1. Signal Processing Laboratory Swiss Federal Institute of Technology, Lausanne 1 Anisotropic Diffusion’s Extension to Constrained Line Processes  Anisotropic Diffusion’s Application in 3D Confocal Microscopy Image Processing Cédric Dufour

2. Signal Processing Laboratory Swiss Federal Institute of Technology, Lausanne 2 Contents  Anisotropic diffusion’s basics  Extension to constrained line processes  Anisotropic diffusion vs. constrained line processes  3D microscopy image processing  Conclusions

3. Signal Processing Laboratory Swiss Federal Institute of Technology, Lausanne 3 Anisotropic diffusion’s basics (1)  Underlying principle: standard heat diffusion Grayscale intensity value Time (iteration) variable Diffusion coefficient  Equivalent to gaussian local meaning  Equivalent to gaussian local meaning (the variance being related unequivocally to the diffusion coefficient)

4. Signal Processing Laboratory Swiss Federal Institute of Technology, Lausanne 4 Anisotropic diffusion’s basics (2)  Problem: diffusion occurs in all direction, regardless of edges  Blurring

5. Signal Processing Laboratory Swiss Federal Institute of Technology, Lausanne 5 Anisotropic diffusion’s basics (3)  Solution: bind the diffusion coefficient to the gradient of the intensity Anisotropic diffusion coefficient (“edge stopping” function)  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. Signal Processing Laboratory Swiss Federal Institute of Technology, Lausanne 6 Anisotropic diffusion’s basics (4)  Results: diffusion is inhibited when the gradient gets more important (edges)  Piecewise smooth image A.D.

7. Signal Processing Laboratory Swiss Federal Institute of Technology, Lausanne 7 Extension to constrained line processes (1)  Anisotropic diffusion can be derived from the minimization of a smoothness functional: Smoothness normSmoothness functional More info: 3D neighborhood for anisotropic diffusion

8. Signal Processing Laboratory Swiss Federal Institute of Technology, Lausanne 8 Extension to constrained line processes (2)  Expressing this minimization problem according to the line process formulation, we have: Fitting constantLine process penalty function Line process  Adding explicit spatial constraints: Spatial constraints More info: line process and penalty function characteristics

9. Signal Processing Laboratory Swiss Federal Institute of Technology, Lausanne 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. Signal Processing Laboratory Swiss Federal Institute of Technology, Lausanne 10 Extension to constrained line processes (4)  Computational results: ImageGradient

11. Signal Processing Laboratory Swiss Federal Institute of Technology, Lausanne 11 Extension to constrained line processes (5)  Adding spatial constraints...  … we obtain the following iterative formula: More info: spatial constraints clique Hysteresis termNon-maximum suppression term

12. Signal Processing Laboratory Swiss Federal Institute of Technology, Lausanne 12 Extension to constrained line processes (6)  Results: the diffusion is inhibited by the spatial constraints  Sharper details and smoother contours C.L.P.

13. Signal Processing Laboratory Swiss Federal Institute of Technology, Lausanne 13 Anisotropic diffusion vs. constrained line processes (7)  Comparative MSE and variance: MSEVariance A.D. C.L.P. A.D. C.L.P.

14. Signal Processing Laboratory Swiss Federal Institute of Technology, Lausanne 14

15. Signal Processing Laboratory Swiss Federal Institute of Technology, Lausanne 15 3D microscopy image processing (1)  Goal: obtain correlation statistics in multi-channel 3D confocal microscopy images CH.1CH.2

16. Signal Processing Laboratory Swiss Federal Institute of Technology, Lausanne 16 3D microscopy image processing (2)  Step 1: de-noising (using anisotropic diffusion)  Smooth image CH.1CH.2

17. Signal Processing Laboratory Swiss Federal Institute of Technology, Lausanne 17 3D microscopy image processing (3)  Step 2: thresholding  Proteins mask CH.1CH.2

18. Signal Processing Laboratory Swiss Federal Institute of Technology, Lausanne 18 3D microscopy image processing (4)  Step 3: skeleton and labeling  Disjointed protein labeled skeleton CH.1CH.2 More info: disjointed clusters

19. Signal Processing Laboratory Swiss Federal Institute of Technology, Lausanne 19 3D microscopy image processing (5)  Step 4: geodesic growth  Disjointed protein labeled mask CH.1CH.2

20. Signal Processing Laboratory Swiss Federal Institute of Technology, Lausanne 20 3D microscopy image processing (6)  Step 5: compute distance table  Distance between proteins CH.1  CH.2

21. Signal Processing Laboratory Swiss Federal Institute of Technology, Lausanne 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. Signal Processing Laboratory Swiss Federal Institute of Technology, Lausanne 22 3D microscopy image processing (8)  Step 7: compute the statistics  Proper correlation statistics and interpretation To do! (IBCM’s biologists task)

23. Signal Processing Laboratory Swiss Federal Institute of Technology, Lausanne 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. Signal Processing Laboratory Swiss Federal Institute of Technology, Lausanne 24 Thank you for your attention !

25. Signal Processing Laboratory Swiss Federal Institute of Technology, Lausanne 25 More info (1)  Edge stopping function (here, Lorenzian): Lorenzian (  =0.25)

26. Signal Processing Laboratory Swiss Federal Institute of Technology, Lausanne 26 More info (2)  3D neighborhood for anisotropic diffusion:

27. Signal Processing Laboratory Swiss Federal Institute of Technology, Lausanne 27 More info (3)  Line process and penalty function characteristics: Lorenzian (  =0.25)

28. Signal Processing Laboratory Swiss Federal Institute of Technology, Lausanne 28 More info (4)  Starting relating axiom between the standard anisotropic formulation and the line process formulation:

29. Signal Processing Laboratory Swiss Federal Institute of Technology, Lausanne 29 More info (5)  Spatial constraints clique:

30. Signal Processing Laboratory Swiss Federal Institute of Technology, Lausanne 30 More info (6)  Step 3: skeleton and labeling  Disjointed clusters Original clusterDisjointed cluster