Searcharxiv⌕ Search

arXiv subjects

Quansheng Liu

Publications and source records attributed to Quansheng Liu.

50 records · Page 3Linked to original sources
arXiv2013

Image Integrity Authentication Scheme Based On Fixed Point Theory

Based on fixed point theory, this paper proposes a new scheme for image integrity authentication, which is different from Digital Signature and Fragile Watermarking. A realization of the new scheme is given based on Gaussian Convolution and Deconvolution (GCD) functions. For a given image, if it is invariant under a GCD function, we call it GCD fixed point image. An existence theorem of fixed points for GCD functions is proved and an iterative algorithm is presented for finding fixed points. Experiments show that GCD fixed point images perform well in transparence, fragility, security and tampering localization.

cs.CR↗
arXiv2013

Convergence rates for a branching process in a random environment

Let $(Z_n)$ be a supercritical branching process in a random environment $ξ$. We study the convergence rates of the martingale $W_n = Z_n/ E[Z_n| ξ]$ to its limit $W$. The following results about the convergence almost sur (a.s.), in law or in probability, are shown. (1) Under a moment condition of order $p\in (1,2)$, $W-W_n = o (e^{-na})$ a.s. for some $a>0$ that we find explicitly; assuming only $EW_1 \log W_1^{α+1} < \infty$ for some $α>0$, we have $W-W_n = o (n^{-α})$ a.s.; similar conclusions hold for a branching process in a varying environment. (2) Under a second moment condition, there are norming constants $a_n(ξ)$ (that we calculate explicitly) such that $a_n(ξ) (W-W_n)$ converges in law to a non-degenerate distribution. (3) For a branching process in a finite state random environment, if $W_1$ has a finite exponential moment, then so does $W$, and the decay rate of $P(|W-W_n| > ε)$ is supergeometric.

math.PR↗
arXiv2012

Convergence Theorems for the Non-Local Means Filter

In this paper, we establish convergence theorems for the Non-Local Means Filter in removing the additive Gaussian noise. We employ the techniques of "Oracle" estimation to determine the order of the widths of the similarity patches and search windows in the aforementioned filter. We propose a practical choice of these parameters which improve the restoration quality of the filter compared with the usual choice of parameters.

stat.AP↗
arXiv2012

Large deviation exponential inequalities for supermartingales

Let $(X_{i}, \mathcal{F}_{i})_{i\geq 1}$ be a sequence of supermartingale differences and let $S_k=\sum_{i=1}^k X_i$. We give an exponential moment condition under which $P(\max_{1\leq k \leq n} S_k \geq n)=O(\exp\{-C_1 n^α\}),$ $n\rightarrow \infty,$ where $α\in (0, 1)$ is given and $C_{1}>0$ is a constant. We also show that the power $α$ is optimal under the given condition. In particular, when $α=1/3$, we recover an inequality of Lesigne and Volný.

math.PR↗
arXiv2012

Hoeffding's inequality for supermartingales

We give an extension of Hoeffding's inequality to the case of supermartingales with differences bounded from above. Our inequality strengthens or extends the inequalities of Freedman, Bernstein, Prohorov, Bennett and Nagaev.

math.PR↗
arXiv2012

Optimal Weights Mixed Filter for Removing Mixture of Gaussian and Impulse Noises

According to the character of Gaussian, we modify the Rank-Ordered Absolute Differences (ROAD) to Rank-Ordered Absolute Differences of mixture of Gaussian and impulse noises (ROADG). It will be more effective to detect impulse noise when the impulse is mixed with Gaussian noise. Combining rightly the ROADG with Optimal Weights Filter (OWF), we obtain a new method to deal with the mixed noise, called Optimal Weights Mixed Filter (OWMF). The simulation results show that the method is effective to remove the mixed noise.

cs.CV↗
arXiv2012

A New Poisson Noise Filter based on Weights Optimization

We propose a new image denoising algorithm when the data is contaminated by a Poisson noise. As in the Non-Local Means filter, the proposed algorithm is based on a weighted linear combination of the bserved image. But in contract to the latter where the weights are defined by a Gaussian kernel, we propose to choose them in an optimal way. First some "oracle" weights are defined by minimizing a very tight upper bound of the Mean Square Error. For a practical application the weights are estimated from the observed image. We prove that the proposed filter converges at the usual optimal rate to the true image. Simulation results are presented to compare the performance of the presented filter with conventional filtering methods.

stat.AP↗
arXiv2012

Support and density of the limit $m$-ary search trees distribution

The space requirements of an $m$-ary search tree satisfies a well-known phase transition: when $m\leq 26$, the second order asymptotics is Gaussian. When $m\geq 27$, it is not Gaussian any longer and a limit $W$ of a complex-valued martingale arises. We show that the distribution of $W$ has a square integrable density on the complex plane, that its support is the whole complex plane, and that it has finite exponential moments. The proofs are based on the study of the distributional equation $ W\egalLoi\sum_{k=1}^mV_k^λW_k$, where $V_1, ..., V_m$ are the spacings of $(m-1)$ independent random variables uniformly distributed on $[0,1]$, $W_1, ..., W_m$ are independent copies of W which are also independent of $(V_1, ..., V_m)$ and $λ$ is a complex number.

math.PR↗
arXiv2011

Limit distributions for multitype branching processes of m-ary search trees

A particular continuous-time multitype branching process is considered, it is the continuous-time embedding of a discrete-time process which is very popular in theoretical computer science: the m-ary search tree (m is an integer). There is a well-known phase transition: when m \leq 26, the asymptotic behavior of the process is Gaussian, but for m \geq 27 it is no more Gaussian and a limit W of a complex-valued martingale arises. Thanks to the branching property it appears as a solution of a smoothing equation of the type Z = e^{-λT}(Z(1) + ... + Z(m)), where λ \in C, the Z(k) are independent copies of Z and T is a R_+-valued random variable, independent of the Z(k). This distributional equation is extensively studied by various approaches. The existence and unicity of solution of the equation are proved by contraction methods. The fact that the distribution of W is absolutely continuous and that its support is the whole complex plane is shown via Fourier analysis. Finally, the existence of exponential moments of W is obtained by considering W as the limit of a complex Mandelbrot cascade.

math.PR↗
arXiv2011

Controlled Total Variation regularization for inverse problems

This paper provides a new algorithm for solving inverse problems, based on the minimization of the $L^2$ norm and on the control of the Total Variation. It consists in relaxing the role of the Total Variation in the classical Total Variation minimization approach, which permits us to get better approximation to the inverse problems. The numerical results on the deconvolution problem show that our method outperforms some previous ones.

cs.CV↗
arXiv2011

Removing Gaussian Noise by Optimization of Weights in Non-Local Means

A new image denoising algorithm to deal with the additive Gaussian white noise model is given. Like the non-local means method, the filter is based on the weighted average of the observations in a neighborhood, with weights depending on the similarity of local patches. But in contrast to the non-local means filter, instead of using a fixed Gaussian kernel, we propose to choose the weights by minimizing a tight upper bound of mean square error. This approach makes it possible to define the weights adapted to the function at hand, mimicking the weights of the oracle filter. Under some regularity conditions on the target image, we show that the obtained estimator converges at the usual optimal rate. The proposed algorithm is parameter free in the sense that it automatically calculates the bandwidth of the smoothing kernel; it is fast and its implementation is straightforward. The performance of the new filter is illustrated by numerical simulations.

stat.OT↗
arXiv2011

Moments, moderate and large deviations for a branching process in a random environment

Let $(Z_{n})$ be a supercritical branching process in a random environment $ξ$, and $W$ be the limit of the normalized population size $Z_{n}/\mathbb{E}[Z_{n}|ξ]$. We show large and moderate deviation principles for the sequence $\log Z_{n}$ (with appropriate normalization). For the proof, we calculate the critical value for the existence of harmonic moments of $W$, and show an equivalence for all the moments of $Z_{n}$. Central limit theorems on $W-W_n$ and $\log Z_n$ are also established.

math.PR↗
arXiv2010

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

Let $(Z_n)$ be a supercritical branching process in a random environment $% ζ$, and $W$ be the limit of the normalized population size $Z_n/\mathbb{E%}(Z_n|ζ)$. We show necessary and sufficient conditions for the existence of weighted moments of $W$ of the form $\E W^α\ell(W)$, where $α\geq 1$, $\ell$ is a positive function slowly varying at $\infty$. In the Galton-Watson case, the results improve those of Bingham and Doney (1974).

math.PR↗