Image cryptosystems based on PottsNICA algorithms

Meng-Hong ChenJiann-Ming WuDepartment of Applied MathematicsNational Donghwa University

Blind Source Separation (BSS)

Sources

Observations

Unknown MixingStructure

BSS by PottsICA

Observations

Recovered sources

PottsNICA

The ICA problem

Unknown mixing structure:

Unkown statistical independent sources: S=

Observations:

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

The criterion on independency of components of y can be quantified by he Kullback-Leibler divergence

The Kullback-Leibler Divergence

Then

The Kullback-Leibler Divergence

Partition the range of each output component

… …

Potts Modeling

Energy function for ICA

To minimize L’ is to solve a mixed integer and linear programming

Annealed neural dynamics

Boltzmann distribution

Use mean field equations to find the mean configuration at each

Derivation of mean field equations

Free energy by

Mean field equations

A hybrid of mean field annealing

MFE

( 1 )

( 2 )

Natural gradient descent method

W’W

W’W ( 3 )

The PottsNICA algorithm

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,

Figure1

Original images

Mixtures of the sources by the mixing matrix A(4x4)

Recovered images by PossNICA

N = 4

Figure2

N = 5

Figure3

N = 8

Performance evaluations by Amari

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