Diffusion Tensor Imaging with Deterministic Error Bounds

Artur Gorokh; Yury Korolev; Tuomo Valkonen

doi:10.1007/s10851-016-0639-7

Diffusion Tensor Imaging with Deterministic Error Bounds

Artur Gorokh, Yury Korolev, Tuomo Valkonen

Department of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

1 Citation (SciVal)

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.

Original language	English
Pages (from-to)	137-157
Number of pages	21
Journal	Journal of Mathematical Imaging and Vision
Volume	56
Issue number	1
DOIs	https://doi.org/10.1007/s10851-016-0639-7
Publication status	Published - 1 Sept 2016

Bibliographical note

Funding Information:
While at the Center for Mathematical Modelling of the Escuela PolitÃ©cnica Nacional in Quito, Ecuador, T.Â Valkonen has been supported by a Prometeo scholarship of the Senescyt (Ecuadorian Ministry of Science, Technology, Education, and Innovation). In Cambridge, T.Â Valkonen has been supported by the EPSRC Grants No.Â EP/J009539/1 "Sparse & Higher-order Image Restoration" and No.Â 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). The authors would also like to thank Karl Koschutnig for the in vivo dataset, Kristian Bredies for scripts used to generate the tractography images and Florian Knoll for many inspiring discussions.

Funding Information:
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 No. EP/J009539/1 “Sparse & Higher-order Image Restoration” and No. 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). The authors would also like to thank Karl Koschutnig for the in vivo dataset, Kristian Bredies for scripts used to generate the tractography images and Florian Knoll for many inspiring discussions.

Publisher Copyright:
© 2016, The Author(s).

Keywords

Deterministic
Diffusion tensor imaging
Error bounds
Noise modelling
Total generalised variation

ASJC Scopus subject areas

Statistics and Probability
Modelling and Simulation
Condensed Matter Physics
Computer Vision and Pattern Recognition
Geometry and Topology
Applied Mathematics

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10.1007/s10851-016-0639-7Licence: CC BY

Cite this

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title = "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.",

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Diffusion Tensor Imaging with Deterministic Error Bounds

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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