The aim of this project is to substantially improve computer algorithms for image analysis in medical imaging. Currently available techniques often require significant applicationspecific tuning and have a limited application scope. This is mostly due to the involvement of feature spaces that involve many physical dimensions and lack a clear group structure. Therefore, we take inspiration by the superior pattern recognition capabilities of the human brain and recent insight on how this is accomplished, and formulate a novel operator design aiming at better results and a wider applicability. This novel operator design combines (partial) 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 starts with previously developed insights of R.Duits on ’invertible orientation scores’, which can be regarded as a single entity in a larger Lie group theoretical framework. Within this 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 structures, 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 (crossing preserving) contextual enhancement via invertible scores and improvements in optimal curve extractions in the Lie group domain of the score will be pursued. We will apply the resulting algorithms to a wide range of medical imaging challenges in neurology, retinal and cardiac applications. Our team, consisting of 2 PhD’s, 1 PostDoc, 1 scientific programmer and the Principal Investigator (R.Duits), will therefore cooperate closely with advanced clinical - as well as mathematical - partners from around the world.