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

Department of InformaticsDepartment of InformaticsAristotle University of Aristotle University of

ThessalonikiThessaloniki

Chaotic Sequences and Chaotic Sequences and Applications in Digital Applications in Digital

WatermarkingWatermarking

Athanasios Nikolaidis

Department of Informatics

Aristotle University of Thessaloniki

Department of InformaticsDepartment of InformaticsAristotle University of ThessalonikiAristotle University of Thessaloniki

ContentsContents

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.


Introduction to chaosIntroduction 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.


Definition of a chaotic mapDefinition 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.


Properties of a chaotic functionProperties 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.


Properties of a chaotic functionProperties 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.


Properties of a chaotic functionProperties of a chaotic functionElement 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.


An example: The quadratic familyAn example: The quadratic family

The functions of the quadratic family are defined as:

fp(x) = px(1-x)

The following hold:

fp(0) = fp(1) = 0 and fp(qp) = qp where qp=(p-1)/p

0 < qp < 1 if p > 1

This means that there is at least one fixed point for

each function of the family.


Properties of the quadratic familyProperties 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.


Properties of the quadratic familyProperties of the quadratic family

Bifurcation diagram of the quadratic map Parameter p controls the orbit of the map.


Piecewise-linear Markov mapsPiecewise-linear Markov mapsA map f : [0, 1] [0, 1] is an eventually expanding, piecewise-linear Markov map when: There is a set of partition points α0, α1,..., αΝ satisfying

0 = α0< α1 <…< αΝ =1 and such that restricted to each of the intervals Vi=(αi-1, αi), the map f is affine.

The map has the Markov property that partition points map to partition points: For each i, f (αi) = αj for some j.

The map has the eventually expanding property, i.e., there is an integer k > 0 such that


Examples of piecewise-linear Examples of piecewise-linear Markov mapsMarkov maps

n-way Bernoulli shift:

B(x) = n · x (mod 1)

skew tent map:


Examples of piecewise-linear Examples of piecewise-linear Markov mapsMarkov maps

Bernoulli shift (n=4) Tent map (α=0.3)


Useful properties of piecewise-Useful properties of piecewise-linear Markov mapslinear Markov maps

Existence of invariant densities under certain operators: Frobenius-Perron operator:

pn(·) = Pf{ pn-1(·)} = Pfn { p0(·)}

Invariant density:

p(·) = Pfn { p(·)}, n

Tunable spectral/correlation properties (parameter n for n-way Bernoulli shift, parameter α for skew tent map).


Watermarking using Watermarking using pseudorandom sequencepseudorandom sequence

Input sequence

Pseudorandom

sequenceWatermark

sequenceLowpass filtering

Signed

sequence

If lowpass attacks are to be coped with, pre-filtering is required since the original pseudorandom sequence has white spectrum.


Watermarking using piecewise Watermarking using piecewise linear Markov sequencelinear 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.


Spectral content of tent mapsSpectral content of tent maps

Power spectral density of several low-pass tent maps (α1 lowpass, α0.5 white)

Power spectral density of several high-pass tent maps (α0 highpass, α0.5 white)


Spectral content of Bernoulli shiftsSpectral content of Bernoulli shifts

Power spectral density of several lowpass Bernoulli maps (n0 lowpass, n white)


Analysis of correlation detector Analysis of correlation detector for pseudorandom sequencesfor pseudorandom sequences

Correlator mean for a pseudorandom sequence:

Correlator variance for a pseudorandom sequence:


Analysis of correlation detector Analysis of correlation detector for Bernoulli shift sequencesfor Bernoulli shift sequences

Correlator mean for a n-way Bernoulli shift sequence:

Correlator variance is a complicated function of p, N, n, k, μfo and σfo and is greater in the case of wrong watermark presence than in the case of watermark absence.


Analysis of correlation detector Analysis of correlation detector for tent map sequencesfor 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.


Graphic representation of mean Graphic representation of mean and variance for Bernoulli shiftand variance for Bernoulli shift

Correlator mean for several values of n. Correlator variance for several values of n.


Graphic representation of mean Graphic representation of mean and variance for tent mapand variance for tent map

a) Correlator mean for lowpass tent. b) Correlator mean for white tent. c) Correlator mean for highpass tent.


Comments on correlator mean Comments on correlator mean and varianceand 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.


ROCs for Bernoulli sequencesROCs for Bernoulli sequences

ROCs for several Bernoulli sequences (different values of n).


ROCs for tent sequencesROCs for tent sequences

ROCs for several tent sequences (different values of α).


Comparison of performance of Comparison of performance of Bernoulli and tent sequencesBernoulli and tent sequences

ROCs for several Bernoulli and tent sequences.


Comments on performance of Comments on performance of Bernoulli and tent sequencesBernoulli and tent sequencesExperimental 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.


ROCs after lowpass attackROCs after lowpass attack

ROCs for several chaotic sequences after mean filtering.


Comments on performance after Comments on performance after lowpass attacklowpass 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.


ReferencesReferences1) 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.


ReferencesReferences6) 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.

Documents

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