Efficient Recursive Algorithms for Computing the Mean Diffusion Tensor and Applications to DTI Segmentation^★

Area of Science:

• Medical Imaging
• Computational Geometry
• Statistics

Background:

• Computing the mean of symmetric positive definite (SPD) matrices is essential for diffusion tensor image (DTI) processing, including segmentation and clustering.
• Existing non-recursive algorithms for calculating the mean of SPD matrices are computationally intensive.

Purpose of the Study:

• To introduce novel recursive algorithms for computing the mean of SPD matrices using Riemannian distance and symmetrized Kullback-Leibler divergence.
• To demonstrate the significant computational efficiency gains of these new recursive methods over existing non-recursive approaches.

Main Methods:

• Development of a recursive estimator for Karcher expectation in the space of SPD matrices.
• Formulation of a recursive version of the symmetrized Kullback-Leibler divergence for SPD matrix mean computation.
• Comparative timing analysis of recursive versus non-recursive algorithms.

Main Results:

• The proposed recursive algorithms achieve computational time savings of several orders of magnitude compared to non-recursive methods.
• Theoretical results establish a recursive estimator for Karcher expectation, akin to a law of large numbers on the SPD manifold.
• Application of recursive algorithms to DTI segmentation tasks shows considerable improvements in computation time.

Conclusions:

• Novel recursive algorithms provide a highly efficient method for computing the mean of SPD matrices.
• These algorithms offer a significant advancement for DTI processing, particularly in segmentation and clustering.
• The findings pave the way for faster and more scalable DTI analysis techniques.