TY - JOUR AB - Ptychography is a computational imaging technique that aims to reconstruct the object of interest from a set of diffraction patterns. Each of these is obtained by a localized illumination of the object, which is shifted after each illumination to cover its whole domain. Because in the resulting measurements the phase information is lost, ptychography gives rise to solving a phase retrieval problem. In this work, we consider ptychographic measurements contaminated by a background signal. Such a background is caused by imperfections in the experimental setup and appears as a signal that is added to the diffraction patterns. The background is assumed to be independent of the shift of the object, i.e., it is the same for all diffraction patterns. Two algorithms are provided, for arbitrary objects and for so-called phase objects that do not absorb the light but only scatter it. For the second type, a uniqueness of reconstruction is established for almost every object. Our approach is based on the Wigner distribution deconvolution, which lifts the object to a higher-dimensional matrix space where the recovery can be reformulated as a linear problem. The background only affects a few equations of the linear system that are therefore discarded. The lost information is then restored using redundancy in the higher-dimensional space. AU - Melnyk, O. AU - Römer, P. C1 - 72506 C2 - 56602 CY - 3600 Univ City Science Center, Philadelphia, Pa 19104-2688 Usa SP - 1978-2014 TI - Background removal for ptychography via wigner distribution deconvolution. JO - SIAM J. Imaging Sci. VL - 17 IS - 3 PB - Siam Publications PY - 2024 SN - 1936-4954 ER - TY - JOUR AB - Ptychography is a lensless imaging technique, which considers reconstruction from a set of far-field diffraction patterns obtained by illuminating small overlapping regions of the specimen. In many cases, the distribution of light inside the illuminated region is unknown and has to be estimated along with the object of interest. This problem is referred to as blind ptychography. While in ptychography the illumination is commonly assumed to have a point spectrum, in this paper we consider an alternative scenario with a nontrivial light spectrum known as blind polychromatic ptychography. First, we show that nonblind polychromatic ptychography can be seen as a recovery from quadratic measurements. Then, a reconstruction from such measurements can be performed by a variant of the Amplitude Flow algorithm, which has guaranteed sublinear convergence to a critical point. Second, we address recovery from blind polychromatic ptychographic measurements by devising an alternating minimization version of Amplitude Flow and showing that it converges to a critical point at a sublinear rate. AU - Filbir, F. AU - Melnyk, O. C1 - 69082 C2 - 53847 CY - 3600 Univ City Science Center, Philadelphia, Pa 19104-2688 Usa SP - 1308-1337 TI - Image recovery for blind polychromatic ptychography. JO - SIAM J. Imaging Sci. VL - 16 IS - 3 PB - Siam Publications PY - 2023 SN - 1936-4954 ER - TY - JOUR AB - In this paper we introduce the notion of second-order total generalized variation (TGV) regularization for manifold-valued data in a discrete setting. We provide an axiomatic approach to formalize reasonable generalizations of TGV to the manifold setting and present two possible concrete instances that fulfill the proposed axioms. We provide well-posedness results and present algorithms for a numerical realization of these generalizations to the manifold setup. Further, we provide experimental results for synthetic and real data to further underpin the proposed generalization numerically and show its potential for applications with manifold-valued data. AU - Bredies, K.* AU - Holler, M.* AU - Storath, M.* AU - Weinmann, A. C1 - 54546 C2 - 45585 CY - 3600 Univ City Science Center, Philadelphia, Pa 19104-2688 Usa SP - 1785-1848 TI - Total generalized variation for manifold-valued data. JO - SIAM J. Imaging Sci. VL - 11 IS - 3 PB - Siam Publications PY - 2018 SN - 1936-4954 ER - TY - JOUR AB - In many image and signal processing applications, such as interferometric synthetic aperture radar (SAR), electroencephalogram (EEG) data analysis, ground-based astronomy, and color image restoration, in HSV or LCh spaces the data has its range on the one-dimensional sphere S-1. Although the minimization of total variation (TV) regularized functionals is among the most popular methods for edge-preserving image restoration, such methods were only very recently applied to cyclic structures. However, as for Euclidean data, TV regularized variational methods suffer from the so-called staircasing effect. This effect can be avoided by involving higher order derivatives into the functional. This is the first paper which uses higher order differences of cyclic data in regularization terms of energy functionals for image restoration. We introduce absolute higher order differences for S-1-valued data in a sound way which is independent of the chosen representation system on the circle. Our absolute cyclic first order difference is just the geodesic distance between points. Similar to the geodesic distances, the absolute cyclic second order differences have only values in [0, pi]. We update the cyclic variational TV approach by our new cyclic second order differences. To minimize the corresponding functional we apply a cyclic proximal point method which was recently successfully proposed for Hadamard manifolds. Choosing appropriate cycles this algorithm can be implemented in an efficient way. The main steps require the evaluation of proximal mappings of our cyclic differences for which we provide analytical expressions. Under certain conditions we prove the convergence of our algorithm. Various numerical examples with artificial as well as real-world data demonstrate the advantageous performance of our algorithm. AU - Bergmann, R.* AU - Laus, F.* AU - Steidl, G.* AU - Weinmann, A. C1 - 43159 C2 - 36036 SP - 2916-2953 TI - Second order differences of cyclic data and applications in variational denoising. JO - SIAM J. Imaging Sci. VL - 7 IS - 4 PY - 2014 SN - 1936-4954 ER - TY - JOUR AB - We propose a fast splitting approach to the classical variational formulation of the image partitioning problem, which is frequently referred to as the Potts or piecewise constant Mumford--Shah model. For vector-valued images, our approach is significantly faster than the methods based on graph cuts and convex relaxations of the Potts model which are presently the state-of-the-art. The computational costs of our algorithm only grow linearly with the dimension of the data space which contrasts the exponential growth of the state-of-the-art methods. This allows us to process images with high-dimensional codomains such as multispectral images. Our approach produces results of a quality comparable with that of graph cuts and the convex relaxation strategies, and we do not need an a priori discretization of the label space. Furthermore, the number of partitions has almost no influence on the computational costs, which makes our algorithm also suitable for the reconstruction of piecewise constant (color or vectorial) images.   AU - Storath, M.* AU - Weinmann, A. C1 - 32610 C2 - 35158 SP - 1826-1852 TI - Fast partitioning of vector-valued images. JO - SIAM J. Imaging Sci. VL - 7 IS - 3 PY - 2014 SN - 1936-4954 ER - TY - JOUR AB - We consider total variation minimization for manifold valued data. We propose a cyclic proximal point algorithm and a parallel proximal point algorithm to minimize TV functionals with $ell^p$-type data terms in the manifold case. These algorithms are based on iterative geodesic averaging which makes them easily applicable to a large class of data manifolds. As an application, we consider denoising images which take their values in a manifold. We apply our algorithms to diffusion tensor images, interferometric SAR images as well as sphere and cylinder valued images. For the class of Cartan-Hadamard manifolds (which includes the data space in diffusion tensor imaging) we show the convergence of the proposed TV minimizing algorithms to a global minimizer. AU - Weinmann, A. AU - Demaret, L. AU - Storath, M.* C1 - 32609 C2 - 35157 SP - 2226-2257 TI - Total variation regularization for manifold-valued data. JO - SIAM J. Imaging Sci. VL - 7 IS - 4 PY - 2014 SN - 1936-4954 ER -