1. Parameter Controlled Volume Thinning Nikhil Gagvani Deborah Silver

2. Introduction Algorithm for the Distance Transform Skeleton Extraction Algorithm for the Centerline Reconstruction

3. The Neighbors of a Voxel F-Neighbors E-NeighborsV-Neighbors A voxel is the smallset unique element of the volume.

4. Voxels object-voxels background-voxels boundary-voxels 000000… 0000… 00… S _S_S BV

5. Distance Transform in 2D (1/2) The distance transform of a simple shape.

6. Distance Transform in 2D (2/2)

7. Distance Transform in 3D (1/6) The distance transform at a voxel p = {x,y,z} is defined as  : the distance from voxel(x,y,z) to voxel(i,j,k)  BV : the set of boundary voxels

8. Distance Transform in 3D (2/6) We compute the distance using a weighted distance metric.  F-neighbor : 3  E-neighbor : 4  V-neighbor : 5

9. Distance Transform in 3D (3/6) Algorithm  Init  First pass Calculate the Distance transform of boundary-voxels.  Second pass Propagate the boundary inward.

10. Distance Transform in 3D (4/6) Init  For all voxels, assign a distance transform S _S_S

11. Distance Transform in 3D (5/6) First pass  Calculate the Distance transform of boundary-voxels For all voxels that have a (F/E/V) neighbor { (3 for face/ 4 for edge/ 5 for vertex); Add p to BV ; } S _S_S BV

12. Distance Transform in 3D (6/6) Second pass  Propagate the boundary inward Repeat for all Find all voxels which are (F/E/V) neighbors of p ; Assign (3 or 4 or 5) } ; Remove p from BV ; Add r to BV ; until no is modified S _S_S BV

13. Skeleton Extraction (1/12) Definition 1  If a voxel p has a distance transform, the Ball B(p) associated with p is the set of object voxels q such that the distance from p to q is less then. p S _S_S

14. Skeleton Extraction (2/12) Definition 2  The ball for an object voxel is maximal if it is not contained in the ball of any other voxel. Observation  The set of voxels whose balls are maximal is sufficient to reconstruct the object.

15. Skeleton Extraction (3/12) Minimal set for reconstruction  The ball of a voxel may not be completely contained in that of another, but may be contained in the union of the balls of several other voxels. abc Black voxel Gray voxel not essential

16. Skeleton Extraction (4/12) Definition 3  The witness for a voxel p is any 26-neighbor q such that the distance transform of q, ( 3 or 4 or 5 ). Thus, if a voxel q is a witness for a voxel p, S _S_S p S _S_S

17. Skeleton Extraction (5/12) Claim 1  The ball of a voxel p must be contained in the ball of one of its 26 neighbor if it is to be contained in the ball of any other voxel in the object. S _S_S p S _S_S

18. Skeleton Extraction (6/12) To find non-witness voxels, the 26-neighbors of all object voxels need to be scanned. (brute force approach) Rather than grow the ball for every neighbor and scan for containment in the union of balls, a simple approach is to average the DT value of the neighbors of p.

19. Skeleton Extraction (7/12) Definition 4  the mean of the neighbors’ distance transform (MNT) is a 26-neighbor of p. If MNTp < DTp, add p to the skeleton. MNTp = 4

20. Skeleton Extraction (8/12) Thinness parameter (TP)  We introduce the TP, that allows control over the removal of non-witness voxels yielding skeletons of varying density. Condition 1  If MNT p < DT p – TP, add p to the skeleton. Low value of TPHigh value of TP Thick Thin

21. Skeleton Extraction (9/12) A maple leaf and it’s skeleton The Complexity of this algorithm is therefore O(N), where N is the number of object-voxels. These skeletal voxels are not generally connected. (Spanning Tree) TP = 0TP = 2TP = 4TP = 6

22. Skeleton Extraction (10/12) Trachea a.The segmented trachea b.Skeleton with a thinness = 2.5 c.The minimum spanning tree of the skeletal voxels

23. Skeleton Extraction (11/12) Effact of the Thinness Parameter

24. Skeleton Extraction (12/12) Skeleton of S shape

25. Algorithm for the Centerline (1/4) A centerline is a curve that is centered with respect to the object boundaries. It can serve as the path for a virtual camera in surgical path planning. It is a semi-automatic algorithm in which the user specifies end-points for centerline generation.

26. Algorithm for the Centerline (2/4) SK : the set of skeleton voxels. p1 and p2 : the end-points of the centerline, p1,p2 SK F : subdivision parameter (“ fineness ”)

27. Algorithm for the Centerline (3/4) Centerlines for Surgical Navigation

28. Algorithm for the Centerline (4/4) Centerlines for Surgical Navigation

29. Reconstruction (1/3) In order to reconstruct the object from the skeletal voxels, balls of radius equal to the distance transform value have to be constructed. A path terminates when its value is less than or equal to 3 but greater than zero. The quality of reconstruction depends upon the thinness of the skeleton.

30. Reconstruction (2/3) Lossless reconstruction

31. Reconstruction (3/3) Lossy reconstruction