This paper considers two nonlocal regularizations for image recovery, which exploit the spatial interactions in images. We get superior results using preprocessed data as input for the weighted functionals. Applications discussed include image deconvolution and tomographic reconstruction. The numerical results show our method outperforms some previous ones.

Coherent ptychographic imaging experiments often discard the majority of the flux from a light source to define the coherence of the illumination. Even when the coherent flux is sufficient, the stability required during an exposure is another important limiting factor. Partial coherence analysis can considerably reduce these limitations. A partially coherent illumination can often be written as the superposition of a single coherent illumination convolved with a separable translational kernel. This article proposes the gradient decomposition of the probe (GDP), a model that exploits translational kernel separability, coupling the variances of the kernel with the transverse coherence. An efficient first-order splitting algorithm (GDP-ADMM) for solving the proposed nonlinear optimization problem is described. Numerical experiments demonstrate the effectiveness of the proposed method with Gaussian and binary kernel functions in fly-scan measurements. Remarkably, GDP-ADMM using nanoprobes produces satisfactory results even when the ratio between the kernel width and the beam size is more than one, or when the distance between successive acquisitions is twice as large as the beam width.

Recovering a signal from its Fourier magnitude is referred to as phase retrieval, which occurs in different fields of engineering and applied physics. This paper gives a new characterization of the phase retrieval problem. Particularly useful is the analysis revealing that the common gradient-based regularization does not restrict the set of solutions to a smaller set. Specifically focusing on binary signals, we show that a box relaxation is equivalent to the binary constraint for Fourier-types of phase retrieval. We further prove that binary signals can be recovered uniquely up to trivial ambiguities under certain conditions. Finally, we use the characterization theorem to develop an efficient denoising algorithm.

In this paper we address the problem of recovering an image from a sequence of distorted versions of it, where the distortion is caused by what is commonly referred to as ground-level turbulence. In mathematical terms, such distortion can be described as the cumulative effect of a blurring kernel and a time-dependent deformation of the image domain. We introduce a statistical dynamic model for the generation of turbulence based on linear dynamical systems (LDS). We expand the model to include the unknown image as part of the unobserved state and apply Kalman filtering to estimate such state. This operation yields a blurry image where the blurring kernel is effectively isoplanatic. Applying blind nonlocal Total Variation (NL-TV) deconvolution yields a sharp final result. © 2013 Springer Science+Business Media New York.

We present a method to enhance the quality of a video sequence captured through a turbulent atmospheric medium, and give an estimate of the radiance of the distant scene, represented as a "latent image," which is assumed to be static throughout the video. Due to atmospheric turbulence, temporal averaging produces a blurred version of the scene's radiance. We propose a method combining Sobolev gradient and Laplacian to stabilize the video sequence, and a latent image is further found utilizing the "lucky region" method. The video sequence is stabilized while keeping sharp details, and the latent image shows more consistent straight edges. We analyze the well- posedness for the stabilizing PDE and the linear stability of the numerical scheme. © 2013 American Institute of Mathematical Sciences.

In this paper, we study the L ₁ /L ₂ minimization on the gradient for imaging applications. Several recent works have demonstrated that L ₁ /L ₂ is better than the L ₁ norm when approximating the L ₀ norm to promote sparsity. Consequently, we postulate that applying L ₁ /L ₂ on the gradient is better than the classic total variation (the L ₁ norm on the gradient) to enforce the sparsity of the image gradient. Numerically, we design a specific splitting scheme, under which we can prove subsequential and global convergence for the alternating direction method of multipliers (ADMM) under certain conditions. Experimentally, we demonstrate visible improvements of L ₁ /L ₂ over L ₁ and other nonconvex regularizations for image recovery from low-frequency measurements and two medical applications of MRI and CT reconstruction. Finally, we reveal some empirical evidence on the superiority of L ₁ /L ₂ over L ₁ when recovering piecewise constant signals from low-frequency measurements to shed light on future works.