# Sumit Kaushik presents Higher Order Tensor Framework for Segmentation and Fiber Tracking in DWMR Images

On 2019-10-01 11:00:00 at G205, Karlovo náměstí 13, Praha 2

Clinically DTI (Diffusion Tensor Imaging) is a widely used image modality for

diagnosis and study of

progression of neuro diseases. In DTI, 2nd order tensors are ensured to have

symmetric positive definite

property. Thus, lie in well studied Riemann geometric space. But these 2nd

order

tensors are unable to

resolve the white matter fibers in regions of crossing/merging. Higher order

tensors appear naturally to deal

this issue. The higher order tensors are comparatively less explored,

specifically for their use in applications

like segmentation and fiber tractography in the crossing/ merging regions.

There

do exist some applications

where the rotation invariants of these tensors gives rise to new set of

bio-markers and are robust than existing

ones due to 2nd order tensors in DTI. In this work, 4th order tensors have been

explored which is extendable

to higher order tensors. The initial idea is to use the known Riemann geometry

for the higher 4th order

tensors. Three different 2nd order projections of 4th order tensors have been

utilized under three anisotropy

preserving similarity measures for the geometric space. Spectral quaternion

similarity measure is known to

better preserve the anisotropy of 2nd order tensors than Log Euclidean metric.

An extension of this measure

called SlerpSQ (spherical linear interpolation spectral quaternion) produces

smoother interpolation curves.

The spectral metric approach is known to behave robustly under noise. It is

observed that the diagonal

components of the flattened fourth-order tensors live in the well-known

Riemannian symmetric space of

symmetric positive-definite matrices. The projection under spectral similarity

measure proved effective for

segmentation of white matter complex structures with high curvatures and

crossings. Experimentally, in

the work it is confirmed that this projection unfolds the geometry and hence,

effective in revealing directions

of underlying fibers than existing method known as cartesian tensor orientation

distribution function (CT-

ODF). In the fiber tracking application, the algorithms relying upon full

tensor

information, are known to

have advantage in tracking curved fibers and in presence of noise. The choice

of

metric tensor for such

geometric space is crucial. So far, inverse of the tensor is widely used as a

metric tensor. It has also been

showed that activation function can also be used for rescaling of this metric

tensor (β scaled), so as to

minimize the Riemann cost in anisotropic regions ( or maximizing the cost in

isotropic regions ) while fiber

tracking. Further, the known ray tracing algorithm is also modified to counter

the deviation of fiber from

actual path under the β scaled metric tensor and compared to the recent

adjugate and standard

inverse metric tensor. Various synthetic and real images are used for testing

the segmentation and fiber

tracking methods.

[Sumit Kaushik is finishing a PhD at Masaryk University and is currently

looking

for a postdoc position.]

diagnosis and study of

progression of neuro diseases. In DTI, 2nd order tensors are ensured to have

symmetric positive definite

property. Thus, lie in well studied Riemann geometric space. But these 2nd

order

tensors are unable to

resolve the white matter fibers in regions of crossing/merging. Higher order

tensors appear naturally to deal

this issue. The higher order tensors are comparatively less explored,

specifically for their use in applications

like segmentation and fiber tractography in the crossing/ merging regions.

There

do exist some applications

where the rotation invariants of these tensors gives rise to new set of

bio-markers and are robust than existing

ones due to 2nd order tensors in DTI. In this work, 4th order tensors have been

explored which is extendable

to higher order tensors. The initial idea is to use the known Riemann geometry

for the higher 4th order

tensors. Three different 2nd order projections of 4th order tensors have been

utilized under three anisotropy

preserving similarity measures for the geometric space. Spectral quaternion

similarity measure is known to

better preserve the anisotropy of 2nd order tensors than Log Euclidean metric.

An extension of this measure

called SlerpSQ (spherical linear interpolation spectral quaternion) produces

smoother interpolation curves.

The spectral metric approach is known to behave robustly under noise. It is

observed that the diagonal

components of the flattened fourth-order tensors live in the well-known

Riemannian symmetric space of

symmetric positive-definite matrices. The projection under spectral similarity

measure proved effective for

segmentation of white matter complex structures with high curvatures and

crossings. Experimentally, in

the work it is confirmed that this projection unfolds the geometry and hence,

effective in revealing directions

of underlying fibers than existing method known as cartesian tensor orientation

distribution function (CT-

ODF). In the fiber tracking application, the algorithms relying upon full

tensor

information, are known to

have advantage in tracking curved fibers and in presence of noise. The choice

of

metric tensor for such

geometric space is crucial. So far, inverse of the tensor is widely used as a

metric tensor. It has also been

showed that activation function can also be used for rescaling of this metric

tensor (β scaled), so as to

minimize the Riemann cost in anisotropic regions ( or maximizing the cost in

isotropic regions ) while fiber

tracking. Further, the known ray tracing algorithm is also modified to counter

the deviation of fiber from

actual path under the β scaled metric tensor and compared to the recent

adjugate and standard

inverse metric tensor. Various synthetic and real images are used for testing

the segmentation and fiber

tracking methods.

[Sumit Kaushik is finishing a PhD at Masaryk University and is currently

looking

for a postdoc position.]