Deconvolution in white noise with a random blurring function

在随机掩码成像中的盲反卷积提升：可识别性与凸松弛

Lifting for Blind Deconvolution in Random Mask Imaging: Identifiability and Convex Relaxation

本文分析了在编码成像系统中图像与未知模糊的盲反卷积问题，通过提升方法将测量转换为矩阵外积的线性方程组，证明在特定条件下通过核范数最小化可实现高概率恢复。

A SURE Approach for Digital Signal/Image Deconvolution Problems

In this paper, we are interested in the classical problem of restoring data degraded by a convolution and the addition of a white Gaussian noise. The originality of the proposed approach is two-fold. Firstly, we formulate the restoration problem as a nonlinear estimation problem leading to the minimization of a criterion derived from Stein's unbiased quadratic risk estimate. Secondly, the deconvolution procedure is performed using any analysis and synthesis frames that can be overcomplete or not. New theoretical results concerning the calculation of the variance of the Stein's risk estimate ar…

Random action of compact Lie groups and minimax estimation of a mean pattern

This paper considers the problem of estimating a mean pattern in the setting of Grenander's pattern theory. Shape variability in a data set of curves or images is modeled by the random action of elements in a compact Lie group on an infinite dimensional space. In the case of observations contaminated by an additive Gaussian white noise, it is shown that estimating a reference template in the setting of Grenander's pattern theory falls into the category of deconvolution problems over Lie groups. To obtain this result, we build an estimator of a mean pattern by using Fourier deconvolution and ha…

线性反问题中的半参数 Bernstein-von Mises 定理

Semi-parametric Bernstein-von Mises in Linear Inverse Problems

本文在信号加白噪声模型中研究反问题标量参数的贝叶斯恢复，建立其边际后验的 Bernstein-von Mises 定理，并应用于热扩散系数恢复和半盲反卷积位置参数恢复。

Minimax detection of localized signals in statistical inverse problems

We investigate minimax testing for detecting local signals or linear combinations of such signals when only indirect data is available. Naturally, in the presence of noise, signals that are too small cannot be reliably detected. In a Gaussian white noise model, we discuss upper and lower bounds for the minimal size of the signal such that testing with small error probabilities is possible. In certain situations we are able to characterize the asymptotic minimax detection boundary. Our results are applied to inverse problems such as numerical differentiation, deconvolution and the inversion of …

Regularization of linear ill-posed problems involving multiplication operators

We study regularization of ill-posed equations involving multiplication operators when the multiplier function is positive almost everywhere and zero is an accumulation point of the range of this function. Such equations naturally arise from equations based on non-compact self-adjoint operators in Hilbert space, after applying unitary transformations arising out of the spectral theorem. For classical regularization theory, when noisy observations are given and the noise is deterministic and bounded, then non-compactness of the ill-posed equations is a minor issue. However, for statistical ill-…

Adaptive estimation of linear functionals in the convolution model and applications

We consider the model $Z_i=X_i+\varepsilon_i$, for i.i.d. $X_i$'s and $\varepsilon_i$'s and independent sequences $(X_i)_{i\in{\mathbb{N}}}$ and $(\varepsilon_i)_{i\in{\mathbb{N}}}$. The density $f_{\varepsilon}$ of $\varepsilon_1$ is assumed to be known, whereas the one of $X_1$, denoted by $g$, is unknown. Our aim is to estimate linear functionals of $g$, $<ψ,g>$ for a known function $ψ$. We propose a general estimator of $<ψ,g>$ and study the rate of convergence of its quadratic risk as a function of the smoothness of $g$, $f_{\varepsilon}$ and $ψ$. Different contexts with dependent data, s…