1. Patricia van Marlen April 12, 2018
Linear and anisotropic diffusion in image processing: A study on implementation, parameters and segmentation
Patricia van Marlen
April 12, 2018

2. Outline Introduction Part 1: diffusion filtering methods
numerical schemes
results: methods and schemes
Part 2:
time step Δt: theory and results
stopping time S: theory and results
Part 3:
results: segmentation
Conclusion and further research

3. Introduction
LUMC/TUD MRI project: MRI scanner for developing countries

4. Introduction LUMC/TUD MRI project:
MRI scanner for developing countries
Advantages:
affordable

5. Introduction LUMC/TUD MRI project:
MRI scanner for developing countries
Advantages:
affordable
transportable

6. Introduction LUMC/TUD MRI project:
MRI scanner for developing countries
Advantages:
affordable
transportable
always ready for use
no cooling necessary

7. Introduction LUMC/TUD MRI project:
MRI scanner for developing countries
Advantages:
affordable
transportable
always ready for use
no cooling necessary
Disadvantage:
low SNR: noise in the images

8. Introduction LUMC/TUD MRI project:
MRI scanner for developing countries
Advantages:
affordable
transportable
always ready for use
no cooling necessary
Disadvantage:
low SNR: noise in the images
Noise removal: diffusion filtering methods

9. Diffusion filtering methods: linear diffusion
Diffusion equation D is the diffusion coefficient u is gray value Neumann boundary conditions:

10. Diffusion filtering methods: linear diffusion
Diffusion equation D is the diffusion coefficient u is gray value Neumann boundary conditions: D=1, leads to the heat equation Disadvantage: space-invariant blurring

11. Diffusion filtering methods: anisotropic diffusion
Diffusion equation D is the diffusion coefficient u is gray value Neumann boundary conditions:

12. Diffusion filtering methods: anisotropic diffusion
Diffusion equation D is the diffusion coefficient u is gray value Neumann boundary conditions: D=c(| u|), leads to

13. Diffusion filtering methods: Anisotropic diffusion
Intraregional blurring: Within a region, u is small So: if u is small, we need c(| u|) to be high Edge preservation: Close to a boundary, u is large So: if u is large, we need c (| u|) to be close to zero

14. Diffusion filtering methods: Anisotropic diffusion
Well-posed method:

15. Diffusion filtering methods: Anisotropic diffusion
Well-posed method: Ill-posed method:

16. Diffusion filtering methods: Anisotropic diffusion
Well-posed method: Ill-posed method: Fulfill the threshold property: Perona-Malik 1: Perona-Malik e:

17. Numerical schemes: Forward Time, Central Space (FTCS)
Time-derivative: Euler forward Space-derivatives: central difference Accuracy: O(Δt)+O(Δx2) Set Δx=Δy=1

18. Numerical schemes: Forward Time, Central Space (FTCS)
Linear diffusion filtering

19. Numerical schemes: Forward Time, Central Space (FTCS)
Linear diffusion filtering Anisotropic diffusion filtering

20. Numerical schemes: Forward Time, Central Space (FTCS)
Linear diffusion filtering Anisotropic diffusion filtering Addition of Gaussian kernel:

21. Numerical schemes: Forward Time, Central Space (FTCS)
Linear diffusion filtering Anisotropic diffusion filtering with Gaussian kernel Addition of Gaussian kernel:

22. Numerical schemes: Additive Operator Splitting (AOS)
The AOS scheme in 2D

23. Numerical schemes: Additive Operator Splitting (AOS)
The AOS scheme in 2D We have two dimensions, so Ax and Ay

24. Numerical schemes: Additive Operator Splitting (AOS)
The AOS scheme in 2D We have two dimensions, so Ax and Ay
Figure 2: a) original numbering b) transposed numbering

25. Results: methods and schemes
Test problem: Gaussian noise with mean 0 and SD 0.06
Figure 3: test problem with and without noise

26. Results: methods and schemes
Test problem: Gaussian noise with mean 0 and SD 0.06 Difference measure is the SD of the noise in the image:
Figure 3: test problem with and without noise

27. Results: methods and schemes FTCS scheme
standard deviation δ
Perona-Malik 1
0.0592
Perona-Malik e
Perona-Malik 1 with GK
0.0594
Perona-Malik e with GK
Well-posed method
0.0595
Ill-posed method
Heat equation
0.0597
Table 1: results of the diffusion filtering methods for FTCS scheme

28. Results: methods and schemes FTCS scheme
standard deviation δ
Perona-Malik 1
0.0592
Perona-Malik e
Perona-Malik 1 with GK
0.0594
Perona-Malik e with GK
Well-posed method
0.0595
Ill-posed method
Heat equation
0.0597
Table 1: results of the diffusion filtering methods for FTCS scheme
Figure 4:
best result of
Perona-Malik 1

29. Results: methods and schemes FTCS scheme
standard deviation δ
Perona-Malik 1
0.0592
Perona-Malik e
Perona-Malik 1 with GK
0.0594
Perona-Malik e with GK
Well-posed method
0.0595
Ill-posed method
Heat equation
0.0597
Table 1: results of the diffusion filtering methods for FTCS scheme
Figure 5:
unstable result
of well-posed
method

30. Results: methods and schemes AOS scheme
standard deviation δ
Perona-Malik 1 with GK
0.0590
Perona-Malik e
Perona-Malik 1
0.0592
Well-posed method
0.0593
Perona-Malik e with GK
0.0596
Ill-posed method
Table 2: results of the diffusion filtering methods for AOS scheme

31. Results: methods and schemes AOS scheme
standard deviation δ
Perona-Malik 1 with GK
0.0590
Perona-Malik e
Perona-Malik 1
0.0592
Well-posed method
0.0593
Perona-Malik e with GK
0.0596
Ill-posed method
Table 2: results of the diffusion filtering methods for AOS scheme
Figure 6:
best result of
Perona-Malik 1
with GK

32. Results: methods and schemes conclusions
the AOS scheme gave slightly better results, but due to calculation time FTCS is used for the following parts

33. Results: methods and schemes conclusions
the AOS scheme gave slightly better results, but due to calculation time FTCS is used for the following parts
Perona-Malik 1 seems the best method and is also chosen for following parts, it is given by:

34. Results: methods and schemes conclusions
the AOS scheme gave slightly better results, but due to calculation time FTCS is used for the following parts
Perona-Malik 1 seems the best method and is also chosen for following parts, it is given by:
the well-posed and ill-posed method showed instability for FTCS scheme, but were stable for AOS scheme

35. Results: methods and schemes conclusions
the AOS scheme gave slightly better results, but due to calculation time FTCS is used for the following parts
Perona-Malik 1 seems the best method and is also chosen for following parts, it is given by:
the well-posed and ill-posed method showed instability for FTCS scheme, but were stable for AOS scheme
parameter choice has a big influence on the outcomes

36. Parameter study Investigation of
parameter K, used in the Perona-Malik functions
time step size Δt
number of time steps T

37. Parameter study Investigation of
parameter K, used in the Perona-Malik functions
time step size Δt
number of time steps T

38. Parameter study time step size Δt
So far Δt was constant Adaptive time step for Perona-Malik methods with α=4, a=0.25 and b=0.75

39. Parameter study time step size Δt
So far Δt was constant Adaptive time step for Perona-Malik methods with α=4, a=0.25 and b=0.75 Initial iterations (large uchange ): Δt ~ Later iterations (small uchange ): Δt ~ 0.25 So: an increasing function

40. Parameter study time step size Δt
Initial iterations: Δt ~ 0.25 Later iterations: Δt ~ 0 So: a decreasing function
Figure 7: Optimal time step for Perona-Malik 1

41. Parameter study time step size Δt
Adaptive time step for Perona-Malik methods with α=4, a=0.25 and b=0.75 Initial iterations: Δt ~ Later iterations: Δt ~ 0.25 So: an increasing function

42. Parameter study time step size Δt
Adaptive time step for Perona-Malik methods with α=4, a=1 and b=1 Initial iterations: Δt ~ 0.25 Later iterations: Δt ~ 0 So: a decreasing function

43. Results: parameters time step size Δt
Behaviour of Blue: optimal Δt Red: original method Green: modified method
Figure 8:
behaviour of various time steps

44. Results: parameters time step size Δt
Figure 9: results for K=0.05
Figure 8:
behaviour of various time steps

45. Results: parameters time step size Δt
Figure 9: results for K=0.05
Figure 8:
behaviour of various time steps
Figure 10: results for K=0.5

46. Parameter study time step size Δt
Adaptive time step for well-posed and ill-posed method if with 0<q<1

47. Parameter study time step size Δt
Adaptive time step for well-posed and ill-posed method if with 0<q<1 Method is based on constant total pixel value (TPV), if TPV starts to increase the time step is reduced

48. Results: parameters time step size Δt
Adaptive time step for well-posed and ill-posed method if with 0<q<1
Figure 11:
results for
adaptive
time step for
various q

49. Parameter study stopping time S
Optimal stopping time: number of time steps at which the image is at its best and the diffusion should stop

50. Parameter study stopping time S
Optimal stopping time: number of time steps at which the image is at its best and the diffusion should stop Correlation stopping time

51. Parameter study stopping time S
Optimal stopping time: number of time steps at which the image is at its best and the diffusion should stop Correlation stopping time λ-stopping time

52. Parameter study stopping time S
Optimal stopping time: number of time steps at which the image is at its best and the diffusion should stop Correlation stopping time λ-stopping time

53. Results: parameters stopping time S
Scorr
Svisual
0.01
841
631
850
0.05
168
126
180
0.1 84 63 90
0.2 41 31 45
Table 3: results for Shepp-Logan, K=0.05 Δt Sλ
Scorr
Svisual
0.01 31 41 70
0.05 6 8 15
0.1 3 4 7
0.2 2 5
Table 4: results for Ella, K=0.05

54. Results: parameters stopping time S
Correlation stopping time λ-stopping time New stopping time Snew

55. Results: parameters stopping time S
Scorr
Svisual
Snew
0.01
841
631
850
0.05
168
126
180
0.1 84 63 90 80
0.2 41 31 45
Table 5: results for Shepp-Logan, K=0.05 Δt Sλ
Scorr
Svisual
Snew
0.01 31 41 70
0.05 6 8 15
0.1 3 4 7
0.2 2 5
Table 6: results for Ella, K=0.05

56. Results: parameters stopping time S
Scorr
Svisual
Snew
0.01
841
631
850
0.05
168
126
180
0.1 84 63 90 80
0.2 41 31 45
Table 5: results for Shepp-Logan, K=0.05 Δt Sλ
Scorr
Svisual
Snew
0.01 31 41 70
0.05 6 8 15
0.1 3 4 7
0.2 2 5
Table 6: results for Ella, K=0.05

57. Results: parameters stopping time S
Figure 12: results when using stopping time Snew

58. Segmentation
Segmentation partitions an image into several regions, useful to identify structures Segmentation method: region growing algorithm Segmentation is not possible on noisy images, the irregularities would lead to incorrect regions

59. Results: segmentation
Figure 13: segmentation of a) noiseless Shepp-Logan b) filtered Shepp-Logan

60. Results: segmentation
Figure 14: segmentation of a) noiseless Ella b) filtered Ella

61. Conclusions Methods and numerical schemes
Perona-Malik 1 seems to give the best results
AOS had slightly better results, but its calculation time was also a little higher
Parameter study
introduced time step and stopping time lead to good results
well-posed and ill-posed method were unstable for FTCS scheme, this can be overcome with an adaptive time step
Segmentation
segmentation showed that images obtained with anisotropic diffusion filtering are of good quality

62. Conclusions Methods and numerical schemes
Perona-Malik 1 seems to give the best results
AOS had slightly better results, but its calculation time was also a little higher
Parameter study
introduced time step and stopping time lead to good results
well-posed and ill-posed method were unstable for FTCS scheme, this can be overcome with an adaptive time step
Segmentation
segmentation showed that images obtained with anisotropic diffusion filtering are of good quality

63. Conclusions Methods and numerical schemes
Perona-Malik 1 seems to give the best results
AOS had slightly better results, but its calculation time was also a little higher
Parameter study
introduced time step and stopping time lead to good results
well-posed and ill-posed method were unstable for FTCS scheme, this can be overcome with an adaptive time step
Segmentation
segmentation showed that images obtained with anisotropic diffusion filtering are of good quality

64. Further research
further investigation of parameters

65. Further research further investigation of parameters
improve the calculation time of the AOS method

66. Further research further investigation of parameters
improve the calculation time of the AOS method
test methods on real data

67. Questions?

68. Figure 15: best result for Perona-Malik 1 with GK, FTCS scheme
Figure 16: best result for Perona-Malik 1, AOS scheme

69. Figure 17: segmentation of normal brain image

70. Figure 18: segmentation of hydrocephalus image