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LIE ANALYSIS Report Summary

Project ID: 335555
Funded under: FP7-IDEAS-ERC
Country: Netherlands

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

The project is on track with the schedule, and has already delivered considerable output (68 output items; 14 high ranked journal publications, 12 peer-reviewed conference articles, 3 book chapters, 7 posters, 6 abstracts, 3 code releases, 6 awards/recognitions, and 17 invited external presentations).

This output includes new mathematical results, with many new algorithms in image analysis producing state-of-the-art results, with benefits and spin-off in clinical applications, and industrial engineering.

Publications range from mathematical journals such as JDCS, ACHA, JMIV, ESAIM-COCV, SIIMS (SIAM), until applied (medical) imaging journals/proceedings such as PAMI, TMI, MICCAI, ISMRM. Each time our new mathematical theory is directly, successfully employed in our image analysis application articles.

In order to facilitate off-spin to other applied image analysis groups we have provided open source implementations (DiPY, Mathematica).

In short, highlights and main new results are:

- New Exact solutions to the 3D version of Mumford’s direction process (NB. In response to an open problem by Field medalist David Mumford in his work “Elastica in Computer Vision’’), and on top of this, we solved for all linear PDE's describing left-invariant Brownian motions on the Lie group SE(3) of 3D rotations and translations.

- Complete comparison of our recent numerical methods to the PI’s exact solutions of Partial Differential Equations (PDE’s) on the roto-translation group SE(d).

- Design of new PDE method for line-tracking of globally optimal data-driven sub-Riemannian geodesics in the roto-translation group with fast-marching implementations and applications. Here our methods generically deals both with crossings and contextual alignment.

- A complete, new theory for locally adaptive frames in Lie groups, with proven value for automatic segmentation, localization and tracking in image analysis applications (with improvement of state-of-the-art methods).

- Three extensions of the Lie group analysis framework, to other Lie group instances have been achieved:
1. Lie group SO(3) with retinal imaging applications (e.g. automatic vessel tracking in spherical retinal images),
2. Lie group H(5) with cardiac imaging applications (e.g. quantification of cardiac wall velocities and deformations via Gabor transforms).
3. Lie group SIM(2) with multi-scale extensions of invertible orientation scores with crossing-preserving enhancement and denoising of multiple scale elongated structures in (medical) images.

- New crossing-preserving PDE-flow techniques on diffusion weighted MRI, which have clear benefits in the applications studied with our clinical partners (Kempenhaeghe, Hospital Clinic Barcelona). They are successfully released in public software environments (e.g. DipY).

- Improved, tangible, Gaussian approximations of heat-kernels on the homogeneous space of positions and orientations.

- Global optimality has been proven for our, cuspfree, simple solutions to association field models for line-perception and propagation.

- Our new automatic template matching via invertible orientation scores for localization of small isotropic objects in environments with many lines/edges.
It performs better than case-specific state-of-the-art methods on 3 case studies:
1. optic nerve head detection in retinal images,
2. fovea detection in retinal images,
3. pupil detection in regular camera images .

- Crossing-preserving vesselness measures have been developed and successfully applied in retinal imaging.

- New and tangible design of invertible 3D orientation scores, released in our public Mathematica software for invertible orientations.

- New exact solutions and proven geometric properties of cuspless sub-Riemannian geodesics in the Euclidean motion group SE(3).

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