1. Department of Informatics Aristotle University of Thessaloniki Chaotic Sequences and Applications in Digital Watermarking Athanasios Nikolaidis Department of Informatics Aristotle University of Thessaloniki

2. Department of Informatics Aristotle University of Thessaloniki Contents Basic definitions and properties of chaotic sequences. The quadratic map as an example. Piecewise-linear Markov maps. Application of Markov maps in digital watermarking. Statistical analysis of correlation detector. Comments on performance of Markov maps.

3. Department of Informatics Aristotle University of Thessaloniki Introduction to chaos Chaos: a state of disorder and irregularity. It describes many physical phenomena with complex behavior by simple laws. Dynamical systems: systems that develop in time in a non-trivial manner. Deterministic chaos: irregular motion generated by nonlinear dynamical systems whose laws determine the time evolution of a state of the system from a knowledge of its previous history.

4. Department of Informatics Aristotle University of Thessaloniki Definition of a chaotic map Let A be a set. A function f : A  A is called chaotic on A if: f has sensitive dependence on initial conditions. f is topologically transitive. Periodic points are dense in A.

5. Department of Informatics Aristotle University of Thessaloniki Properties of a chaotic function Unpredectability A function f : A  A has sensitive dependence on initial conditions if there exists δ > 0 such that, for any x  A and any neighborhood N of x, there exists y  N and n  0 such that | f n (x)-f n (y)| > δ. Intuition: For each point x there is at least one point y in any neighborhood of it, which will eventually separate from x by a distance of at least δ after a certain number n of iterations of the function.

6. Department of Informatics Aristotle University of Thessaloniki Properties of a chaotic function Indecomposability A function f : A  A is said to be topologically transitive if for any pair of open sets B, C  A there exists k > 0 such that f k (B)  C  . Intuition: Points belonging to an arbitrarily small neighborhood will eventually move to any other neighborhood after a certain number of iterations.

7. Department of Informatics Aristotle University of Thessaloniki Properties of a chaotic function Element of regularity The point x is a fixed point for f if f (x)=x. The point x is a periodic point of period n if f n (x)=x. The least positive integer n for which f n (x)=x is called the prime period of x. Intuition: There are points in set A that are finally mapped onto themselves after a number of iterations. When these points are dense in set A, an element of regularity is introduced.

8. Department of Informatics Aristotle University of Thessaloniki An example: The quadratic family The functions of the quadratic family are defined as: f p (x) = px(1-x) The following hold: f p (0) = f p (1) = 0 and f p (q p ) = q p where q p =(p-1)/p 0 1 This means that there is at least one fixed point for each function of the family.

9. Department of Informatics Aristotle University of Thessaloniki Properties of the quadratic family A periodic point q of prime period n is called a hyperbolic periodic point if |(f n )(q)|  1. The number (f n )(q) is called the multiplier of the periodic point. A hyperbolic periodic point q of prime period n is called an attractor (or attractive periodic point) if |(f n )(q)| < 1. A hyperbolic fixed point q is called a repellor (or a repelling fixed point) if |(f n )(q)| > 1.

10. Department of Informatics Aristotle University of Thessaloniki Properties of the quadratic family Bifurcation diagram of the quadratic map  Parameter p controls the orbit of the map.

11. Department of Informatics Aristotle University of Thessaloniki Piecewise-linear Markov maps A map f : [0, 1]  [0, 1] is an eventually expanding, piecewise-linear Markov map when: There is a set of partition points α 0, α 1,..., α Ν satisfying 0 = α 0 < α 1 <…< α Ν =1 and such that restricted to each of the intervals V i =(α i-1, α i ), the map f is affine. The map has the Markov property that partition points map to partition points: For each i, f (α i ) = α j for some j. The map has the eventually expanding property, i.e., there is an integer k > 0 such that

12. Department of Informatics Aristotle University of Thessaloniki Examples of piecewise-linear Markov maps n-way Bernoulli shift: B(x) = n · x (mod 1) skew tent map:

13. Department of Informatics Aristotle University of Thessaloniki Examples of piecewise-linear Markov maps Bernoulli shift (n=4)Tent map (α=0.3)

14. Department of Informatics Aristotle University of Thessaloniki Useful properties of piecewise- linear Markov maps Existence of invariant densities under certain operators: Frobenius-Perron operator: p n (·) = P f { p n-1 (·)} = P f n { p 0 (·)} Invariant density: p(·) = P f n { p(·)},  n Tunable spectral/correlation properties (parameter n for n- way Bernoulli shift, parameter α for skew tent map).

15. Department of Informatics Aristotle University of Thessaloniki Watermarking using pseudorandom sequence Input sequence Pseudorandom sequence Watermark sequence Lowpass filtering Signed sequence  If lowpass attacks are to be coped with, pre-filtering is required since the original pseudorandom sequence has white spectrum.

16. Department of Informatics Aristotle University of Thessaloniki Watermarking using piecewise linear Markov sequence Input sequence Piecewise linear Markov sequence = Watermark sequence Signed sequence  In this case, no pre-filtering is required, since the chaotic sequence can be tuned to have the desired spectral content.

17. Department of Informatics Aristotle University of Thessaloniki Spectral content of tent maps  Power spectral density of several low- pass tent maps (α  1 lowpass, α  0.5 white)  Power spectral density of several high-pass tent maps (α  0 highpass, α  0.5 white)

18. Department of Informatics Aristotle University of Thessaloniki Spectral content of Bernoulli shifts  Power spectral density of several lowpass Bernoulli maps (n  0 lowpass, n  white)

19. Department of Informatics Aristotle University of Thessaloniki Analysis of correlation detector for pseudorandom sequences Correlator mean for a pseudorandom sequence: Correlator variance for a pseudorandom sequence:

20. Department of Informatics Aristotle University of Thessaloniki Analysis of correlation detector for Bernoulli shift sequences Correlator mean for a n-way Bernoulli shift sequence: Correlator variance is a complicated function of p, N, n, k, μ fo and σ fo and is greater in the case of wrong watermark presence than in the case of watermark absence.

21. Department of Informatics Aristotle University of Thessaloniki Analysis of correlation detector for tent map sequences Correlator mean for a skew tent map sequence: Correlator variance is again provided by a complicated function and proves to be greater in the case of watermark presence than in the case of watermark absence.

22. Department of Informatics Aristotle University of Thessaloniki Graphic representation of mean and variance for Bernoulli shift  Correlator mean for several values of n.  Correlator variance for several values of n.

23. Department of Informatics Aristotle University of Thessaloniki Graphic representation of mean and variance for tent map a) Correlator mean for lowpass tent.b) Correlator mean for white tent.c) Correlator mean for highpass tent.

24. Department of Informatics Aristotle University of Thessaloniki Comments on correlator mean and variance Correlator variance has the lowest value for the correct watermark (for the tent sequences). Correlator mean and variance converge always to a constant value. Mean and variance converge faster for white sequences than either for highpass or for lowpass ones. The probability of a wrong (shifted) watermark being detected as the correct one reduces as the map tends to have a white spectrum.

25. Department of Informatics Aristotle University of Thessaloniki ROCs for Bernoulli sequences ROCs for several Bernoulli sequences (different values of n).

26. Department of Informatics Aristotle University of Thessaloniki ROCs for tent sequences ROCs for several tent sequences (different values of α).

27. Department of Informatics Aristotle University of Thessaloniki Comparison of performance of Bernoulli and tent sequences ROCs for several Bernoulli and tent sequences.

28. Department of Informatics Aristotle University of Thessaloniki Comments on performance of Bernoulli and tent sequences Experimental curves are nearly identical to theoretical ones. Performance of highpass sequences is superior to that of white sequences and especially to that of lowpass ones. White tent sequences perform somewhat better than white random ones and so do lowpass tent compared to lowpass Bernoulli. In order to resist lowpass attacks, the best choice is to put highpass watermarks in low-frequency coefficients of the original signal.

29. Department of Informatics Aristotle University of Thessaloniki ROCs after lowpass attack ROCs for several chaotic sequences after mean filtering.

30. Department of Informatics Aristotle University of Thessaloniki Comments on performance after lowpass attack Sequences attaining lowpass characteristics perform better after a lowpass attack than sequences of white or highpass spectrum. Tent sequences with α=0.7 exhibit the best performance among all white, lowpass and highpass sequences that were tested. Tent sequences can simulate the performance of Bernoulli or pseudorandom sequences by properly choosing the value of α.

31. Department of Informatics Aristotle University of Thessaloniki References 1) G. Voyatzis and I. Pitas, Chaotic Watermarks for Embedding in the Spatial Digital Image Domain, IEEE Int. Conference on Image Processing (ICIP'98), Chicago, Illinois, USA, 4-7 October 1998, vol. II, pp. 432-436. 2) A. Nikolaidis and I. Pitas, Comparison of different chaotic maps with application to image watermarking, IEEE Int. Symposium on Circuits and Systems (ISCAS 2000), Geneva, Switzerland, 28-31 May 2000, vol. V, pp. 509-512. 3) S. Tsekeridou, V. Solachidis, N. Nikolaidis, A. Nikolaidis, A. Tefas and I. Pitas, Statistical Analysis of a Watermarking System based on Bernoulli Chaotic Sequences, Signal Processing, Elsevier, Special Issue on Information Theoretic Issues in Digital Watermarking, accepted for publication 2000. 4) S. Tsekeridou, V. Solachidis, N. Nikolaidis, A. Nikolaidis, A. Tefas, I. Pitas, Theoretic Performance Analysis of a Watermarking System based on Bernoulli Chaotic Sequences, Communications and Multimedia Security Conf. (CMS 2001), accepted for publication, Darmstadt, Germany, 21-22 May 2001. 5) S. Tsekeridou, V. Solachidis, N. Nikolaidis, A. Nikolaidis, A. Tefas, I. Pitas, Bernoulli Shift Generated Chaotic Watermarks: Theoretic Invcestigation, IEEE Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP 2001), accepted for publication, Salt Lake City, Utah, USA, 7-11 May 2001.

32. Department of Informatics Aristotle University of Thessaloniki References 6) S. Tsekeridou, V. Solachidis, N. Nikolaidis, A. Nikolaidis, A. Tefas, I. Pitas, Theoretic Investigation of the Use of Watermark Signals derived from Bernoulli Chaotic Sequences, 12th Scandinavian Conference on Image Analysis 2001 (SCIA2001), accepted for publication, Bergen, Norway, 11-14 June 2001. 7) A. Tefas, A. Nikolaidis, N. Nikolaidis, V. Solachidis, S. Tsekeridou, and I. Pitas, Statistical Analysis of Markov Chaotic Sequences for Watermarking Applications, IEEE Int. Symposium on Circuits and Systems (ISCAS 2001), accepted for publication, Sydney, Australia, 6 - 9 May 2001. 8) A. Tefas, A. Nikolaidis, N. Nikolaidis, V. Solachidis, S. Tsekeridou, and I. Pitas, Performance Analysis of Watermarking Schemes based on Skew Tent Chaotic Sequences, IEEE-EURASIP Wor. on Nonlinear Signal and Image Processing (NSIP 2001), accepted for publication, Baltimore, Maryland, USA, 3-6 June 2001. 9) S.H. Isabelle and G.W. Wornell, Statistical analysis and spectral estimation techniques for one-dimensional chaotic signals, IEEE Trans. on Signal Processing, vol. 45, no. 6, pp. 1495-1506, June 1997. 10) G. Depovere, T. Kalker, and J.-P. Linnartz, Improved watermark detection reliability using filtering before correlation, IEEE Int. Conference on Image Processing (ICIP'98), Chicago, Illinois, USA, 4-7 October 1998, vol. I, pp. 430-434.