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Lie Group Analysis for Medical Image Processing

Final Report Summary - LIE ANALYSIS (Lie Group Analysis for Medical Image Processing)

The project achieved a new mathematical framework for medical image processing with high output, awards and recognition. This output includes new mathematical results, with many new algorithms in image analysis producing state-of-the-art results, with spin-off in many clinical applications, and industrial engineering. To deal with complex structures in images where existing geometric algorithms fail, we established a new computational framework that extends the domain of images to higher dimensional spaces (Lie groups) and then applies the correct geometric image processing.
To facilitate and underpin this new geometric image processing, we proved many new mathematical theorems and derived new exact and numeric PDE-solutions on the Lie groups of interest.

Highlights:

- We found new exact solutions to all linear PDEs describing left-invariant Brownian motions on the roto-translation group SE(d) for d=2,3, incl. Mumford's direction process (solving an open problem by a Field medalist) and (hypo)elliptic diffusions.
- We developed a new Fourier transform on the homogeneous space of positions and positions and used it to derive new exact solutions to many linear PDEs defined on this space.
We also provide new tangible analytic approximation of the PDE-kernels with correct symmetries.
- We derived new numerical algorithms for PDE-based image processing on scores and Lie groups, and we compared these and existing numerical solutions to our new exact solutions for validation.
- We derived and analyzed many new exact solutions to shortest paths in Lie groups. We exploited data-driven extensions in vessel/fiber tracking in 2D and 3D medical images.
- We set up a complete, new theory for locally adaptive frames in Lie groups, with proven value for automatic segmentation, localization and tracking algorithms in image analysis applications.
- We generalized our Lie group analysis framework to other Lie groups than SE(d):
1. Lie group SO(3) with retinal imaging applications (vessel tracking in spherical images),
2. Lie group H(5) with cardiac imaging applications (quantification of cardiac wall deformations).
3. Lie group SIM(2) with crossing-preserving enhancement and denoising of multiple scale elongated structures in (medical) images.
- We achieved new crossing-preserving PDE-flow techniques on Lie group SE(3) that we applied to diffusion weighted MRI images of the brain, with a
a) better tracking of neural fibers,
b) better quantification of structural connectivity in brain white matter,
c) better localization of anatomical bundles in brain white matter (crucial for epilepsy surgery).
- We proved global optimality of our cuspfree solutions to sub-Riemannian (association field) models for line-perception and propagation.
The results are generalized by formal theorems to data-driven extensions of these models, that show that backtracking by steepest descent on distance maps produces only global minimizers. The distance maps are viscosity solutions to sub-Riemannian eikonal PDEs that we solve efficiently.
- We developed a new framework for geometric machine learning algorithms on Lie groups with proven value (receiving research awards), we obtained state-of-the-art results in
1. optic nerve head detection in retinal images,
2. fovea detection in retinal images,
3. pupil detection in regular camera images.
4. detection of mitotic figures in histopathology for cancer detection.
5. blood vessel segmentation in retinal images.
6. cell-boundary segmentation algorithm in electron microscopy images.
- New Crossing-preserving vesselness measures are successfully applied in retinal imaging.
- New tangible design and processing of invertible orientation scores with public software implementations (cf. www.lieanalysis.nl)