1. Radial Thickness Calculation and Visualization for Volumetric Layers
Defeng Wang
The Chinese University of Hong Kong

2. Overview
Motivation
Algorithm
User’s Guide
Results
Conclusion od

3. Motivation (1)
Volumetric layers are often encountered in medical image analysis
E.g., skull vault; myocardium of the left ventricle, etc.
Automatic landmarking method for structures with coupled surfaces [1]
Consider thickness in landmarking volumetric layers
Higher quality compared with landmark optimization on two surfaces separately.
[1] L. Shi, D. Wang, et al. Landmark Correspondence Optimization for Coupled Surfaces, MICCAI 2007, Brisbane, Australia

4. Motivation (2) Definitions for layer thickness
Closest thickness (Tclose)
Normal thickness (Tnormal)
We propose to use radial thickness (Tradial)
Distance between each pair of corresponding points on two surfaces with the same polar coordinates
In comparison
Tclose and Tnormal depend on the starting surface
Tradial is unique and landmarks are grouped in pairs

5. Motivation (3) Thickness definitions illustration
Figure 1: Different thickness definitions illustrated on a part of one axial plane of the skull boundary: (a) the coupled surfaces; (b) the closest thickness measure; (c) the normal thickness measure; (d) the radial thickness

6. Algorithm description (1)
Two coupled triangle meshes are treated as
A master mesh
A supplementary mesh
Each ray is generated from the center to each vertex in the master mesh
Radial thickness calculation is to determine the distance between
Each vertex on the master mesh
The intersection point of the compatible ray with the supplementary surface

7. Algorithm description (2)
Specifically, the radial thickness calculation is reduced to
Checking if there is any intersection of a ray and a triangle in the supplementary mesh
Determine the coordinates if the intersection exists
We adopt the fast and minimum-storage algorithm of ray/triangle intersection examining [2]
[2] T. Moller and B. Trumbore. Fast, minimum storage ray/triangle intersection. Journal of Graphics Tools, 2(1):21–28, 1997.

8. Algorithm description (3)
A ray emitted from center C0 to one vertex V in the master mesh
A point T(u,v), on a triangle defined by 3 verticesV0, V1, V2,
(u,v) are the barycentric coordinates
The intersection is equivalent to

9. Algorithm description (4)
By using the Cramer’s rule and defining A B C
The final speeded-up computations are in the following form

10. User’s guide (1)
The vtkRadialThicknessCalculate class take VTK meshes as inputs
One master mesh
One supplementary mesh
Output can be radial thickness values, or the normalized ones in [0,1]
Ray directions can be
Calculated from the master mesh,
Specified in a text file containing each direction as 3 real number in a row

11. User’s guide (2) Include the header file Declare an instance
Set master and supplementary meshes

12. User’s guide (3) Specify the file name to save radial thickness values
Two output files, one is “thicknessFile.txt” containing radial thickness values
The other is “thicknessFile_normalize.txt” saving the normalized ones
Start calculation
The directions can also be set with a direction file

13. Results (1) In order to reproduce the results, require packages
CMake 2.4.6
Visualization Toolkit VTK 5.0.3
KWMeshvisu [3], is adopted to visualize the calculated radial distances on the master mesh
Neurolib [4], the VTK2Meta tool in the MetaMeshTools project from the Neurolib package is adopted to convert the mesh data from VTK to the Meta format, which can be loaded in KWMeshvisu for visualization
[3] Ipek Oguz, et al. KWMeshvisu: A mesh visualization tool for shape analysis. In IJ MICCAI Open Science Workshop, Kopenhagen, 2006.
[4]

14. Results (2) Results on human skull vaults
Skull vault is defined as the upper part of the skull
Skull vault is an open coupled-surface structure
The skull volume was segmented from the head CT data,
Acquired at the Prince of Wales Hospital, Hong Kong
Field of view of the CT data is 512 * 512
Voxel size is 0.49mm * 0.49mm * 0.63mm

15. Results (3)
Figure 2: Coupled surfaces of the skull vault: (a) the back view; (b) the top view.

16. Results (4) Inner surface is taken as the master surface
Figure 3: The color-coded thickness values plotted on the surface of the inner skull: (a) the back view; (b) the top view; (c) the color bar used to code the normalized thickness values.

17. Results (5) Outer surface is taken as the master surface
Figure 4: The color-coded thickness values plotted on the surface of the outer skull: (a) the back view; (b) the top view; (c) the color bar used to code the normalized thickness values.

18. Results (6)
Note that, if there is no intersection between the ray and the supplementary surface, we artificially set the thickness value 0
This explains Figure 3 shows a ribbon of zero values in the inner surface of the skull vault
From the results, the radial thickness values represent the layer thickness reasonably
No matter whether the master surface is inner surface, or outer surface, the resultant thickness value will not be altered, except marginal regions

19. Conclusion
We implement a new thickness definition for volume layer structures, radial thickness
This implementation is encapsulated in the vtkRadialThicknesscalculate class
This class contains a realization of the fast ray/triangle intersection algorithm
Experimental results demonstrate that radial thickness is a suitable thickness measurement for the structures with a near-spherical morphology.

20. Acknowledgment
The work described in this paper was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CUHK4453/06M) and CUHK Shun Hing Institute of Advanced Engineering.