LIE ANALYSIS · Lie Group Analysis for Medical Image Processing

The aim of this project is to substantially improve computer algorithms for image analysis in medical imaging. Currently available techniques often require significant application-specific tuning and have a limited application scope. This is mostly due to the use of non-generic feature spaces that involve many physical dimensions and lack mathematical foundation.Instead, we derive inspiration from the superior generic pattern recognition capabilities of the human brain and propose a novel operator design aiming at better results and wider applicability.This novel operator design combines (partial and ordinary) differential equations on non-compact Lie groups (induced by stochastic processes and sub-Riemannian geometric control) with wavelet transforms. Many mathematical challenges arise in the analysis and (numerical) solutions of these operators.The research departs from previously developed insights of the PI on 'invertible orientation scores', which can be regarded as a specific instance in a general Lie group theoretical framework. Within this general framework one obtains a comprehensive invertible score defined on a higher dimensional Lie group beyond position space. The key challenge is to appropriately exploit these scores, their survey of multiple features per position, their underlying group structure, and their invertibility. We will tackle this via left-invariant evolutions and left-invariant sub-Riemannian optimal control within the score.The orientation score approach will be systematically extended towards multi-scale-and-orientation, multi-velocity and multi-frequency encoding and processing, widening the application scope. Moreover, improvements in contextual enhancement via invertible scores and improvements in optimal curve extractions in the Lie group domain of the score will be pursued.We will develop and apply the resulting algorithms to a wide range of medical imaging challenges in neurological, retinal and cardiac applications.