By Voloshynovskiy, Herrigel, Baumgaertner, Pun

**Read Online or Download A Stochastic Approach to Content Adaptive Digital Image Watermarking PDF**

**Best probability books**

**Introduction to Probability and Statistics for Engineers and Scientists (3rd Edition)**

This up-to-date vintage presents a superb creation to utilized chance and facts for engineering or technological know-how majors. writer Sheldon Ross indicates how chance yields perception into statistical difficulties, leading to an intuitive figuring out of the statistical systems ordinarilly utilized by working towards engineers and scientists.

**Applied Bayesian Modelling (2nd Edition) (Wiley Series in Probability and Statistics)**

This publication offers an obtainable method of Bayesian computing and knowledge research, with an emphasis at the interpretation of genuine information units. Following within the culture of the profitable first variation, this ebook goals to make a variety of statistical modeling purposes available utilizing established code that may be with no trouble tailored to the reader's personal functions.

**Meta analysis : a guide to calibrating and combining statistical evidence**

Meta research: A advisor to Calibrating and mixing Statistical Evidence acts as a resource of simple tools for scientists eager to mix facts from assorted experiments. The authors goal to advertise a deeper realizing of the idea of statistical facts. The ebook is created from components – The guide, and the speculation.

- Calcul des probabilités
- Photo-Electric Emission and Thermionic Emission Once More
- A. A. Markovs work on probability
- Fluctuation Theory for Lévy Processes: Ecole d'Eté de Probabilités de Saint-Flour XXXV - 2005

**Additional info for A Stochastic Approach to Content Adaptive Digital Image Watermarking**

**Example text**

Z ³ © ª´ (2π)−d/2 det (Σ1 )−1/2 exp − 12 (x − µ1 )T Σ−1 (x − µ ) 1 1 ³ Rd ´ © ª × (2π)−d/2 det (Σ2 )−1/2 exp − 12 (x − µ2 )T Σ−1 dx. 2 (x − µ2 ) A= © 2006 by Taylor & Francis Group, LLC 48 Statistical Inference based on Divergence Measures Therefore, ½ ¾ Z ¢ 1¡ T −1 T −1 exp − (x − µ1 ) Σ1 (x − µ1 ) + (x − µ1 ) Σ1 (x − µ2 ) A=L 2 Rd with L = (2π)−d det (Σ1 )−1/2 det (Σ2 )−1/2 . Now we shall write the expression T −1 (x − µ1 )T Σ−1 1 (x − µ1 ) + (x − µ2 ) Σ2 (x − µ2 ) as (x − µ∗ )T C −1 (x − µ∗ ) + B.

4. Let X be a random variable with probability density function f(x). Show Z 1 x2 f (x) dx ≥ exp (2H (X)) . 2πe R 5. f. f. gθ (y), if there exists a nonnegative function h on the product space X × Y for which the following relations are satisfied: Z h(x, y)fθ (x)dµ(x) i) gθ (y) = X Z Z ii) h(x, y) ≥ 0, h(x, y)dµ(x) = h(x, y)dµ(y) = 1. X Y Show that H(Y ) ≥ H(X). 6. Derive the expression of Shannon’s entropy for the following random variables: Beta, Cauchy, Chi-square, Erlang, Exponential, F-Snedecor, Gamma, Laplace, Logistic, Lognormal, Maxwell-Normal, Normal, Normal-generalized, Pareto, Rayleigh and T-Student.

This author also presented a simple measure of divergence: the J-divergence among k populations. , k} . , xk ) = − kj=1 xj j , aj ≥ 0 with Pk j=1 aj = 1 the f-dissimilarity is the negative aﬃnity introduced by Toussaint, (1974). More examples can be seen in Gyorfi and Nemetz (1978) and Zografos (1998a). The f-dissimilarity leads also to the Csiszar’s φ-divergence if f(x1 , x2 ) = x2 φ(x1 /x2 ). Other interesting families of divergence measures among k populations can be seen in Kapur (1988), Sahoo and Wong (1988), Rao (1982a), Toussaint (1978).