Efficient Total Variation Algorithm for Fetal Brain MRI Reconstruction S´ ebastien Tourbier 1,2 , Xavier Bresson 1,2 , Patric Hagmann 2 , Jean-Philippe Thiran 3,2 , Reto Meuli 2 , and Meritxell Bach Cuadra 1,2,3,⋆ 1 Centre d’Imagerie Biom´ edicale, Switzerland 2 Department of Radiology, Lausanne University Hospital Center (CHUV) and University of Lausanne (UNIL), Switzerland 3 Signal Processing Laboratory (LTS5), Ecole Polytechnique F´ ed´ erale de Lausanne (EPFL), Switzerland Abstract. Fetal MRI reconstruction aims at finding a high-resolution image given a small set of low-resolution images. It is usually modeled as an inverse problem where the regularization term plays a central role in the reconstruction quality. Literature has considered several regular- ization terms s.a. Dirichlet/Laplacian energy [1], Total Variation (TV)- based energies [2,3] and more recently non-local means [4]. Although TV energies are quite attractive because of their ability in edge preserva- tion, standard explicit steepest gradient techniques have been applied to optimize fetal-based TV energies. The main contribution of this work lies in the introduction of a well-posed TV algorithm from the point of view of convex optimization. Specifically, our proposed TV optimization algorithm for fetal reconstruction is optimal w.r.t. the asymptotic and iterative convergence speeds O(1/n 2 ) and O(1/ √ ε), while existing tech- niques are in O(1/n) and O(1/ε). We apply our algorithm to (1) clinical newborn data, considered as ground truth, and (2) clinical fetal acquisi- tions. Our algorithm compares favorably with the literature in terms of speed and accuracy. 1 Introduction The aim of fetal brain MRI reconstruction is to reconstruct a high-resolution (HR) image X given a set of low-resolution (LR) images X LR k , which may be highly corrupted with noise, blurring, intensity bias and (large-scale) mo- tion. The fetal reconstruction problem can be cast as an inverse problem s.t. min X ∑ K k=1 ‖H k X − X LR k ‖ 2 , where H k are supposedly linear operators that transform the HR image X into a LR (corrupted) image X LR k . The above least square problem is said to be ill-posed, meaning that it has generally no meaningful solutions. The natural solution is to regularize, a la Tikhonov, that is to say (1) making assumptions about the HR image X s.a. smoothness, edge recovery, histogram conservation, etc, and (2) adding a regularization term R(X ) that projects X into some spaces that hold the desired properties, i.e. ⋆ This work is supported by Grant SNSF-141283. P. Golland et al. (Eds.): MICCAI 2014, Part II, LNCS 8674, pp. 252–259, 2014. c  Springer International Publishing Switzerland 2014