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Quasi-isometric Representation of Three Dimensional Triangulated Surfaces

Quasi-isometric Representation of Three Dimensional Triangulated Surfaces. Project Summary Efrat Barak Amiad Segal. Introduction. Project objective What are CT scans Data processing The new mathematical method Code structure design Results Suggestions for future projects.

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Quasi-isometric Representation of Three Dimensional Triangulated Surfaces

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  1. Quasi-isometric Representation of Three Dimensional Triangulated Surfaces Project Summary Efrat Barak Amiad Segal

  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: • 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 • 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 ResultsFront part: 407 trianglesBack 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 anchortriangle 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 Back PartC(f) = 1.1405151 planes Front PartC(f) = 1.140570 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 planes Back Part: 495 planes C(f) = 1.0175

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

  24. 2. A Weak Distortion Bound Rotating…

  25. 2. A Weak Distortion Bound

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

  27. Simulation with many CT slices The results of the surface flattening, Front PartC(f) = 1.140588 planes Back PartC(f) = 1.1405173 planes C(f) = 1.1405

  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

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