Otmar Scherzer

A. De Cezaro, M. Haltmeier, A. Leitao et al.

We investigate modified steepest descent methods coupled with a loping Kaczmarz strategy for obtaining stable solutions of nonlinear systems of ill-posed operator equations. We show that the proposed method is a convergent regularization method. Numerical tests are presented for a linear problem related to photoacoustic tomography and a non-linear problem related to the testing of semiconductor devices.

Vinicius Albani, Peter Elbau, Maarten V. de Hoop et al.

In this paper, we prove optimal convergence rates results for regularisation methods for solving linear ill-posed operator equations in Hilbert spaces. The result generalises existing convergence rates results on optimality to general source conditions, such as logarithmic source conditions. Moreover, we also provide optimality results under variational source conditions and show the connection to approximative source conditions.

G. Zangerl, O. Scherzer, M. Haltmeier

A method for photoacoustic tomography is presented that uses circular integrals of the acoustic wave for the reconstruction of a three-dimensional image. Image reconstruction is a two-step process: In the first step data from a stack of circular integrating are used to reconstruct the circular projection of the source distribution. In the second step the inverse circular Radon transform is applied. In this article we establish inversion formulas for the first step, which involves an inverse problem for the axially symmetric wave equation. Numerical results are presented that show the validity and robustness of the resulting algorithm.

Roman Andreev, Peter Elbau, Maarten V. de Hoop et al.

In recent years, a series of convergence rates conditions for regularization methods has been developed. Mainly, the motivations for developing novel conditions came from the desire to carry over convergence rates results from the Hilbert space setting to generalized Tikhonov regularization in Banach spaces. For instance, variational source conditions have been developed and they were expected to be equivalent to standard source conditions for linear inverse problems in a Hilbert space setting. We show that this expectation does not hold. However, in the standard Hilbert space setting these novel conditions are optimal, which we prove by using some deep results from Neubauer, and generalize existing convergence rates results. The key tool in our analysis is a novel source condition, which we put into relation to the existing source conditions from the literature. As a positive by-product, convergence rates results can be proven without spectral theory, which is the standard technique for proving convergence rates for linear inverse problems in Hilbert spaces.

Maarten V. de Hoop, Lingyun Qiu, Otmar Scherzer

We consider nonlinear inverse problems described by operator equations in Banach spaces. Assuming conditional stability of the inverse problem, that is, assuming that stability holds on a closed, convex subset of the domain of the operator, we introduce a novel nonlinear projected steepest descent iteration and analyze its convergence to an approximate solution given limited accuracy data. We proceed with developing a multi-level algorithm based on a nested family of closed, convex subsets on which stability holds and the stability constants are ordered. Growth of the stability constants is coupled to the increase in accuracy of approximation between neighboring levels to ensure that the algorithm can continue from level to level until the iterate satisfies a desired discrepancy criterion, after a finite number of steps.

Thomas Glatz, Otmar Scherzer, Thomas Widlak

Elastographic imaging is a widely used technique which can in principle be implemented on top of every imaging modality. In elastography, the specimen is exposed to a force causing local displacements in the probe, and imaging is performed before and during the displacement experiment. From the computed displacements material parameters can be deduced, which in turn can be used for clinical diagnosis. Photoacoustic imaging is an emerging image modality, which exhibits functional and morphological contrast. However, opposed to ultrasound imaging, for instance, it is considered a modality which is not suited for elastography, because it does not reveal speckle patterns. However, this is somehow counter-intuitive, because photoacoustic imaging makes available the whole frequency spectrum as opposed to single frequency standard ultrasound imaging. In this work, we show that in fact artificial speckle patterns can be introduced by using only a band-limited part of the measurement data. We also show that after introduction of artificial speckle patterns, deformation estimation can be implemented more reliably in photoacoustic imaging.

Julian Schmid, Thomas Glatz, Behrooz Zabihian et al.

To obtain the initial pressure from the collected data on a planar sensor arrangement in Photoacoustic tomography, there exists an exact analytic frequency domain reconstruction formula. An efficient realization of this formula needs to cope with the evaluation of the datas Fourier transform on a non-equispaced mesh. In this paper, we use the non-uniform fast Fourier transform to handle this issue and show its feasibility in 3D experiments. This is done in comparison to the standard approach that uses polynomial interpolation. Moreover, we investigate the effect and the utility of flexible sensor location on the quality of photoacoustic image reconstruction. The computational realization is accomplished by the use of a multi-dimensional non-uniform fast Fourier algorithm, where non-uniform data sampling is performed both in frequency and spatial domain. We show that with appropriate sampling the imaging quality can be significantly improved. Reconstructions with synthetic and real data show the superiority of this method.

Guozhi Dong, Bert Juettler, Otmar Scherzer et al.

We study Tikhonov regularization for solving ill--posed operator equations where the solutions are functions defined on surfaces. One contribution of this paper is an error analysis of Tikhonov regularization which takes into account perturbations of the surfaces, in particular when the surfaces are approximated by spline surfaces. Another contribution is that we highlight the analysis of regularization for functions with range in vector bundles over surfaces. We also present some practical applications, such as an inverse problem of gravimetry and an imaging problem for denoising vector fields on surfaces, and show the numerical verification.

Guozhi Dong, Otmar Scherzer

In this paper we derive nonlinear evolution equations associated with a class of non-convex energy functionals which can be used for correcting displacement errors in imaging data. We study properties of these filtering flows and provide experiments for correcting angular perturbations in tomographical data.

Fabian Parzer, Prashin Jethwa, Alina Boecker et al.

Context. Blob detection is a common problem in astronomy. One example is in stellar population modelling, where the distribution of stellar ages and metallicities in a galaxy is inferred from observations. In this context, blobs may correspond to stars born in-situ versus those accreted from satellites, and the task of blob detection is to disentangle these components. A difficulty arises when the distributions come with significant uncertainties, as is the case for stellar population recoveries inferred from modelling spectra of unresolved stellar systems. There is currently no satisfactory method for blob detection with uncertainties. Aims. We introduce a method for uncertainty-aware blob detection developed in the context of stellar population modelling of integrated-light spectra of stellar systems. Methods. We develop theory and computational tools for an uncertainty-aware version of the classic Laplacian-of-Gaussians method for blob detection, which we call ULoG. This identifies significant blobs considering a variety of scales. As a prerequisite to apply ULoG to stellar population modelling, we introduce a method for efficient computation of uncertainties for spectral modelling. This method is based on the truncated Singular Value Decomposition and Markov Chain Monte Carlo sampling (SVD-MCMC). Results. We apply the methods to data of the star cluster M54. We show that the SVD-MCMC inferences match those from standard MCMC, but are a factor 5-10 faster to compute. We apply ULoG to the inferred M54 age/metallicity distributions, identifying between 2 or 3 significant, distinct populations amongst its stars.

Otmar Scherzer, Cong Shi, Thi Lan Nhi Vu

The Chan-Vese functionals have proven to by a first-class method for segmentation and classification. Previously they have been implemented with level-set methods based on a pixel-wise representation of the level-sets. Later parametrized level-set approximations, such as splines, have been studied. In this paper we consider neural networks as parametrized approximations of level-set functions. We show in particular, that parametrized two-layer networks are most efficient to approximate polyhedral segments and classes. We also prove the efficiency for segmentation and classification.

Peter Gangl, Bochra Mejri, Otmar Scherzer

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.

Simon Hubmer, Lukas Weissinger, Ronny Ramlau et al.

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.

Ekaterina Sherina, Lisa Krainz, Wolfgang Drexler et al.

We present a novel multi-modal optical coherence photoacoustic elastography (OCPE) framework, which combines two imaging modalities, optical coherence tomography (OCT) and photoacoustic tomography (PAT), 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 OCT and PAT-based elastography measurements, and perform systematic evaluations to assess the impact of hybrid elastography data on strain and stiffness reconstructions. Studies on a silicone elastomer phantom demonstrate that the combined OCT-PAT approach outperforms single-modality OCT elastography and PAT elastography, yielding higher strain signal-to-noise ratio 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.

Sonia Foschiatti, Axel Kittenberger, Otmar Scherzer

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.

Leon Frischauf, Otmar Scherzer, Cong Shi

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.

Aniello Raffaele Patrone, Christian Valuch, Ulrich Ansorge et al.

Saliency maps are used to understand human attention and visual fixation. However, while very well established for static images, there is no general agreement on how to compute a saliency map of dynamic scenes. In this paper we propose a mathematically rigorous approach to this prob- lem, including static saliency maps of each video frame for the calculation of the optical flow. Taking into account static saliency maps for calculating the optical flow allows for overcoming the aperture problem. Our ap- proach is able to explain human fixation behavior in situations which pose challenges to standard approaches, such as when a fixated object disappears behind an occlusion and reappears after several frames. In addition, we quantitatively compare our model against alternative solutions using a large eye tracking data set. Together, our results suggest that assessing optical flow information across a series of saliency maps gives a highly accurate and useful account of human overt attention in dynamic scenes.

Lukas F. Lang, Otmar Scherzer

In this work we consider optical flow on evolving Riemannian 2-manifolds which can be parametrised from the 2-sphere. Our main motivation is to estimate cell motion in time-lapse volumetric microscopy images depicting fluorescently labelled cells of a live zebrafish embryo. We exploit the fact that the recorded cells float on the surface of the embryo and allow for the extraction of an image sequence together with a sphere-like surface. We solve the resulting variational problem by means of a Galerkin method based on vector spherical harmonics and present numerical results computed from the aforementioned microscopy data.

Aniello Raffale Patrone, Otmar Scherzer

In this paper we present a decomposition algorithm for computation of the spatial-temporal optical flow of a dynamic image sequence. We consider several applications, such as the extraction of temporal motion features and motion detection in dynamic sequences under varying illumination conditions, such as they appear for instance in psychological flickering experiments. For the numerical implementation we are solving an integro-differential equation by a fixed point iteration. For comparison purposes we use a standard time dependent optical flow algorithm, which in contrast to our method, constitutes in solving a spatial-temporal differential equation.

Clemens Kirisits, Lukas F. Lang, Otmar Scherzer

We extend the concept of optical flow with spatiotemporal regularisation to a dynamic non-Euclidean setting. Optical flow is traditionally computed from a sequence of flat images. The purpose of this paper is to introduce variational motion estimation for images that are defined on an evolving surface. Volumetric microscopy images depicting a live zebrafish embryo serve as both biological motivation and test data.