1. Enhancement, Completion and Detection of Elongated Structures in Medical Imaging via Evolutions on Lie Groups muscle cells bone-structure retinal bloodvessels catheters neural fibers in brain collagen fibresDNA strains hart Remco Duits

2. Completion & Enhancement muscle cells bone-structure EP-catheters Detection

3. muscle cells bone-structure EP-catheters Completion & Enhancement Detection

4. Invertible Orientation Scores imagekernelorientation score invertible

5. Invertible Orientation Scores imagekernelorientation score invertible orientations are disentangled in the orientation score image orientation score

6. Processing via Scores

7. The Practical Advantage of Left Invariant Diffusions via Orientation Scores

8. Brownian motion of water molecules along fibers fibertracking DTIHARDI Extend to New Medical Image Modalities

9. Challenges 1.Extension and analysis of non-linear diffusion on orientation scores. 2. Diffusion on HARDI-images. 3. Erosion on HARDI-images. 4. Derivation Green’s functions of underlying stochastic processes for an automated graphical sketcher. 5. Extension to other groups than SE(d) such as diffusion and erosion on Gabor transforms of signals / images. 6. Detection of elongated structures via Hamilton Jacobi equations on SE(2). Important for bi-plane navigation in treatments of cardiac arrhythmias. 7. Isomorphism between diffusion and erosion on SE(d) ? 8. Erosion towards optimal curves/modes. Compare different probabilistic approaches to optimal paths: contour completion elastica curves contour enhancement geodesics

10. Challenge II : Crossing Preserving Diffusion on HARDI : Left-invariant vector field on : Diffusion matrix

11. Challenge II : Crossing Preserving Diffusion on HARDI

12. Convolution (Precompute) : spherical surface measure : Green’s function diffusion : any rotation such that : Reflected Green’s function Left Invariant Finite Differences Goal : non-linear adaptive crossing preserving HARDI-diffusion (adaptive curvature & torsion)

13. 1. Angular erosion 2. Spatial erosion Challenge III : Left Invariant HJB-Equations on HARDI Combine the 2 evolution processes below to single erosion process on Goal: Improve fibertracking

14. Challenge V : Extension to other groups 14 Goals : 1.Create a musical score from music 2. Texture enhancement in medical imaging

15. Strengths of the Proposal Mathematical Skills. Both theoretically & practically. - “The PI has undoubtly quite a range of theoretical knowledge (Lie groups, group representations, partial differential equations, wavelet transforms) ” (ref. 1) - “The PI is widely regarded as an expert in the field in which he operates. ” (ref.2) Generic Solution to Relevant Problem in Image Processing. - “Detection, enhancement and completion of elongated structures is a very active field of image processing. The proposal is very ambitious and covers a wide set of new investigations as well theoretically and practically oriented, which are extremely interesting in the whole. ” (ref.1) - “General framework with numerous applications not only limited to biomedical imaging. ”(ref.2) Realistic New Goals which build on cum laude Previous Work. - “Overall, the workplan is appropriate and realistic….The demand of two PhD students as well as a PostDoc fellow is justifed…. Excellent track record “ (ref.1 & 2 ) Strong Multi-disciplinary Embedding. Both at W&I and BME. - “Good "Embedding" environment and well-established collaborations both at the national and international level. ” (ref.1) Ref. Evaluation: A+ / A+

16. Strong Embedding. I have initiated (together with Luc Florack) a close and enthousiastic collaboration between W&I department and BME department at the TU/e. TU/e W&I : - Mathematical Image Processing (Florack) - LIME Imaging Applications (Matheij & Janssen). - Variational Methods and Probability Theory (Peletier & Wittich) TU/e BME : - Biomedical Applications (ter Haar Romeny & Platel) - Visualization (Vilanova). - Inviso b.v professional FPGA-design for real-time parallel computation. International : - Prof. Führ, Lehrstuhl A Für die Mathematik, RWTH Aachen, Germany. - Prof. Felsberg, Computer Vision Laboratory, Linköping University, Sweden. - Prof. Mumford, Dep. of Applied Mathematics, Brown University, Providence, USA.