Quasi-isometric Representation of Three

Dimensional Triangulated Surfaces

Project Summary

Introduction

• Project objective

• What are CT scans

• Data processing

• The new mathematical method

• Code structure design

• Results

• Suggestions for future projects

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.

Background

• Why do we want to represent three dimensional surfaces as two dimensional images?

• Why did we choose the large intestine for the demonstrations?

CT Scans

What is a CT machine?

CT Scans

Data Processing

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

The Mathematical Method

2 3max{ ( , ), ( , )}e N e N

The Projection Condition:

The Length Distortion

( ) cot 1C f

TriangulationRepresentation of the

raw data:

The Triangulation Algorithm1. Cutting the cylinder

of samples

The Triangulation Algorithm

2. Projecting each of the halves of the cylinder on the x-z plane

The Triangulation Algorithm

3. Triangulating the samples points on the plane

4. Reshaping the plane to it’s former form

The Triangulation Algorithm

Results

Front part: 407 triangles

Back part: 624 triangles

Designed Algorithm

• Triangulation Algorithm

• Finding Neighbor Triangles Algorithm

• Spreading Algorithm

• Single Triangle Projection Algorithm

The Spreading Algorithm1. 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.

Flow Chart of the ProgramProcessing CT Data

Triangulating the Surfaces

Repeated for each surface:Random choice a of new triangle, starting a new patch.

For all the neighbors of the triangle:

Check if the neighbor fulfills the projection condition

If Yes: Project the neighbor on the plane, add it to the patch,

and check his neighbors(recursive call)

If Not: Continue tothe next neighbor

Results

Results For

Front PartC(f) = 1.1405

70 planes

Back PartC(f) = 1.1405151 planes

82

ResultsThe two parts together, C(f) = 1.1405

Testing the Program by Simulating Edge Situations

89 1. A Strict Distortion Bound

-200

2040

60

-20

0

20

40-5

0

5

The front part after flattening, C(f) = 1.0175, Number of planes = 292

-200

2040

60

-40

-20

0

20-5

0

5

The back part after flattening, C(f) = 1.0175 , Number of planes = 495

Front Part: 292 planes Back Part: 495 planes

C(f) = 1.0175

2. A Weak Distortion Bound

Testing the Program by Simulating Edge Situations

3

1 Plane,C(f) = 20.0811

Rotating…

2. A Weak Distortion Bound

2. A Weak Distortion Bound

Simulation with many CT slices

-40 -20 0 20 40 60

0

20

40

60

-4

-2

0

2

4

The front part after triangulation, and before projection on surfaces

-40-20

020

4060

-20

-10

0

10

20-4

-2

0

2

4

The back part after triangulation, and before projection on surfaces

Back part: 1329 triangles Front part: 728 triangles

The results of the triangulation:

Simulation with many CT slices

-40-20

020

4060

-20

-10

0

10

20-5

0

5

-40-20

020

4060

0

20

40

60-6

-4

-2

0

2

4

The front part after flattening, C(f) = 1.1405, Number of planes = 73

The results of the surface flattening,

C(f) = 1.1405

82

Front PartC(f) = 1.140588 planes

Back PartC(f) = 1.1405173 planes

Simulation with many CT slices

-40-20

020

4060

-20

0

20

40

60-6

-4

-2

0

2

4

The two parts together, C(f) = 1.1405

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

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

THE END