Topology Simplification Algorithm for Segmentation of Medical Scans

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Topology Simplification Algorithm for
Segmentationof Medical Scans
Université catholique de Louvain
Committee Prof B. Macq (Advisor) Prof L.
Vandendorpe (Chairman) Prof A.H. Barr Prof S.K.
Warfield Dr C. De Vleeschouwer
Sylvain Jaume Ph.D. Private Defense December 17,
2003
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Agenda

• Introduction
• Theory of Topology
• Related Work
• Algorithm
• Results
• Conclusion

PhD Defense Topology Simplification Algorithm
for Segmentation of Medical Scans S. Jaume Dec
17, 2003 2
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Introduction I Data

• MRI, CT, PET, US provide
• volumetric images

• 3D grid of voxels

• 1 voxel 1 intensity

256 x 256 x 124 voxels
PhD Defense Topology Simplification Algorithm
for Segmentation of Medical Scans S. Jaume Dec
17, 2003 3
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Introduction II Application

• Neuroscience maps brain functions

to brain boundary (surface)

• Mean brain scan brain surface

PhD Defense Topology Simplification Algorithm
for Segmentation of Medical Scans S. Jaume Dec
17, 2003 4
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Introduction III Goal

• Segmentation classification of

voxels representing the brain

• Problems

- errors difficult to find
- holes at brain boundary

• Goal

- correction of segmentation
- automated correction
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Topology I Continuous Topology

• Study of shape properties preserved

through deformations, twistings,
and stretchings (but no tearings)

• Homeomorphism equivalence

- sphere ellipsoid
- torus cup
Massey 1967
PhD Defense Topology Simplification Algorithm
for Segmentation of Medical Scans S. Jaume Dec
17, 2003 6
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Topology II Non-Separating Loops
Non-separating loops

• Closed lines on the surface

• Surface still in one piece

• Move from one side to the other

Topology simplification
B

• Fill loop A

A
A

• Empty loop B

B
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Topology III Discrete Topology

• Euler characteristic

X -C F E V
g ( 2 K X ) / 2
C cubes
V vertices
F faces
K components
E edges
g genus, holes

• Examples

F 6 , E 12, V 8, X 2, g 0 hole
F 16 , E 32, V 16, X 0, g 1 hole

• But no localization, no simplification

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Related Work I Various Methods

• Conquering voxels Mangin95, Han03

Problem large unconquered regions

• Deformable surfaces Davatzikos95, Bischoff03

Problem do not enter into brain folds

• Mesh methods Axen98, Guskov01, Fischl01

Problem complexity
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Related Work II Reeb Graph
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Related Work II Reeb Graph
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Related Work II Reeb Graph
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Related Work II Reeb Graph
hole between 2 planes
no cycle in the graph
Need another method to detect holes between 2
planes
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Algorithm Outline

• Topology Detection

Have we enclosed a hole?
Contribution single exploration of image

• Topology Localization

What is the extent of the hole?
Contribution more accurate localization

• Topology simplification

How to remove the hole from the image?
Contribution less complex rasterization
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Algorithm I Topology Detection
Have we enclosed a hole?
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Algorithm I Topology Detection
Have we enclosed a hole?

• wavefront splits

• Hole starts? or 2 branches?

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Algorithm I Topology Detection
Have we enclosed a hole?
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Algorithm I Topology Detection
Have we enclosed a hole?

• Wavefronts merge

• we have enclosed a hole

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Algorithm I Topology Detection
Have we enclosed a hole?
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Algorithm II Topology Localization
destination contour

• Dijkstras algorithm

• shortest path around hole

start contour
Reeb loop
cross loop
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Algorithm III Topology Simplification
How to remove the hole from the image?
Rasterization transform loop surface
into voxels
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Results I Visualization
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Results II Statistics
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Discussion Comparison
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Conclusion

• Algorithm to provide brain surface from
  segmentation

• Reduced complexity with wavefront traversal

• Better accuracy with a shorter loop

• Reduced complexity for rasterization of loop

• Software available for doctors

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Perspectives

• Automated segmentation is possible

• Other applications data compression, texture
  mapping

• Understanding topology recognition, indexation
  search

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Agenda

• Medical Scans
• Segmentation
• Continuous Topology
• Generating Loops
• Discrete Topology
• Various Methods
• Reeb Graph Methods
• Contributions
• Topology Detection
• Topology Localization
• Topology Simplification
• Visualization of Results
• Statistics
• Conclusion

1. Introduction
2. Theory of Topology
3. Related Work
4. Algorithm
5. Results
6. Conclusions
PhD Defense Topology Simplification Algorithm
for Segmentation of Medical Scans S. Jaume Dec
17, 2003 27