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Seminars
Event
- Title:
- A. Ghosh - Applications of Tensors in Diffusion MRI
- When:
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29 Mar 2012 - 29 Mar 2012 10:30 - 11:30
- Where:
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- Category:
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Seminars
Description
Diffusion MRI (dMRI) is a promising new modality of MRI that provides an unprecedented insight into the structure of cerebral and spinal white matter. dMRI can measure the diffusion of water molecules in the brain in vivo and non-invasively at scales comparable to axonal connectivities of neurons. Since the fine and organized structure of the white matter hinders and constrains the diffusion of water, it therefore becomes possible to infer the microstructure of the white matter from dMRI measurements. Tensors have played a central role in modelling this microstructure since the early days of dMRI. Diffusion Tensor Imaging (DTI) was the first comprehensive mathematical model proposed that allows to reconstruct a surprisingly great amount of the white matter connectivity. DTI employs a simple 3x3 2nd order symmetric positive definite tensor. However, DTI's simplicity, which has allowed for its phenomenal success and wide acceptance also poses limitations. Higher order tensors were introduced to model the dMRI signal since almost a decade now to both generalize DTI and as an alternate basis for spherical harmonics to represent spherical diffusion functions. This talk will first present an introduction to dMRI and then a number of problems related to tensors of order higher than 2, mainly 4, that we have attempted to tackle in the Athena Project Team. First we will consider the problem of estimating 4th order tensors with a positivity constraint to satisfy the physics of the problem. We will present two solutions -- a (not truly) Riemannian approach and a polynomial solution based on Hilbert's theorem on ternary quartics. Next we will consider the problem of extracting the maxima of spherical diffusion functions by using higher order tensor or polynomial representations which allows to correctly estimate axon fiber bundle directions from dMRI models. We worked on this problem with the GALAAD Project Team. And finally, time permitting, we will touch upon computing 4th order tensor invariants, which we have recently begun exploring.
Venue
- Place:
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Y106
- Street:
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INRIA, Byron Building
Description
Byron Building, first floor.
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