Shape spaces via medial axis transforms for segmentation of complex geometry in 3D voxel data
Jochen Abhau Oswin Aichholzer Sebastian Colutto Bernhard Kornberger Otmar Scherzer
Inverse Problems & Imaging 2013, 7(1): 1-25 doi: 10.3934/ipi.2013.7.1
In this paper we construct a shape space of medial ball representations from given shape training data using methods of Computational Geometry and Statistics. The ultimate goal is to employ the shape space as prior information in supervised segmentation algorithms for complex geometries in 3D voxel data. For this purpose, a novel representation of the shape space (i.e., medial ball representation) is worked out and its implications on the whole segmentation pipeline are studied. Such algorithms have wide applications for industrial processes and medical imaging, when data are recorded under varying illumination conditions, are corrupted with high noise or are occluded.
keywords: skin surfaces medial axis transform Medial ball representation image segmentation EM algorithm. Procrustes analysis
Convergence of Tikhonov regularization for solving ill-posed operator equations with solutions defined on surfaces
Guozhi Dong Bert Jüttler Otmar Scherzer Thomas Takacs
Inverse Problems & Imaging 2017, 11(2): 221-246 doi: 10.3934/ipi.2017011

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.

keywords: Tikhonov regularization convergence vector field ambient space surface approximation magnetization computation
Kaczmarz methods for regularizing nonlinear ill-posed equations II: Applications
Markus Haltmeier Richard Kowar Antonio Leitão Otmar Scherzer
Inverse Problems & Imaging 2007, 1(3): 507-523 doi: 10.3934/ipi.2007.1.507
In part I we introduced modified Landweber--Kaczmarz methods and established a convergence analysis. In the present work we investigate three applications: an inverse problem related to thermoacoustic tomography, a nonlinear inverse problem for semiconductor equations, and a nonlinear problem in Schlieren tomography. Each application is considered in the framework established in the previous part. The novel algorithms show robustness, stability, computational efficiency and high accuracy.
keywords: Ill-posed systems; Landweber--Kaczmarz; Regularization.
Optical flow on evolving sphere-like surfaces
Lukas F. Lang Otmar Scherzer
Inverse Problems & Imaging 2017, 11(2): 305-338 doi: 10.3934/ipi.2017015

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.

keywords: Optical flow sphere-like surfaces vector spherical harmonics variational methods biomedical imaging computer vision
A variational algorithm for the detection of line segments
Elena Beretta Markus Grasmair Monika Muszkieta Otmar Scherzer
Inverse Problems & Imaging 2014, 8(2): 389-408 doi: 10.3934/ipi.2014.8.389
In this paper we propose an algorithm for the detection of edges in images that is based on topological asymptotic analysis. Motivated from the Mumford--Shah functional, we consider a variational functional that penalizes oscillations outside some approximate edge set, which we represent as the union of a finite number of thin strips, the width of which is an order of magnitude smaller than their length. In order to find a near optimal placement of these strips, we compute an asymptotic expansion of the functional with respect to the strip size. This expansion is then employed for defining a (topological) gradient descent like minimization method. As opposed to a recently proposed method by some of the authors, which uses coverings with balls, the usage of strips includes some directional information into the method, which can be used for obtaining finer edges and can also result in a reduction of computation times.
keywords: topological minimization Line segment detection image segmentation.
On a spatial-temporal decomposition of optical flow
Aniello Raffaele Patrone Otmar Scherzer
Inverse Problems & Imaging 2017, 11(4): 761-781 doi: 10.3934/ipi.2017036

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.

keywords: Spatial-temporal optical flow decomposition variational methods
Identifiability and reconstruction of shapes from integral invariants
Thomas Fidler Markus Grasmair Otmar Scherzer
Inverse Problems & Imaging 2008, 2(3): 341-354 doi: 10.3934/ipi.2008.2.341
Integral invariants have been proven to be useful for shape matching and recognition, but fundamental mathematical questions have not been addressed in the computer vision literature. In this article we are concerned with the identifiability and numerical algorithms for the reconstruction of a star-shaped object from its integral invariants. In particular we analyse two integral invariants and prove injectivity for one of them. Additionally, numerical experiments are performed.
keywords: Integral invariants object recognition. Landweber iteration
A variational setting for volume constrained image registration
Christiane Pöschl Jan Modersitzki Otmar Scherzer
Inverse Problems & Imaging 2010, 4(3): 505-522 doi: 10.3934/ipi.2010.4.505
We consider image registration, which is the determination of a geometrical transformation between two data sets. In this paper we propose constrained variational methods which aim for controlling the change of area or volume under registration transformation. We prove an existence result, convergence of a finite element method, and present a simple numerical example for volume-preserving registration.
keywords: image registration area and volume preserving registration. constrained regularization
Kaczmarz methods for regularizing nonlinear ill-posed equations I: convergence analysis
Markus Haltmeier Antonio Leitão Otmar Scherzer
Inverse Problems & Imaging 2007, 1(2): 289-298 doi: 10.3934/ipi.2007.1.289
In this article we develop and analyze novel iterative regularization techniques for the solution of systems of nonlinear ill-posed operator equations. The basic idea consists in considering separately each equation of this system and incorporating a loping strategy. The first technique is a Kaczmarz-type method, equipped with a novel stopping criteria. The second method is obtained using an embedding strategy, and again a Kaczmarz-type approach. We prove well-posedness, stability and convergence of both methods.
keywords: Ill-posed systems; Landweber--Kaczmarz; Regularization.

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