wave let domain image water marking

7/30/2019 Wave Let Domain Image Water Marking

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Wavelet Domain Image

WatermarkingThis numerical tour explores robust watermarking over the wavelet domain.

Contents

Installing toolboxes and setting up the path.

Digital Image Watermarking

Watermark Embedding

Watermark Detection

Quantization Attack

Many thanks to Patrick Bas and Teddy Furon for their usefuladvices on digital image watermarking.

Installing toolboxes and setting up the

path.

You need to download the following files: signaltoolbox and general toolbox.

You need to unzip these toolboxes in your working directory,so that you havetoolbox_signal and toolbox_general in your

directory.

For Scilab user: you must replace the Matlab comment '%'by its Scilab counterpart '//'.

Recommandation: You should create a text file named forinstance numericaltour.sce (in Scilab) or numericaltour.m (in Matlab)

to write all the Scilab/Matlab command you want to execute.Then, simply run exec('numericaltour.sce'); (in Scilab)

or numericaltour; (in Matlab) to run the commands.

Execute this line only if you are using Matlab.

getd = @(p)path(p,path); % scilab users must *not* execute this
http://www.ceremade.dauphine.fr/~peyre/numerical-tour/tours/coding_5_watermarking/#2http://www.ceremade.dauphine.fr/~peyre/numerical-tour/tours/coding_5_watermarking/#9http://www.ceremade.dauphine.fr/~peyre/numerical-tour/tours/coding_5_watermarking/#17http://www.ceremade.dauphine.fr/~peyre/numerical-tour/tours/coding_5_watermarking/#38http://www.ceremade.dauphine.fr/~peyre/numerical-tour/tours/coding_5_watermarking/#50http://www.ceremade.dauphine.fr/~peyre/numerical-tour/tours/toolbox_signal.ziphttp://www.ceremade.dauphine.fr/~peyre/numerical-tour/tours/toolbox_signal.ziphttp://www.ceremade.dauphine.fr/~peyre/numerical-tour/tours/toolbox_general.ziphttp://www.ceremade.dauphine.fr/~peyre/numerical-tour/tours/coding_5_watermarking/#2http://www.ceremade.dauphine.fr/~peyre/numerical-tour/tours/coding_5_watermarking/#9http://www.ceremade.dauphine.fr/~peyre/numerical-tour/tours/coding_5_watermarking/#17http://www.ceremade.dauphine.fr/~peyre/numerical-tour/tours/coding_5_watermarking/#38http://www.ceremade.dauphine.fr/~peyre/numerical-tour/tours/coding_5_watermarking/#50http://www.ceremade.dauphine.fr/~peyre/numerical-tour/tours/toolbox_signal.ziphttp://www.ceremade.dauphine.fr/~peyre/numerical-tour/tours/toolbox_signal.ziphttp://www.ceremade.dauphine.fr/~peyre/numerical-tour/tours/toolbox_general.zip


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Then you can add the toolboxes to the path.

getd('toolbox_signal/');getd('toolbox_general/');

Digital Image WatermarkingDigital media watermarking is a popular image forensicproblem. It requires to embed a signature into a sound, image,video, 3D mesh, etc.

An good source of information regarding digital watermarkingis the book

Digital Watermarking and Steganography, 2nd Ed, Ingemar J.Cox, Matthew L. Miller, Jeffrey Bloom, Morgan Kaufmann,2007.

One can also visit the BOWS-2 challenge homepage for a stateof the art digital watermarking implementation.

We consider here a robust watermarking embedding, i.e. thegoal is to embed a watermark that is both impercevable anddifficult to remove (by attack such as compression, denoising,

adding noise, blurring, etc).

This is somehow conflicting goals since impercevableinformation is likely to be removed by an efficient compressionor denoising algorithm. An efficient watermarking schemeshould thus use more clever tools than state of the artdenoising/compression algorithms.

Note also that we perform here "0 bit" watermarking, i.e. wedo not embed a meaningful message within the watermarking.

We are only interested in testing the presence of a givenwatermark.

Here we bench a wavelet method for the embedding of asingle watermark. We check how much the watermark can bedetected after various attack. Depending on a probability of
http://bows2.gipsa-lab.inpg.fr/http://bows2.gipsa-lab.inpg.fr/


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false alarm, we compute the probability of detecting thewatermark.

Watermark Embedding

A watermark is computed as a weighted random vector that isadded to the wavelet coefficient.

The weighting of the watermark vector takes into account theamplitude of the host coefficient in order to reduce visualdistortion. This also increases the robustness to denoising andcompression attacks.

Load an image of pixels.

n = 256;name = 'hibiscus';f = load_image(name, n);f = rescale(sum(f,3));

Display the original image.

clf;imageplot(f);


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Shortcut for the wavelet transform that maps an image towavelet coefficients . We note its inverse using theshortcut PsiS.Jmin = log2(n)-2;Psi = @(f)perform_wavelet_transf(f, Jmin, +1);PsiS = @(a)perform_wavelet_transf(a, Jmin, -1);

Compute the wavelet coefficients.

a = Psi(f);

Display the wavelet coefficients.

clf;

plot_wavelet(a,Jmin);


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The coefficients to be watermarked is only a subset of thetotal set of coefficients, where .

We select here only the fine scale wavelets.

A = ones(n); A(1:2^Jmin,1:2^Jmin) = 0;I = find(A(:));P = length(I);

Extract the coefficients .x0 = a(I);

The watermarking is embedded using a multiplicative rule as

where is a random Gaussian vector and where is a constant that ensure that is a given

deviation value.

Generate the base watermark vector .w = randn(P,1);

Target embedding PSNR (should be quite large for theembedding to be unoticeable).

psnr_embedding = 50;


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Exercice 1: (the solution is exo1.m) Compute rho so

that PSNR(y,x0,1)=snr_embedding.

exo1;rho = 0.0785.

Exercice 2: (the solution is exo2.m) According to you, forwhich PSNR the watermark becomes unoticeable?

exo2;

Perform the embedding .x = x0 + rho*abs(x0).*w;

The distortion of the embedding is measured using the PSNR

Check the PSNR of embedding.

disp(['PSNR(x,x0) = ' num2str(psnr(x,x0,1), 3) 'dB.']);PSNR(x,x0) = 50dB.

Given the watermarked coefficients , a watermarked image isreconstructed using the inverse wavelet transform as

Compute the image with the watermark embedded.

a1 = a; a1(I) = x;f1 = PsiS(a1);

Display the watermark over the spacial domain (with contrastboosting).delta = f-f1;clf;imageplot( clamp(delta/std(delta(:)),-3,3) );
http://www.ceremade.dauphine.fr/~peyre/numerical-tour/tours/missing-exo.htmlhttp://www.ceremade.dauphine.fr/~peyre/numerical-tour/tours/missing-exo.htmlhttp://www.ceremade.dauphine.fr/~peyre/numerical-tour/tours/missing-exo.htmlhttp://www.ceremade.dauphine.fr/~peyre/numerical-tour/tours/missing-exo.html


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Watermark Detection

The watermark is detected (or not detected) from an input

vector using a detector function where is the basewatermark vector. Usually, a large value of means that islikely to come from a watermarked content.

The detection is carried over by a simple thresholding, and thewatermark is declared to be present if

where is a threshold that should be set to guarantee a given probability of false alarms

(i.e. ratio of contents declared to be watermarked whereas they were not watermarked).

The detection corresponds to an hypothesis testing. Oneassumes that is obtained by attacking some vector , and onehas the following alternative depending on wether thecontent is watermarked or not:

The two important quantities to monitor is the probability offalse alarms


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and the probability of true positives

Note that here refers to the probability of an event with respect to the randomization of .

The goal is to design a watermarking scheme (i.e. anembedding strategy and a detection strategy) in order tomaximize for a given .

The simplest detector is a normalized correlation

C = @(y,w)sum(w.*y)./sqrt( sum(w.^2).*sum(y.^2) );

To estimate easily the probability of false alarm, we make theasumption that is close enough to to estimate on the cleanoriginal signalExercice 3: (the solution is exo3.m) Using a Monte Carlosimulation (generation of the order of watermarks, display thehistogram of the repartition of . Compute the variance of thisdistribution.exo3;

We make another approximation : we approximate this densityprobability with a Gaussian density of mean 0 and variance .Under this assumption, one haswhere is the cumulative density function of the Gaussian of variance .
http://www.ceremade.dauphine.fr/~peyre/numerical-tour/tours/missing-exo.htmlhttp://www.ceremade.dauphine.fr/~peyre/numerical-tour/tours/missing-exo.html


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Hence one can use the threshold

This is an example of determination of threshold given a value of .pfa = 1e-3;T = sqrt(2)/2 * sigma0 * erfinv(1-2*pfa);

Actually, it is possible to compute exactly this probability offalse alarm as

where is the incomplete beta function (use betainc function) and is the dimension.

Exercice 4: (the solution is exo4.m) Compare, for variousvalues of the estimation obtained by the Gaussianapproximation with the true value obtained with theincomplete beta function.exo4;

Quantization Attack

A compression attack is simulated by quantizing the waveletcoefficients. We consider here a dead zone quantization attack.

Quantization step (the larger, the more aggressive thecompression.tau = .2;

A quantization attack reads:

Quantization/Dequantization operators, and attack operator.

Quant = @(x)floor(abs(x/tau)).*sign(x);DeQuant = @(v)sign(v) .* (abs(v)+.5) * tau;A = @(x)DeQuant(Quant(x));

Display it.

t = linspace(-2,2,500);plot(t, A(t));

axis('equal');


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Attacked watermarked signal

y = A(x);

Display the attacked image.

a1 = a; a1(I) = y;f1 = PsiS(a1);

Display the watermarked image.

clf;imageplot(clamp(f1));


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Compute the detection .disp(['C(y,w) = ' num2str(C(y,w), 2) '.']);C(y,w) = 0.015.

The probability of true positive is

Exercice 5: (the solution is exo5.m) Compute, by Monte Carlosampling (i.e. draw at random many ) the distribution of for .Store the different realization of in a vector c. Note: the value

of should be recomputed for each .exo5;


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Exercice 6: (the solution is exo6.m) Compute, for a varyingvalue of , the corresponding value of . Display the resultingcurve (ROC curve). This computation should be performedexperimentally using e.g. 1000 random sampling.exo6;


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Exercice 7: (the solution is exo7.m) Try different attackstrengths, by changing the value of . For a , determine thevalue of for witch drops bellow .exo7;

Exercice 8: (the solution is exo8.m) Try different attacks, forinstance on the image itself (blurring, denoising, etc.).

exo8;

Copyright (c) 2010 Gabriel Peyre
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