1. 3D Image Segmentation and Multigrid Method

2. Contents 3D Image Segmentation I. Variational level sets method for 3D
II. Energy function
III. Implementation steps
IV. Demo
Multigrid Method
I. Introduction
II. Implementation steps
III. Advantages and disadvantages

3. 3D Image Segmentation
Variational level sets method for image segmentation ：
ﻼ Use n level set functions to represent n+1 regions.
ﻼ Energy model.
ﻼ Compute the evolution equation through semi-implicit finite difference scheme.

4. In order to represent different regions, we can define:
To improve the numerical performance, the above functions are usually regularized as follows (here,εis a small constant value).

5. Region representation：define n level set functions to represent n+1 regions as follows.
To avoid overlap and vacuum, the following constraint must be enforced:

6. The energy model for image segmentation
Data term
Regularized volume term
Regularized area term
Constraint term to constrain the level set function as signed distance function

7. Implementation Steps 1. Read a sequence of 2D images into a 3D matrix.
2. Initialize the level set function.
3. Compute the evolution equation.
4. Smooth the results if necessary.

8. 1. Read a sequence of 2D images
ﻼ Read a image, store it as a 2D matrix.
For instance: matrix=imread(‘filename.bmp’)
ﻼ Then read a sequence of images
One image 2D matrix
images 3D matrix

9. 2. Initialize the level set function
The zero level set function is initialized to be a spherical surface.
inside>0
outside<0
Calculate the signed distance of each pixel of the 3D matrix and store the value into the Φ matrix.
on the surface=0

10. 3. The evolution equation
1> Energy model
2> Evolution equation
set
3> The semi-implicit finite difference scheme of evolution equation
Let h be the space step, △t be the time step.
The point (i, j, k) can be discreted as:
The Approximation of point of the level set function
can be denoted as on the nth step.

12. 4>The equation can be discretized as follows

13. 4.1 Demo

15. 4.2 Demo

16. Multigrid Method （level 1） Coarse grid (level 2) Coarser grid11 12 13 14 15 16 17 18 21 22 23 24 25 26 27 28 31 32 33 34 35 36 37 38 41 42 43 44 45 46 47 48 51 52 53 54 55 56 57 58 61 62 63 64 65 66 67 68 71 72 73 74 75 76 77 78 81 82 83 84 85 86 87 88
Fine grid
（level 1） 11 13 15 17 31 33 35 37 51 53 55 57 71 73 75 77
Coarse grid
(level 2) 11 15 51 55
Coarser grid
（level 3）

17. Implementation Steps While（condition is true） end
1. Compute on the fine grid for m steps. (Error rectification)
2. Transform from fine grid to coarse grid. (Select values)
3. Compute on the coarse grid for n steps. (Error rectification)
4. Transform from coarse grid to fine grid. (Interpolation)
5. Compute on the fine grid for m steps. (Error rectification)
end

18. x 1. Fine to Coarse: Select the points.
Selection Strategy: Direct Selection, Weighted Matrix,
Others.
2. Coarse to Fine:
Simple linear interpolation x
(xk, yk)
(xk+1, yk+1)
Lagrange interpolation

19. Design data structure Method one V Iteration steps dx dy dz L1 5 1 L210 2 L3 15 W
Iteration steps dx dy dz L1 5 1 L2 10 2 L3 15 20 16

20. Method two Data structure W L1 L2 L3 V L1 L2 L3 dx dy dz L1 5 1 L2 2
Level
Iteration steps dx dy dz L1 5 1 L2 2 L3 10 L1 L2 L3

21. Fine Grid to Coarse Grid (Direct Selection)
L1 (u) L2 (u1) L3(u2)
endx 11 12 13 14 15 16 17 18 21 22 23 24 25 26 27 28 31 32 33 34 35 36 37 38 41 42 43 44 45 46 47 48 51 52 53 54 55 56 57 58 61 62 63 64 65 66 67 68 71 72 73 74 75 76 77 78 81 82 83 84 85 86 87 88 11 13 15 17 31 33 35 37 51 53 55 57 71 73 75 77 11 15 51 55
L1 → L L2 → L3
Level
Iteration steps dx dy dz L1 5 1 L2 2 L3 10

22. Coarse Grid to Fine Grid (linear interpolation)
Interpolation order:
edge → plane → solid

23. 1 . Edge Interpolation x direction We can get： Dx Dx Dx 1 row 2 row …
N row A E B C D x
i Dx Dx endx end （h）
We can get：

24. x direction
y direction
z direction

25. 2. Plane Interpolation XOY: The values of points M、N、E、I
can be get from the edges (MN, EI).
Seen from the illustration, two lines
come across the point Q (MN, EI).
So, one Q value on the line MN and another Q value on the line EI can be get. Then calculate the average value of the two Q values.

26. xoz：
xoy：
yoz：

27. 3. Solid Interpolation The inner points interpolation of solid
is similar to the plane interpolation.
The values of points Q, R, S, T, U, V
can be get through the plane
Interpolation, and the inner point W can be calculated by interpolated points’ values on the three lines (QR, TS, UV) .

29. 4. Excess Boundary (extrapolation)
Endx i 11 12 13 14 15 16 17 18 21 22 23 24 25 26 27 28 31 32 33 34 35 36 37 38 41 42 43 44 45 46 47 48 51 52 53 54 55 56 57 58 61 62 63 64 65 66 67 68 71 72 73 74 75 76 77 78 81 82 83 84 85 86 87 88
A B C
Calculate C with the values
of A and B. x y z

30. Thank you！