Following the pioneering works of Rudin, Osher and Fatemi on total variation (TV) and of Buades, Coll and Morel on non-local means (NL-means), the last decade has seen a large number of denoising methods mixing these two approaches, starting with the nonlocal total variation (NLTV) model. The present article proposes an analysis of the NLTV model for image denoising as well as a number of improvements, the most important of which being to apply the denoising both in the space domain and in the Fourier domain, in order to exploit the complementarity of the representation of image data. A local version obtained by a regionwise implementation followed by an aggregation process, called Local Spatial-Frequency NLTV (L-SFNLTV) model, is finally proposed as a new reference algorithm for image denoising among the family of approaches mixing TV and NL operators. The experiments show the great performance of L-SFNLTV in terms of image quality and of computational speed, comparing with other recently proposed NLTV-related methods.

Let ${\left(Z_n\right)}_{n\ge 0}$ be a supercritical branching process with offspring distribution $p_r=\mathbb{P}(Z_1=r),r\ge 0,$ such that $p_0=p_1=\dots =p_{k_0-1}=0$ and $p_{k_0}>0$, where $k_0\ge 2$. The Lotka–Nagaev estimator $Z_{n+1}/Z_n$ is an important estimator for the offspring mean $m={\sum }_{i=k_0}^{\infty }i{p}_i$. In this paper, we establish a self-normalized large deviation result and self-normalized Cramér type moderate deviations for the Lotka–Nagaev estimator. An application to constructing confidence intervals for m is also discussed.

We first establish a law of large numbers and a convergence theorem in distribution to show the rate of convergence of the non-local means filter for removing Gaussian noise. We then introduce the notion of degree of similarity to measure the role of similarity for the non-local means filter. Based on the convergence theorems, we propose a patch-based weighted means filter for removing impulse noise and its mixture with Gaussian noise by combining the essential idea of the trilateral filter and that of the non-local means filter. Our experiments show that our filter is competitive compared to recently proposed methods.