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. 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.

We consider a problem of quantitative static elastography, the estimation of the Lamé parameters from internal displacement field data. This problem is formulated as a nonlinear operator equation. To solve this equation, we investigate the Landweber iteration both analytically and numerically. The main result of this paper is the verification of a nonlinearity condition in an infinite dimensional Hilbert space context. This condition guarantees convergence of iterative regularization methods. Furthermore, numerical examples for recovery of the Lamé parameters from displacement data simulating a static elastography experiment are presented.