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 fibres DNA strains hart

Remco Duits

Completion & Enhancementmuscle cells bone-structure EP-catheters

Detection

muscle cells bone-structure EP-catheters

Completion & Enhancement Detection

Invertible Orientation Scores

image kernel orientation score

invertible

Invertible Orientation Scores

image kernel orientation score

invertible

orientations are disentangled in the orientation score

image orientation score

Processing via Scores

The Practical Advantage of Left Invariant Diffusions via Orientation Scores

Brownian motion of water molecules along fibers

fibertracking fibertracking

DTI HARDI

Extend to New Medical Image Modalities

Challenges1. 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

Challenge II : Crossing Preserving Diffusion on HARDI

: Left-invariant vector field on

: Diffusion matrix

Challenge II : Crossing Preserving Diffusion on HARDI

Challenge II : Crossing Preserving Diffusion on HARDI

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)

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

Challenge V : Extension to other groups 14

Goals : 1. Create a musical score from music2. Texture enhancement in medical imaging

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+

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.