In multi-modal electron tomography, tilt series of several signals such as X-ray spectra, electron energyloss spectra, annular dark-field, or bright-field data are acquired at the same time in a transmission electron microscope and subsequently reconstructed in three dimensions. However, the acquired data are often incomplete and suffer from noise, and generally each signal is reconstructed independently of all other signals, not taking advantage of correlation between different datasets. This severely limits both the resolution and validity of the reconstructed images. In this paper, we show how image quality in multimodal electron tomography can be greatly improved by employing variational modeling and multichannel regularization techniques. To achieve this aim, we employ a coupled Total Generalized Variation (TGV) regularization that exploits correlation between different channels. In contrast to other regularization methods, coupled TGV regularization allows to reconstruct both hard transitions and gradual changes inside each sample, and links different channels at the level of first and higher order derivatives.This favors similar interface positions for all reconstructions, thereby improving the image quality for all data, in particular, for 3D elemental maps. We demonstrate the joint multi-channel TGV reconstruction on tomographic energy-dispersive X-ray spectroscopy (EDXS) and high-angle annular dark field (HAADF) data, but the reconstruction method is generally applicable to all types of signals used in electron tomography, as well as all other types of projection-based tomographies.
We consider a class of regularization methods for inverse problems where a coupled regularization is employed for the simultaneous reconstruction of data from multiple sources. Applications for such a setting can be found in multi-spectral or multimodality inverse problems, but also in inverse problems with dynamic data. We consider this setting in a rather general framework and derive stability and convergence results, including convergence rates. In particular, we show how parameter choice strategies adapted to the interplay of different data channels allow to improve upon convergence rates that would be obtained by treating all channels equally. Motivated by concrete applications, our results are obtained under rather general assumptions that allow to include the Kullback-Leibler divergence as data discrepancy term. To simplify their application to concrete settings, we further elaborate several practically relevant special cases in detail. To complement the analytical results, we also provide an algorithmic framework and source code that allows to solve a class of jointly regularized inverse problems with any number of data discrepancies. As concrete applications, we show numerical results for multi-contrast MR and joint MR-PET reconstruction.
In this paper we are concerned with the learnability of energies from data obtained by observing time evolutions of their critical points starting at random initial equilibria. As a byproduct of our theoretical framework we introduce the novel concept of mean-eld limit of critical point evolutions and of their energy balance as a new form of transport. We formulate the energy learning as a variational problem, minimizing the discrepancy of energy competitors from fullling the equilibrium condition along any trajectory of critical points originated at random initial equilibria. By Γ-convergence arguments we prove the convergence of minimal solutions obtained from nite number of observations to the exact energy in a suitable sense. The abstract framework is actually fully constructive and numerically implementable. Hence, the approximation of the energy from a nite number of observations of past evolutions allows one to simulate further evolutions, which are fully data-driven. As we aim at a precise quantitative analysis, and to provide concrete examples of tractable solutions, we present analytic and numerical results on the reconstruction of an elastic energy for a one-dimensional model of thin nonlinear-elastic rod. 1. Introduction. 1.1. Evolutions of critical points. Many time-dependent phenomena in physics, biology, social, and economical sciences as well as iterative algorithms in machine learning can be modelled by a function x : [0, T ] → H , where H represents the space of states of the physical, biological, social system, or digital data, which evolves from an initial conguration x(0) = x 0 towards a more convenient state
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