1. Quasi-isometric Representation of Three Dimensional Triangulated Surfaces Project Summary

2. Introduction Project objective What are CT scans Data processing The new mathematical method Code structure design Results Suggestions for future projects

3. Project Objectives: 1.Implement of a new algorithm for minimal distortion flattening of three dimensional surfaces in MATLAB 2. Confirm the abilities of the algorithm and the program by processing data from CT scans of the large intestine.

4. Background Why do we want to represent three dimensional surfaces as two dimensional images? Why did we choose the large intestine for the demonstrations?

5. CT Scans What is a CT machine?

6. CT Scans

7. Data Processing

8. Introduction to the Surface Flattening Problem The idea: Representing a 3D surface as a set of 2D images The goal: implementing a surface flattening algorithm with a minimal distortion

9. The Mathematical Method The Projection Condition:

10. The Length Distortion

11. Triangulation Representation of the raw data:

12. The Triangulation Algorithm 1.Cutting the cylinder of samples

13. The Triangulation Algorithm 2. Projecting each of the halves of the cylinder on the x-z plane

14. The Triangulation Algorithm 3. Triangulating the samples points on the plane 4. Reshaping the plane to it’s former form

15. The Triangulation Algorithm Results Front part: 407 triangles Back part: 624 triangles

16. Designed Algorithm Triangulation Algorithm Finding Neighbor Triangles Algorithm Spreading Algorithm Single Triangle Projection Algorithm

17. The Spreading Algorithm 1. Randomly choose a triangle – it is the anchor triangle 2. Project it on itself. It’s plane is the anchor plane. 3. For each of the triangle’s neighbors: Check whether the neighbor fulfills the projection condition: If Yes – project it on the anchor plane. Add it to the patch. Check the neighbor’s neighbors (Recursive calling). If not – move on to the next neighbor.

18. Flow Chart of the Program

19. Results

20. Results For Front Part C(f) = 1.1405 70 planes Back Part C(f) = 1.1405 151 planes

21. Results The two parts together, C(f) = 1.1405

22. Testing the Program by Simulating Edge Situations 1. A Strict Distortion Bound Front Part: 292 planesBack Part: 495 planes C(f) = 1.0175

23. 2. A Weak Distortion Bound Testing the Program by Simulating Edge Situations 1 Plane, C(f) = 20.0811

24. Rotating… 2. A Weak Distortion Bound

26. Simulation with many CT slices Back part: 1329 trianglesFront part: 728 triangles The results of the triangulation:

27. Simulation with many CT slices The results of the surface flattening, C(f) = 1.1405 Front Part C(f) = 1.1405 88 planes Back Part C(f) = 1.1405 173 planes

28. Simulation with many CT slices

29. Summary The simulation results corresponded to the theoretical results, for both normal cases and edge cases The results show that the algorithm is highly suited for complex surfaces

30. Suggestions for Future Projects Implement a surface flattening algorithm with a curvature based choice of triangles Globalize the abilities of the program by creating a function that can perform a three dimensional surface triangulation

31. THE END