Diffusion tensor imaging with deterministic error bounds

Errors in the data and the forward operator of an inverse problem can be handily modelled using partial order in Banach lattices. We present some existing results of the theory of regularisation in this novel framework, where errors are represented as bounds by means of the appropriate partial order. We apply the theory to diffusion tensor imaging (DTI), where correct noise modelling is challenging: it involves the Rician distribution and the nonlinear Stejskal-Tanner equation. Linearisation of the latter in the statistical framework would complicate the noise model even further. We avoid this using the error bounds approach, which preserves simple error structure under monotone transformations.While at the Center for Mathematical Modelling of the Escuela Politécnica Nacional in Quito, Ecuador, T. Valkonen has been supported by a Prometeo scholarship of the Senescyt (Ecuadorian Ministry of Science, Technology, Education, and Innovation). In Cambridge, T. Valkonen has been supported by the EPSRC grants Nr. EP/J009539/1 “Sparse & Higher-order Image Restoration”, and Nr. EP/M00483X/1 “Efficient computational tools for inverse imaging problems”. A. Gorokh and Y. Korolev are grateful to the RFBR (Russian Foundation for Basic Research) for partial financial support (projects 14-01-31173 and 14-01-91151)