While current state of the art MR-PET scanners enable simultaneous MR and PET measurements, the acquired data sets are still usually reconstructed separately. We propose a new multi-modality reconstruction framework using second order Total Generalized Variation (TGV) as a dedicated multi-channel regularization functional that jointly reconstructs images from both modalities. In this way, information about the underlying anatomy is shared during the image reconstruction process while unique differences are preserved. Results from numerical simulations and in-vivo experiments using a range of accelerated MR acquisitions and different MR image contrasts demonstrate improved PET image quality, resolution, and quantitative accuracy.

The regularization properties of the total generalized variation (TGV) functional for the solution of linear inverse problems by means of Tikhonov regularization are studied. Considering the associated minimization problem for general symmetric tensor fields, the well-posedness is established in the space of symmetric tensor fields of bounded deformation, a generalization of the space of functions of bounded variation. Convergence for vanishing noise level is shown in a multiple regularization parameter framework in terms of the naturally arising notion of TGV-strict convergence. Finally, some basic properties, in particular non-equivalence for different parameters, are discussed for this notion.

A variational model for image reconstruction is introduced and analyzed in function space. Specific to the model is the data fidelity, which is realized via a basis transformation with respect to a Riesz basis followed by interval constraints. This setting in particular covers the task of reconstructing images constrained to data obtained from JPEG or JPEG 2000 compressed files. As image prior, the total generalized variation (TGV) functional of arbitrary order is employed. The present paper, the first of two works that deal with both analytical and numerical aspects of the model, provides a comprehensive analysis in function space and defines concrete instances for particular applications. A new, noncoercive existence result and optimality conditions, including a characterization of the subdifferential of the TGV functional, are obtained in the general setting.

The infimal convolution of total (generalized) variation-type functionals and its application as regularization for video and image reconstruction is considered. The definition of this particular type of regularization functional is motivated by the need of suitably combining spatial and temporal regularity requirements for video processing. The proposed functional is defined in an infinite dimensional setting, and important analytical properties are established. As applications, the reconstruction of compressed video data and of noisy still images is considered. The resulting problem settings are posed in function space, and suitable numerical solution schemes are established. Experiments confirm a significant improvement compared to standard total variation-type methods, which originates from the introduction of spatio-temporal and spatial anisotropies. Introduction.Motivated by the aim of defining a suitable spatio-temporal regularization for image sequences we consider the infimal convolution of an arbitrary number of modified total (generalized) variation functionals. We discuss analytical properties and propose a numerical realization for two concrete applications.The combination of different regularization functionals by infimal convolution was mentioned early in [15] and is discussed in [31] in a more general, discrete setting, both motivated by combining first and higher order derivatives in order to reduce staircasing effects for still images. Given two functionals J 1 and J 2 , their infimal convolution is defined by

In this article, we evaluate Parallel Level Sets (PLS) and Bowsher's method as segmentation-free anatomical priors for regularized brain positron emission tomography (PET) reconstruction. We derive the proximity operators for two PLS priors and use the EM-TV algorithm in combination with the first order primal-dual algorithm by Chambolle and Pock to solve the non-smooth optimization problem for PET reconstruction with PLS regularization. In addition, we compare the performance of two PLS versions against the symmetric and asymmetric Bowsher priors with quadratic and relative difference penalty function. For this aim, we first evaluate reconstructions of 30 noise realizations of simulated PET data derived from a real hybrid positron emission tomography/magnetic resonance imaging (PET/MR) acquisition in terms of regional bias and noise. Second, we evaluate reconstructions of a real brain PET/MR data set acquired on a GE Signa time-of-flight PET/MR in a similar way. The reconstructions of simulated and real 3D PET/MR data show that all priors were superior to post-smoothed maximum likelihood expectation maximization with ordered subsets (OSEM) in terms of bias-noise characteristics in different regions of interest where the PET uptake follows anatomical boundaries. Our implementation of the asymmetric Bowsher prior showed slightly superior performance compared with the two versions of PLS and the symmetric Bowsher prior. At very high regularization weights, all investigated anatomical priors suffer from the transfer of non-shared gradients.