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

2. Blind Source Separation (BSS) Sources Observations Unknown Mixing Structure

3. BSS by PottsICA Observations Recovered sources PottsNICA

4. The ICA problem Unknown mixing structure: Unkown statistical independent sources: S= Observations:

5. 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

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

7. Then The Kullback-Leibler Divergence

8. Partition the range of each output component …… Potts Modeling

9. Energy function for ICA To minimize L’ is to solve a mixed integer and linear programming

10. Annealed neural dynamics Boltzmann distribution Use mean field equations to find the mean configuration at each

11. Derivation of mean field equations Free energy by

12. Mean field equations

13. A hybrid of mean field annealing MFE ( 1 ) ( 2 )

14. Natural gradient descent method W’W ( 3 )

15. The PottsNICA algorithm

16. 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,

17. Figure1 Original images Mixtures of the sources by the mixing matrix A(4x4) Recovered images by PossNICA N = 4

18. Figure2 N = 5

19. Figure3 N = 8

20. Performance evaluations by Amari

21. Table The performance of the three algorithms for tests by Amari evaluation JadeICAFastICAPottsICA K=10 PottsNICA K=10 N=34.29216.51127.29421.5360 N=49.724011.822011.87633.3244 N=5 15.474315.169910.33924.8253 N=838.784135.3410sigularity 19.2109