arXiv subjects

Otmar Scherzer

Publications and source records attributed to Otmar Scherzer.

At least 19 recordsLinked to original sources

arXiv2026

On Fourier Phase Retrieval from Differential Intensity Measurements with Applications to Wavefront Sensing

In this paper, we consider Fourier phase retrieval from differential intensity measurements, i.e., the problem of determining the phase of a complex-valued function from a series of intensity measurements differing only by slight modulations in Fourier domain. These modulations may be induced by optical elements such as prisms or phase plates, or via spatial-light modulators. Generalizing the principles behind the transport of intensity equation, we show that given such differential intensity measurements, the phase can be determined as the solution of certain partial differential or integro-differential equations. This is then used to design efficient reconstruction algorithms for a number of Fourier-type wavefront sensors, such as the pyramid wavefront sensor, commonly used in adaptive optics. Numerical experiments illustrate the usefulness of our proposed approach.

math.NA↗

arXiv2026

Quantitative Multi-Modal Optical Coherence Photoacoustic Elastography

We present a novel multi-modal optical coherence photoacoustic elastography framework, which combines two imaging modalities, optical coherence tomography and photoacoustic tomography, to enable complementary absorption-scattering measurements for the extraction of quantitative tissue features via quasi-static elastography. For this, we develop a sophisticated hybrid inversion algorithm for merging the complementary information layers contained in both optical coherence and photoacoustic-based elastography measurements, and perform systematic evaluations to assess the impact of hybrid elastography data on strain and stiffness reconstructions. An extension to a photoacoustic tomography and optical coherence tomography imaging system allows precise elastographic experiments. Studies on a silicone elastomer phantom demonstrate that the combined optical coherence - photoacoustic approach outperforms single-modality optical coherence elastography and photoacoustic elastography, yielding higher strain signal-to-noise ratios and improved stiffness estimates. These results establish the advantage of multi-modal complementary imaging and data merging for accurate, high-resolution elastographic strain and stiffness mapping in both scattering and absorbing materials.

physics.med-ph↗

arXiv2026

Neural network parametrized level sets for image segmentation

Chan-Vese algorithms have proven to be a first-class method for image segmentation. Early implementations used level set methods with a pixelwise representation of the level set function. Later, parametrized level set approximations, such as splines, have been studied and computationally developed to improve efficiency. In this paper, we use neural networks as parametrized approximations of level set functions for implementing the Chan-Vese methods. We show that this approach is efficient because of the equivalence between two layer neural networks and polygonal approximations of level set-based segmentations. In turn, this allows the two-layer network architecture to be interpreted as an ansatz function for the approximate minimization of Chan-Vese functionals. Based on these theory, we extend the classical Chan-Vese algorithm to a data-driven setting, where prior parameters of the network are obtained through unsupervised training on representative image data. These learned parameters encode geometric structures of the data, leading to improved initialization and faster convergence of the Chan-Vese image segmentation.

math.NA↗

arXiv2026

Pressure beneath a periodic travelling water-wave in constant-vorticity flow over a flat bed

We investigate within the framework of linear theory the behaviour of the total (hydrodynamic) pressure and of the dynamic pressure in a regular wave train which propagates at the surface of water with a flat bed in a flow with constant vorticity. We show that nonzero vorticity, the hallmark of a non-uniform underlying current, may strongly alter the behaviour with respect to the case of irrotational flows, for which the maximum and minimum of the dynamic pressure always occur at the wave crest and at the wave trough, respectively (the extrema of the dynamic pressure may occur along the flat bed or along the critical level, depending on the vorticity strength). While vorticity does not modify the increase of the hydrodynamic pressure with depth, it can significantly alter the location of the extrema of the hydrodynamic pressure at a fixed depth level.

physics.flu-dyn↗

arXiv2026

Raster Scan Diffraction Tomography

Diffraction tomography is a widely used inverse scattering technique for quantitative imaging of weakly scattering media. In its conventional formulation, diffraction tomography assumes monochromatic plane wave illumination. This assumption, however, represents a simplification that often fails to reflect practical imaging systems such as medical ultrasound, where focused beams are used to scan a region of interest of the human body. Such measurement setups, combining focused illumination with scanning, have not yet been incorporated into the diffraction tomography framework. To bridge this gap, we extend diffraction tomography by modeling incident fields as Herglotz waves, thereby incorporating focused beams into the theory. Within this setting, we derive a new Fourier diffraction relation, which forms the basis for quantitative tomographic reconstruction from scanning data. Using this result, we systematically analyze how different scan geometries influence the reconstruction.

math.NA↗

arXiv2025

Neural operators for solving nonlinear inverse problems

We consider solving a probably infinite dimensional operator equation, where the operator is not modeled by physical laws but is specified indirectly via training pairs of the input-output relation of the operator. Neural operators have proven to be efficient to approximate infinite dimensional operators. In this paper we analyze Tikhonov regularization with neural operators as surrogates for solving ill-posed operator equations. The analysis is based on balancing approximation errors of neural operators, regularization parameters, and noise. Moreover, we extend the approximation properties of neural operators from sets of continuous functions to Sobolev and Lebesgue spaces, which is crucial for solving inverse problems and we discuss the problem of finding an appropriate network structure of neural operators (training). Finally, we present some numerical experiments.

math.NA↗

arXiv2025

Addendum on data driven regularization by projection

We study the stability of regularization by projection for solving linear inverse problems if the forward operator is given indirectly but specified via some input-output training pairs. We extend the approach in "Data driven regularization by projection" (Aspri, Korolev, and Scherzer; Inverse Problems; 36 (2020), 125009) to data pairs, which are noisy and, possibly, linearly dependent.

math.NA↗

arXiv2025

Regularization of Nonlinear Inverse Problems -- From Functional Analysis to Data-Driven Approaches

The focus of this book is on the analysis of regularization methods for solving \emph{nonlinear inverse problems}. Specifically, we place a strong emphasis on techniques that incorporate supervised or unsupervised data derived from prior experiments. This approach enables the integration of data-driven insights into the solution of inverse problems governed by physical models. \emph{Inverse problems}, in general, aim to uncover the \emph{inner mechanisms} of an observed system based on indirect or incomplete measurements. This field has far-reaching applications across various disciplines, such as medical or geophysical imaging, as well as, more broadly speaking, industrial processes where identifying hidden parameters is essential.

math.OC↗

arXiv2025

Deciphering scrolls with tomography: A training experiment

The recovery of severely damaged ancient written documents has proven to be a major challenge for many scientists, mainly due to the impracticality of physical unwrapping them. Non-destructive techniques, such as X-ray computed tomography (CT), combined with computer vision algorithms, have emerged as a means of facilitating the virtual reading of the hidden contents of the damaged documents. This paper proposes an educational laboratory aimed at simulating the entire process of acquisition and virtual recovery of the ancient works. We have developed an experimental setup that uses visible light to replace the detrimental X-rays, and a didactic software pipeline that allows students to virtually reconstruct a transparent rolled sheet with printed text on it, the wrapped scroll.

eess.IV↗

arXiv2024

Vertex characterization via second-order topological derivatives

This paper focuses on identifying vertex characteristics in 2D images using topological asymptotic analysis. Vertex characteristics include both the location and the type of the vertex, with the latter defined by the number of lines forming it and the corresponding angles. This problem is crucial for computer vision tasks, such as distinguishing between fore- and background objects in 3D scenes. We compute the second-order topological derivative of a Mumford-Shah type functional with respect to inclusion shapes representing various vertex types. This derivative assigns a likelihood to each pixel that a particular vertex type appears there. Numerical tests demonstrate the effectiveness of the proposed approach.

math.NA↗

arXiv2024

Spectral Function Space Learning and Numerical Linear Algebra Networks for Solving Linear Inverse Problems

We consider solving a probably ill-conditioned linear operator equation, where the operator is not modeled by physical laws but is specified via training pairs (consisting of images and data) of the input-output relation of the operator. We derive a stable method for computing the operator, which consists of first a Gram-Schmidt orthonormalization of images and a principal component analysis of the data. This two-step algorithm provides a spectral decomposition of the linear operator. Moreover, we show that both Gram-Schmidt and principal component analysis can be written as a deep neural network, which relates this procedure to de-and encoder networks. Therefore, we call the two-step algorithm a linear algebra network. Finally, we provide numerical simulations showing the strategy is feasible for reconstructing spectral functions and for solving operator equations without explicitly exploiting the physical model.

math.NA↗

arXiv2024

Using the Transport of Intensity and the Transport of Phase Equation for Phase Retrieval

We investigate the transport of intensity equation (TIE) and the transport of phase equation (TPE) for solving the phase retrieval problem. Both the TIE and the TPE are derived from the paraxial Helmholtz equation and relate phase information to the intensity. The TIE is usually favored since the TPE is nonlinear. The main contribution of this paper is a discussion of preferential use of either one of the two equations or potential benefits of a hybrid use. Moreover, we discuss the solution of the TPE with the method of characteristics and with viscosity methods. Both the TIE and the viscosity method are numerically implemented with finite element methods.

math.NA↗

arXiv2024

Computational inverse scattering with internal sources: a reproducing kernel Hilbert space approach

We present a method to reconstruct the dielectric susceptibility (scattering potential) of an inhomogeneous scattering medium, based on the solution to the inverse scattering problem with internal sources. We employ the theory of reproducing kernel Hilbert spaces, together with regularization to recover the susceptibility of two- and three-dimensional scattering media. Numerical examples illustrate the effectiveness of the proposed reconstruction method.

math.NA↗

arXiv2024

Classification with neural networks with quadratic decision functions

Neural networks with quadratic decision functions have been introduced as alternatives to standard neural networks with affine linear ones. They are advantageous when the objects or classes to be identified are compact and of basic geometries like circles, ellipses etc. In this paper we investigate the use of such ansatz functions for classification. In particular we test and compare the algorithm on the MNIST dataset for classification of handwritten digits and for classification of subspecies. We also show, that the implementation can be based on the neural network structure in the software Tensorflow and Keras, respectively.

cs.LG↗

arXiv2024

Diffraction Tomography for a Generalized Incident Field

Diffraction tomography is an inverse scattering technique used to reconstruct the spatial distribution of the material properties of a weakly scattering object. The object is exposed to radiation, typically light or ultrasound, and the scattered waves induced from different incident field angles are recorded. In conventional diffraction tomography, the incident wave is assumed to be a monochromatic plane wave, an unrealistic simplification in practical imaging scenarios. In this article, we extend conventional diffraction tomography by introducing the concept of customized illumination scenarios, with a pronounced emphasis on imaging with focused beams. We present a new forward model that incorporates a generalized incident field and extends the classical Fourier diffraction theorem to the use of this incident field. This yields a new two-step reconstruction process which we comprehensively evaluate through numerical experiments.

math.NA↗

arXiv2023

Quadratic neural networks for solving inverse problems

In this paper we investigate the solution of inverse problems with neural network ansatz functions with generalized decision functions. The relevant observation for this work is that such functions can approximate typical test cases, such as the Shepp-Logan phantom, better, than standard neural networks. Moreover, we show that the convergence analysis of numerical methods for solving inverse problems with shallow generalized neural network functions leads to more intuitive convergence conditions, than for deep affine linear neural networks.

math.NA↗

arXiv2023

Analysis of Generalized Iteratively Regularized Landweber Iterations driven by data

We investigate generalized versions of the Iteratively Regularized Landweber Method, initially introduced in [Appl. Math. Optim., 38(1):45-68, 1998], to address linear and nonlinear ill-posed problems. Our approach is inspired by the data-driven perspective emphasized in the introduction by Aspri et al. [Numer. Funct. Anal. Optim., 41(10):1190-1227, 2020]. We provide a rigorous analysis establishing convergence and stability results and present numerical outcomes for linear operators, with the Radon transform serving as a prototype.

math.NA↗

arXiv2023

On convergence rates of adaptive ensemble Kalman inversion for linear ill-posed problems

In this paper we discuss a deterministic form of ensemble Kalman inversion as a regularization method for linear inverse problems. By interpreting ensemble Kalman inversion as a low-rank approximation of Tikhonov regularization, we are able to introduce a new sampling scheme based on the Nyström method that improves practical performance. Furthermore, we formulate an adaptive version of ensemble Kalman inversion where the sample size is coupled with the regularization parameter. We prove that the proposed scheme yields an order optimal regularization method under standard assumptions if the discrepancy principle is used as a stopping criterion. The paper concludes with a numerical comparison of the discussed methods for an inverse problem of the Radon transform.

math.NA↗