Image cryptosystems based on PottsNICA algorithms Meng-Hong Chen Jiann-Ming Wu Department of Applied Mathematics National Donghwa University

Blind Source Separation (BSS) Sources Observations Unknown Mixing Structure

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 ( 3 )

The PottsNICA algorithm

Simulations We test the PottsICA method using facial images where the last one is a noise image. The parameters for the PottsICA algorithm are K=10, c ₁ =8, c ₂ =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 JadeICAFastICAPottsICA K=10 PottsNICA K=10 N= N= N= N= sigularity