Diffusion Tensor Imaging with Deterministic Error Bounds

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 non-linear 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.

Variational regularisation for inverse problems with imperfect forward operators and general noise models

“…For comparison, with the data term ( 1.7 ) and without an operator error, ( 1.2 ) translates into where the constraint is equivalent to ‖ Au − f δ ‖ ∞ ⩽ δ . (In [ 25 ], a connection is made between the lower and upper bounds f l , f u and confidence intervals.)…”

“…where the constraint is equivalent to Au − f δ ∞ δ. (In [25], a connection is made between the lower and upper bounds f l , f u and confidence intervals.) One can show that the partial order based condition (1.5) implies the norm based condition (1.4).…”

Variational regularisation for inverse problems with imperfect forward operators and general noise models

“…A drawback of the formulation (1.8), however, is that the data term (1.7) is rather noninformative and does not take into account statistical properties of the noise. In [24], a connection is made between the lower and upper bounds f l , f u in (1.8) and confidence intervals; however, this still does not capture full information about the statistics of the noise.…”