Abstract. In this paper, we derive a new class of methods for the classic 2D phase unwrapping problem of recovering a phase function from its wrapped form. For this, we consider the wrapped phase as a wavefront aberration in an optical system, and use reconstruction methods for (digital) wavefront sensors for its recovery. The key idea is that mathematically, common wavefront sensors are insensitive to whether an incoming wavefront is wrapped or not. However, typical reconstructors for these sensors are optimized to compute smooth wavefronts. Thus, digitally “propagating" a wrapped phase through such a sensor and then applying one of these reconstructors results in a smooth unwrapped phase. First, we show how this principle can be applied to derive phase unwrapping algorithms based on digital Shack–Hartmann and Fourier-type wavefront sensors. Then, we numerically test our approach on an unwrapping problem appearing in a free-space optical communications project currently under development, and compare the results to those obtained with other state-of-the-art algorithms.

Abstract. Cardiac pulsations in the human brain have received recent interest due to their possible role in the pathogenesis of neurodegenerative diseases. Further interest stems from their possible application as an endogenous signal source that can be utilized for brain imaging in general. The (pulse-)wave describing the blood flow velocity along an intracranial artery consists of a forward (anterograde) and a backward (retrograde, reflected) part, but measurements of this wave usually consist of a superposition of these components. In this paper, we provide a mathematical framework for the inverse problem of estimating the pulse wave velocity, as well as the forward and backward components of the pulse wave separately from MRI measurements on intracranial arteries. After a mathematical analysis of this problem, we consider possible reconstruction approaches and derive an alternate direction approach for its solution. The resulting methods provide estimates for anterograde/retrograde wave forms and the pulse wave velocity under specified assumptions on a cerebrovascular model system. Numerical experiments on simulated and experimental data demonstrate the applicability and preliminary in vivo feasibility of the proposed methods.

Abstract. In this paper, we investigate regularization of linear inverse problems with irregular noise. In particular, we consider the case that the noise can be preprocessed by certain adjoint embedding operators. By introducing the consequent preprocessed problem, we provide convergence analysis for general regularization schemes under standard assumptions. Furthermore, for a special case of Tikhonov regularization in computerized tomography, we show that our approach leads to a novel (Fourier-based) filtered backprojection algorithm. Numerical examples with different parameter choice rules verify the efficiency of our proposed algorithm.

Abstract. Ground-based astronomy utilizes modern telescopes to obtain information on the universe by analyzing recorded signals. Due to atmospheric turbulence, the reconstruction process requires solving a deconvolution problem with an unknown point spread function (PSF). The crucial step in PSF estimation is to obtain a high-resolution phase from low-resolution phase gradients, which is a challenging problem. In this paper, when multiple frames of low-resolution phase gradients are available, we introduce PhaseNet, a deep learning approach based on the Taylor frozen flow hypothesis. Our approach incorporates a data-driven residual regularization term, of which the gradient is parameterized by a network, into the Laplacian regularization based model. To solve the model, we unroll the Nesterov accelerated gradient algorithm so that the network can be efficiently and effectively trained. Finally, we evaluate the performance of PhaseNet under various atmospheric conditions and demonstrate its superiority over TV and Laplacian regularization based methods.

Abstract. In this paper, we consider the problem of reconstructing a galaxy’s stellar population-kinematic distribution function from optical integral field unit measurements. These quantities are connected via a high-dimensional integral equation. To solve this problem, we propose a projected Nesterov–Kaczmarz reconstruction method, which efficiently leverages the problem structure and incorporates physical prior information such as smoothness and nonnegativity constraints. To test the performance of our reconstruction approach, we apply it to a dataset simulated from a known ground truth density, and validate it by comparing our recoveries to those obtained by the widely used pPXF software.

Abstract. In this paper, we consider subaperture-based approaches for the digital aberration correction (DAC) of optical coherence tomography (OCT) images. In particular, we introduce a mathematical framework for describing this class of approaches, leading to new insights for the subaperture-correlation method. Furthermore, we propose a novel DAC approach requiring only minimal statistical assumptions on the spectral phase of the scanned object. Finally, we demonstrate the applicability of our novel DAC approach via numerical examples based on both simulated and experimental OCT data.

Adaptive optics is a commonly used technique to correct the phase distortions caused by the Earth's atmosphere to improve the image quality of the ground-based imaging systems. However, the observed images still suffer from the blur caused by the adaptive optics residual wavefront. In this paper, we propose a model for reconstructing the residual phase in high resolution from a sequence of deformable mirror data. Our model is based on the turbulence statistics and the Taylor frozen flow hypothesis with knowledge of the wind velocities in atmospheric turbulence layers. A tomography problem for the phase distortions from different altitudes is solved in order to get a high quality phase reconstruction. We also consider inexact tomography operators resulting from the uncertainty in the wind velocities. The wind velocities are estimated from the deformable mirror data and, additionally, by including them as unknowns in the objective function. We provide a theoretical analysis on the existence of a minimizer of the objective function. To solve the associated joint optimization problem, we use an alternating minimization method which results in a high resolution reconstruction algorithm with adaptive wind velocities. Numerical simulations are carried out to show the effectiveness of our approach.

In this paper we consider the simultaneous reconstruction and segmentation of a function $f$ from measurements $g=Kf$, where $K$ is a linear operator. Assuming that the inversion of $K$ is ill-posed, regularization methods have to be used for the inversion process in case of inexact data. We propose using a Mumford--Shah-type functional for the stabilization of the inversion. Restricting our analysis to the recovery of piecewise constant functions, we investigate the existence of minimizers, their stability, and the regularization properties of our approach. Finally, we present a numerical example from single photon emission computed tomography (SPECT).