No Arabic abstract
In order to determine the 3D structure of a thick sample, researchers have recently combined ptychography (for high resolution) and tomography (for 3D imaging) in a single experiment. 2-step methods are usually adopted for reconstruction, where the ptychography and tomography problems are often solved independently. In this paper, we provide a novel model and ADMM-based algorithm to jointly solve the ptychography-tomography problem iteratively, also employing total variation regularization. The proposed method permits large scan stepsizes for the ptychography experiment, requiring less measurements and being more robust to noise with respect to other strategies, while achieving higher reconstruction quality results.
A qualitative comparison of total variation like penalties (total variation, Huber variant of total variation, total generalized variation, ...) is made in the context of global seismic tomography. Both penalized and constrained formulations of seismic recovery problems are treated. A number of simple iterative recovery algorithms applicable to these problems are described. The convergence speed of these algorithms is compared numerically in this setting. For the constrained formulation a new algorithm is proposed and its convergence is proven.
Here we present new joint reconstruction and regularization techniques inspired by ideas in microlocal analysis and lambda tomography, for the simultaneous reconstruction of the attenuation coefficient and electron density from X-ray transmission (i.e., X-ray CT) and backscattered data (assumed to be primarily Compton scattered). To demonstrate our theory and reconstruction methods, we consider the parallel line segment acquisition geometry of Webber and Miller (Compton scattering tomography in translational geometries. Inverse Problems 36, no. 2 (2020): 025007), which is motivated by system architectures currently under development for airport security screening. We first present a novel microlocal analysis of the parallel line geometry which explains the nature of image artefacts when the attenuation coefficient and electron density are reconstructed separately. We next introduce a new joint reconstruction scheme for low effective $Z$ (atomic number) imaging ($Z<20$) characterized by a regularization strategy whose structure is derived from lambda tomography principles and motivated directly by the microlocal analytic results. Finally we show the effectiveness of our method in combating noise and image artefacts on simulated phantoms.
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.
We study inverse problems for the Poisson equation with source term the divergence of an $mathbf{R}^3$-valued measure, that is, the potential $Phi$ satisfies $$ Delta Phi= text{div} boldsymbol{mu}, $$ and $boldsymbol{mu}$ is to be reconstructed knowing (a component of) the field grad $Phi$ on a set disjoint from the support of $boldsymbol{mu}$. Such problems arise in several electro-magnetic contexts in the quasi-static regime, for instance when recovering a remanent magnetization from measurements of its magnetic field. We develop methods for recovering $boldsymbol{mu}$ based on total variation regularization. We provide sufficient conditions for the unique recovery of $boldsymbol{mu}$, asymptotically when the regularization parameter and the noise tend to zero in a combined fashion, when it is uni-directional or when the magnetization has a support which is sparse in the sense that it is purely 1-unrectifiable. Numerical examples are provided to illustrate the main theoretical results.
In this paper we extend the state-of-the-art filtered backprojection (FBP) method with application of the concept of Total Variation regularization. We compare the performance of the new algorithm with the most common form of regularizing in the FBP image reconstruction via apodizing functions. The methods are validated in terms of cross-correlation coefficient between reconstructed and real image of radioactive tracer distribution using standard Derenzo-type phantom. We demonstrate that the proposed approach results in higher cross-correlation values with respect to the standard FBP method.