Cédric Dufour

1. Anisotropic Diffusion’s Extension toConstrained Line ProcessesAnisotropic Diffusion’s Application in3D Confocal Microscopy Image Processing Cédric Dufour

2. Contents • Anisotropic diffusion’s basics • Extension to constrained line processes • Anisotropic diffusion vs. constrained line processes • 3D microscopy image processing • Conclusions

3. Grayscale intensity value Diffusion coefficient Time (iteration) variable Anisotropic diffusion’s basics (1) • Underlying principle: standardheat diffusion • Equivalent to gaussian local meaning(the variance being related unequivocally to the diffusion coefficient)

4. Anisotropic diffusion’s basics (2) • Problem: diffusion occurs in all direction, regardless of edges • Blurring

5. Anisotropic diffusion coefficient(“edge stopping” function) Anisotropic diffusion’s basics (3) • Solution: bind the diffusion coefficient to the gradient of the intensity • Care must be taken in the choice of the edge stopping function for the problem to be well-posed More info: edge stopping function

6. A.D. Anisotropic diffusion’s basics (4) • Results: diffusion is inhibited when the gradient gets more important (edges) • Piecewise smooth image

7. Smoothness functional Smoothness norm Extension to constrained line processes (1) • Anisotropic diffusion can be derived from the minimization of a smoothness functional: More info: 3D neighborhood for anisotropic diffusion

8. Line process Spatial constraints Fitting constant Line process penalty function Extension to constrained line processes (2) • Expressing this minimization problem according to the line process formulation, we have: • Adding explicit spatial constraints: More info: line process and penalty function characteristics

9. Extension to constrained line processes (3) • The line process formulation is related to the standard anisotropic formulation through: More info: starting relating axiom between the standard anisotropic formulation and the line process formulation

10. Extension to constrained line processes (4) • Computational results: Image Gradient

11. Hysteresis term Non-maximum suppression term Extension to constrained line processes (5) • Adding spatial constraints... • … we obtain the following iterative formula: More info: spatial constraints clique

12. C.L.P. Extension to constrained line processes (6) • Results: the diffusion is inhibited by the spatial constraints • Sharper details and smoother contours

13. A.D. C.L.P. C.L.P. A.D. Anisotropic diffusion vs. constrained line processes (7) • Comparative MSE and variance: MSE Variance

15. 3D microscopy image processing (1) • Goal: obtain correlation statistics in multi-channel 3D confocal microscopy images CH.1 CH.2

16. 3D microscopy image processing (2) • Step 1: de-noising (using anisotropic diffusion) • Smooth image CH.1 CH.2

17. 3D microscopy image processing (3) • Step 2: thresholding • Proteins mask CH.1 CH.2

18. 3D microscopy image processing (4) • Step 3: skeleton and labeling • Disjointed protein labeled skeleton CH.1 CH.2 More info: disjointed clusters

19. 3D microscopy image processing (5) • Step 4: geodesic growth • Disjointed protein labeled mask CH.1 CH.2

20. 3D microscopy image processing (6) • Step 5: compute distance table • Distance between proteins CH.1  CH.2

21. 3D microscopy image processing (7) • Step 6: clustering • Group proteins according to the separating distance D = 5 x ‘mean size’ D = 7.5 x ‘mean size’

22. 3D microscopy image processing (8) • Step 7: compute the statistics • Proper correlation statistics and interpretation To do! (IBCM’s biologists task)

23. Conclusions • Anisotropic diffusion is a powerful tool for de-noising • Spatial constraints (added through the line process formulation) allow to obtain better quality denoising • Application of the anisotropic diffusion along with other morphological and clustering tools allowed efficient segmentation and classification of proteins appearing in 3D confocal microscopy images.

24. Thank you for your attention !