An Inf-Convolution BV type model for dynamic reconstruction

Maïtine Bergounioux (1) , Evangelos Papoutsellis (1)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans

We are interested in a spatial temporal variational model for image sequences. The model involves a fitting data term to be adapted to different modalities such as denoising, debluring or emission tomography. The regularizing term acts as an infimal-convolution type operator that takes into account the respective influence of time and space variables. We give existence and uniqueness results and provide optimality conditions via duality analysis. In a forthcoming paper, see [5], we deal with the numerical realisation of the proposed model and focus on dynamic Positron Emission Tomography (PET) reconstruction. 1. Introduction. In this paper, we are interested in describing a variational model for denoising/debluring and/or emission tomography (ET) reconstruction of vector-valued images. What we call vector-valued images are usually color images, muti-spectral images, images acquired at different time intervals as videos. We focus on dynamic medical imaging as PET or functional MR images. There are many methods that handle videos or dyna-mical processes and the majority of them focus on the numerics. Here we aim to describe a variational model in an infinite dimensional setting, which, to our knowledge, has not been done yet. In a the dynamical setting, spatial and temporal components contribute differently, therefore we proceed with a non global description for both of them. Though, we have in mind a specific application to dynamic PET, we present a model flexible enough to address many applications. We cannot quote the numerous papers on videos, since there is a huge literature, even if we restrict ourselves to variational methods. Let us mention however the paper by Holler and Kunisch [15], where the authors consider the model we investigate in a semi-discrete setting. In addition, for dynamical PET applications, an active contour method with gradient vector flow has been developped in [17, 18] but the underlying variational model has not been explored. In this work, we do not address numerical issues since our concern is purely theoretical: we aim to describe a powerful variational model and perform mathematical analysis (well-posedness and optimality conditions). We will present numerical tests together with comparison with classical semi-discrete models/methods in the PET context in a forthcoming paper [5]. Let us denote u : (0, T) × Ω → R, the dynamic image with respect to the space domain Ω ⊂ R d with d ≥ 1 and with T > 0. In the following, we write Q ⊂ R d+1 instead of (0, T) × Ω. We focus on variational methods applied on a spatial-temporal domain and more precisely we

Mots clés: variational model; Inf-convolution; Total variation; PET; 4D imaging

Domaines: Optimisation et contrôle [math.OC]; Analyse fonctionnelle [math.FA]

Maïtine Bergounioux, Evangelos Papoutsellis. An Inf-Convolution BV type model for dynamic reconstruction. 2016. ⟨hal-01401408v1⟩