Diffusion tensor imaging with deterministic error bounds

TL;DR

This paper introduces a novel approach to model errors in inverse problems, specifically in Diffusion Tensor Imaging, using partial order bounds in Banach lattices to handle complex noise models without linearisation.

Contribution

It applies the partial order error bounds framework to Diffusion Tensor Imaging, enabling robust error modeling amidst complex noise distributions.

Findings

Error bounds effectively model noise in DTI

Preserves simple error structure under transformations

Avoids complex linearisation of nonlinear equations

Abstract

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, 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.