Fast and simple calculus on tensors in the log-Euclidean framework

Area of Science:

• Medical Imaging
• Computational Geometry
• Differential Geometry

Background:

• Diffusion Tensor Imaging (DTI) MRI generates complex tensor data.
• Classical Euclidean tensor computations have limitations and defects.
• Affine-invariant Riemannian metrics improve tensor analysis but are computationally intensive.

Purpose of the Study:

• Introduce novel Log-Euclidean metrics for tensor computations in DTI.
• Address the computational complexity and slowness of existing Riemannian metrics.
• Provide a simpler and faster alternative for tensor analysis in DTI.

Main Methods:

• Developed Log-Euclidean metrics based on matrix logarithms.
• Performed tensor computations in the domain of matrix logarithms.
• Applied metrics to multilinear interpolation and regularization of tensor fields.

Main Results:

• Log-Euclidean metrics demonstrate excellent theoretical properties.
• Achieved significantly simpler and faster computations compared to Riemannian metrics.
• Experimental results on synthetic and real DTI data show comparable performance.

Conclusions:

• Log-Euclidean metrics offer a practical and efficient approach for DTI tensor analysis.
• These metrics maintain theoretical rigor while improving computational feasibility.
• Log-Euclidean methods are suitable for applications like interpolation and regularization in DTI.