithms](https://reader036.vdocument.in/reader036/viewer/2022062314/5681320e550346895d9862e1/html5/thumbnails/18.jpg)
Figure2
N = 5
![Page 19: Image cryptosystems based on PottsNICA algorithms](https://reader036.vdocument.in/reader036/viewer/2022062314/5681320e550346895d9862e1/html5/thumbnails/19.jpg)
Figure3
N = 8
![Page 20: Image cryptosystems based on PottsNICA algorithms](https://reader036.vdocument.in/reader036/viewer/2022062314/5681320e550346895d9862e1/html5/thumbnails/20.jpg)
Performance evaluations by Amari
![Page 21: Image cryptosystems based on PottsNICA algorithms](https://reader036.vdocument.in/reader036/viewer/2022062314/5681320e550346895d9862e1/html5/thumbnails/21.jpg)
Table
The performance of the three algorithms for tests by Amari evaluation
JadeICA FastICA PottsICA
K=10
PottsNICA
K=10
N=3 4.2921 6.5112 7.2942 1.5360
N=4 9.7240 11.8220 11.8763 3.3244
N=5 15.4743 15.1699 10.3392 4.8253
N=8 38.7841 35.3410 sigularity 19.2109