Quansheng Liu

Steganalysis of LSB matching using differences between nonadjacent pixels

This paper models the messages embedded by spatial least significant bit (LSB) matching as independent noises to the cover image, and reveals that the histogram of the differences between pixel gray values is smoothed by the stego bits despite a large distance between the pixels. Using the characteristic function of difference histogram (DHCF), we prove that the center of mass of DHCF (DHCF COM) decreases after messages are embedded. Accordingly, the DHCF COMs are calculated as distinguishing features from the pixel pairs with different distances. The features are calibrated with an image generated by average operation, and then used to train a support vector machine (SVM) classifier. The experimental results prove that the features extracted from the differences between nonadjacent pixels can help to tackle LSB matching as well.

On generalized multiplicative cascades

We consider a generalized Mandelbrots martingale {Yn} and the associated Mandelbrots measure [mu][omega] on marked trees. If the limit variable Z=lim Yn is not degenerate, we study the asymptotic behavior at infinity of its distribution; in the contrary case, we prove that there is an associated natural martingale Yn* converging to a non-negative random variable Z* with infinite mean. Both Z and Z* lead to non-trivial solution of a distributional equation which extends the notion of stable laws. Precise results are obtained about Hausdorff measures and packing measures of the support of the Mandelbrots measure.

Asymptotic properties and absolute continuity of laws stable by random weighted mean

We study properties of stable-like laws, which are solutions of the distributional equation where (N,A1,A2,...) is a given random variable with values in {0,1,...}x[0,[infinity])x[0,[infinity])x..., and Z,Z1,Z2,... are identically distributed positive random variables, independent of each other and independent of (N,A1,A2,...). Examples of such laws contain the laws of the well-known limit random variables in: (a) the Galton-Watson process or general branching processes, (b) branching random walks, (c) multiplicative processes, and (d) smoothing processes. For any solution Z (with finite or infinite mean), we find asymptotic properties of the distribution function P(Z[less-than-or-equals, slant]x) and those of the characteristic function EeitZ; we prove that the distribution of Z is absolutely continuous on (0,[infinity]), and that its support is the whole half-line [0,[infinity]). Solutions which are not necessarily positive are also considered.

Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in a random environment

We first obtain exponential inequalities for martingales. Let be a sequence of martingale differences relative to a filtration and set Sn=X1+...+Xn. We prove that if for some [delta]>0,Q>=1, K>0 and all k, a.s., then for some constant c>0 (depending only on [delta],Q and K) and all x>0, , where c(x)=cx2 if x[set membership, variant]]0,1] and c(x)=cxQ if x>1; the converse also holds if (Xi) are independent and identically distributed. This extends Bernsteins inequality for Q=1 and Hoeffdings inequality for Q=2. We then apply the preceding result to establish exponential concentration inequalities for the free energy of directed polymers in a random environment and obtain upper bounds for its rates of convergence (in probability, almost surely and in Lp); we also give an expression for the free energy in terms of those of some multiplicative cascades, which improves an inequality of Comets and Vargas [Francis Comets, Vincent Vargas, Majorizing multiplicative cascades for directed polymers in random media, ALEA Lat. Am. J. Probab. Math. Stat. 2 (2006), 267-277 (electronic)] to an equality.

Local dimensions of the branching measure on a Galton–Watson tree

Let μ=μω be the branching measure on the boundary ∂T of a supercritical Galton–Watson tree T=T(ω). Denote by d(μ,u) and d(μ,u) the lower and upper local dimensions of μ at u∈∂T. It is well known that almost surely, d(μ,u)=d(μ,u)=logm for μ-almost all u∈∂T, where m is the expected value of the offspring distribution. Here we find exactly when the result holds for all u∈∂T, and obtain some limit theorems about the uniform local dimensions of μ. We also find the exact local dimension of μ at u∈∂T for μ-almost all u.

Asymptotic properties of supercritical age-dependent branching processes and homogeneous branching random walks

Let (Z(t): t[greater-or-equal, slanted]0) be a supercritical age-dependent branching process and let {Yn} be the natural martingale arising in a homogeneous branching random walk. Let Z be the almost sure limit of Z(t)/EZ(t)(t-->[infinity]) or that of Yn (n-->[infinity]). We study the following problems: (a) the absolute continuity of the distribution of Z and the regularity of the density function; (b) the decay rate (polynomial or exponential) of the left tail probability P(Z[less-than-or-equals, slant]x) as x-->0, and that of the characteristic function EeitZ and its derivative as t-->[infinity]; (c) the moments and decay rate (polynomial or exponential) of the right tail probability P(Z>x) as x-->[infinity], the analyticity of the characteristic function [phi](t)=EeitZ and its growth rate as an entire characteristic function. The results are established for non-trivial solutions of an associated functional equation, and are therefore also applicable for other limit variables arising in age-dependent branching processes and in homogeneous branching random walks.

On Two Measures Defined on the Boundary of a Branching Tree

Replying to a question of A. Joffe, we show that two random measures defined on the boundary of a Galton-Watson tree are mutually singular. We compare them in a precise way, and we extend this result to marked trees in the framework of random fractals.

Exact packing measure on a Galton-Watson tree

We find a dimension function [phi]* such that the [phi]*-packing measure of the boundary of a Galton-Watson tree is strictly positive and finite.

Image Integrity Authentication Scheme Based on Fixed Point Theory

Based on the fixed point theory, this paper proposes a new scheme for image integrity authentication, which is very different from digital signature and fragile watermarking. By the new scheme, the sender transforms an original image into a fixed point image (very close to the original one) of a well-chosen transform and sends the fixed point image (instead of the original one) to the receiver; using the same transform, the receiver checks the integrity of the received image by testing whether it is a fixed point image and locates the tampered areas if the image has been modified during the transmission. A realization of the new scheme is based on Gaussian convolution and deconvolution (GCD) transform, for which an existence theorem of fixed points is proved. The semifragility is analyzed via commutativity of transforms, and three commutativity theorems are found for the GCD transform. Three iterative algorithms are presented for finding a fixed point image with a few numbers of iterations, and for the whole procedure of image integrity authentication; a fragile authentication system and a semifragile one are separately built. Experiments show that both the systems have good performance in transparence, fragility, security, and tampering localization. In particular, the semifragile system can perfectly resist the rotation by a multiple of 90° flipping and brightness attacks.

Weighted moments of the limit of a branching process in a random environment

Let (Zn) be a supercritical branching process in an independent and identically distributed random environment ζ = (ζ0, ζ1,…), and let W be the limit of the normalized population size Zn/