Computing Topology Preservation of RBF Transformations for Landmark-Based Image Registration

In image registration, a proper transformation should be topology preserving. Especially for landmark-based image registration, if the displacement of one landmark is larger enough than those of neighbourhood landmarks, topology violation will be occurred. This paper aim to analyse the topology preservation of some Radial Basis Functions (RBFs) which are used to model deformations in image registration. Mat\'{e}rn functions are quite common in the statistic literature (see, e.g. \cite{Matern86,Stein99}). In this paper, we use them to solve the landmark-based image registration problem. We present the topology preservation properties of RBFs in one landmark and four landmarks model respectively. Numerical results of three kinds of Mat\'{e}rn transformations are compared with results of Gaussian, Wendland's, and Wu's functions

Convergence rates in expectation for Tikhonov-type regularization of Inverse Problems with Poisson data

In this paper we study a Tikhonov-type method for ill-posed nonlinear operator equations \gdag = F( ag) where \gdag is an integrable, non-negative function. We assume that data are drawn from a Poisson process with density t\gdag where $t>0$ may be interpreted as an exposure time. Such problems occur in many photonic imaging applications including positron emission tomography, confocal fluorescence microscopy, astronomic observations, and phase retrieval problems in optics. Our approach uses a Kullback-Leibler-type data fidelity functional and allows for general convex penalty terms. We prove convergence rates of the expectation of the reconstruction error under a variational source condition as $t\to\infty$ both for an a priori and for a Lepski{\u\i}-type parameter choice rule

CANVAS: case report on a novel repeat expansion disorder with late-onset ataxia

This article presents the case of a 74-year-old female patient who first developed a progressive disease with sensory neuropathy, cerebellar ataxia and bilateral vestibulopathy at the age of 60 years. The family history was unremarkable. Magnetic resonance imaging (MRI) showed atrophy of the cerebellum predominantly in the vermis and atrophy of the spinal cord. The patient was given the syndromic diagnosis of cerebellar ataxia, neuropathy, vestibular areflexia syndrome (CANVAS). In 2019 the underlying genetic cause of CANVAS was discovered to be an intronic repeat expansion in the RFC1 gene with autosomal recessive inheritance. The patient exhibited the full clinical picture of CANVAS and was tested positive for this repeat expansion on both alleles. The CANVAS is a relatively frequent cause of late-onset hereditary ataxia (estimated prevalence 5‑13/100,000). In contrast to the present patient, the full clinical picture is not always present. Therefore, testing for the RFC1 gene expansion is recommended in the work-up of patients with otherwise unexplained late-onset sporadic ataxia. As intronic repeat expansions cannot be identified by next generation sequencing methods, specific testing is necessary

Necessary conditions for variational regularization schemes

We study variational regularization methods in a general framework, more precisely those methods that use a discrepancy and a regularization functional. While several sets of sufficient conditions are known to obtain a regularization method, we start with an investigation of the converse question: How could necessary conditions for a variational method to provide a regularization method look like? To this end, we formalize the notion of a variational scheme and start with comparison of three different instances of variational methods. Then we focus on the data space model and investigate the role and interplay of the topological structure, the convergence notion and the discrepancy functional. Especially, we deduce necessary conditions for the discrepancy functional to fulfill usual continuity assumptions. The results are applied to discrepancy functionals given by Bregman distances and especially to the Kullback-Leibler divergence.Comment: To appear in Inverse Problem

Iteratively regularized Newton-type methods for general data misfit functionals and applications to Poisson data

We study Newton type methods for inverse problems described by nonlinear operator equations $F(u)=g$ in Banach spaces where the Newton equations $F'(u_n;u_{n+1}-u_n) = g-F(u_n)$ are regularized variationally using a general data misfit functional and a convex regularization term. This generalizes the well-known iteratively regularized Gauss-Newton method (IRGNM). We prove convergence and convergence rates as the noise level tends to 0 both for an a priori stopping rule and for a Lepski{\u\i}-type a posteriori stopping rule. Our analysis includes previous order optimal convergence rate results for the IRGNM as special cases. The main focus of this paper is on inverse problems with Poisson data where the natural data misfit functional is given by the Kullback-Leibler divergence. Two examples of such problems are discussed in detail: an inverse obstacle scattering problem with amplitude data of the far-field pattern and a phase retrieval problem. The performence of the proposed method for these problems is illustrated in numerical examples

Sparse Regularization with lql^q Penalty Term

We consider the stable approximation of sparse solutions to non-linear operator equations by means of Tikhonov regularization with a subquadratic penalty term. Imposing certain assumptions, which for a linear operator are equivalent to the standard range condition, we derive the usual convergence rate $O(\sqrt{\delta})$ of the regularized solutions in dependence of the noise level $\delta$. Particular emphasis lies on the case, where the true solution is known to have a sparse representation in a given basis. In this case, if the differential of the operator satisfies a certain injectivity condition, we can show that the actual convergence rate improves up to $O(\delta)$.Comment: 15 page

A combined first and second order variational approach for image reconstruction

In this paper we study a variational problem in the space of functions of bounded Hessian. Our model constitutes a straightforward higher-order extension of the well known ROF functional (total variation minimisation) to which we add a non-smooth second order regulariser. It combines convex functions of the total variation and the total variation of the first derivatives. In what follows, we prove existence and uniqueness of minimisers of the combined model and present the numerical solution of the corresponding discretised problem by employing the split Bregman method. The paper is furnished with applications of our model to image denoising, deblurring as well as image inpainting. The obtained numerical results are compared with results obtained from total generalised variation (TGV), infimal convolution and Euler's elastica, three other state of the art higher-order models. The numerical discussion confirms that the proposed higher-order model competes with models of its kind in avoiding the creation of undesirable artifacts and blocky-like structures in the reconstructed images -- a known disadvantage of the ROF model -- while being simple and efficiently numerically solvable.Comment: 34 pages, 89 figure

PDEs for tensor image processing

Methods based on partial differential equations (PDEs) belong to those image processing techniques that can be extended in a particularly elegant way to tensor fields. In this survey paper the most important PDEs for discontinuity-preserving denoising of tensor fields are reviewed such that the underlying design principles becomes evident. We consider isotropic and anisotropic diffusion filters and their corresponding variational methods, mean curvature motion, and selfsnakes. These filters preserve positive semidefiniteness of any positive semidefinite initial tensor field. Finally we discuss geodesic active contours for segmenting tensor fields. Experiments are presented that illustrate the behaviour of all these methods

Endothelin-1 Predicts Hemodynamically Assessed Pulmonary Arterial Hypertension in HIV Infection.

BackgroundHIV infection is an independent risk factor for PAH, but the underlying pathogenesis remains unclear. ET-1 is a robust vasoconstrictor and key mediator of pulmonary vascular homeostasis. Higher levels of ET-1 predict disease severity and mortality in other forms of PAH, and endothelin receptor antagonists are central to treatment, including in HIV-associated PAH. The direct relationship between ET-1 and PAH in HIV-infected individuals is not well described.MethodsWe measured ET-1 and estimated pulmonary artery systolic pressure (PASP) with transthoracic echocardiography (TTE) in 106 HIV-infected individuals. Participants with a PASP ≥ 30 mmHg (n = 65) underwent right heart catheterization (RHC) to definitively diagnose PAH. We conducted multivariable analysis to identify factors associated with PAH.ResultsAmong 106 HIV-infected participants, 80% were male, the median age was 52 years and 77% were on antiretroviral therapy. ET-1 was significantly associated with higher values of PASP [14% per 0.1 pg/mL increase in ET-1, p = 0.05] and PASP ≥ 30 mmHg [PR (prevalence ratio) = 1.24, p = 0.012] on TTE after multivariable adjustment for PAH risk factors. Similarly, among the 65 individuals who underwent RHC, ET-1 was significantly associated with higher values of mean pulmonary artery pressure and PAH (34%, p = 0.003 and PR = 2.43, p = 0.032, respectively) in the multivariable analyses.ConclusionsHigher levels of ET-1 are independently associated with HIV-associated PAH as hemodynamically assessed by RHC. Our findings suggest that excessive ET-1 production in the setting of HIV infection impairs pulmonary endothelial function and contributes to the development of PAH