The document proposes calculating and visualizing radial thickness for volumetric layers. It defines radial thickness as the distance between corresponding points on two surfaces with the same polar coordinates. An algorithm is described that uses ray tracing to calculate the intersection between rays emitted from a master surface and a supplementary surface to determine radial thickness values at corresponding points. Results are presented visualizing calculated radial thickness values on human skull vault surfaces to demonstrate the method represents layer thickness reasonably.

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

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 ( T close ) Normal thickness ( T normal ) We propose to use radial thickness ( T radial ) Distance between each pair of corresponding points on two surfaces with the same polar coordinates In comparison T close and T normal depend on the starting surface T radial 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 C 0 to one vertex V in the master mesh A point T(u,v), on a triangle defined by 3 verticesV 0 , V 1 , V 2 , (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 - 2006 MICCAI Open Science Workshop , Kopenhagen, 2006. [4] http://www.ia.unc.edu/dev/

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) 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. Inner surface is taken as the master surface

17. Results (5) 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. Outer surface is taken as the master surface

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.