Wei Zeng Joseph Marino Xianfeng Gu Arie Kaufman

Stony Brook University, New York, USA

The MICCAI 2010 Workshop on Virtual Colonoscopy and Abdominal Imaging

2010-09-20

Conformal Geometry Based Supine and Prone Colon Registration

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• Problem - Supine and Prone Colon Registration– Challenge: Non-rigid deformation and substantial distortion,

due to position shifting

• Solution - Conformal Mapping Based Registration– Formulation: Matching between 3D topological cylinders– Key: 3D => 2D matching problem– Goal: One-to-one map

• Contribution - Diffeomorphism between Surfaces– Advantage: Guarantee one-to-one map of whole surface– Efficiency: Linear time complexity

Overview

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Algorithm

Anatomical Landmark Extraction

Constraints: FeatureCorrespondence of (S1, S2)

Harmonic Energy

Linear System Optimization

Conformal Mapping(φ1, φ2)

Supine & Prone ColonSurfaces (S1, S2)

Internal Feature Detection & Matching

Harmonic Map Registration

Holomorphic Differentials

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• Idea: Extract anatomical landmarks using existing methods– Taenia coli – Slicing the colon surface open– Flexures – Dividing the colon to 5 segments

Anatomical Landmarks Extraction

Taenia Coli Flexures

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• Idea: Solve harmonic functions with Dirichlet boundary conditions.– Colon segment: topological cylinder, denoted as triangular mesh

Conformal Map - Holomorphic Differentials

3D SurfaceNon-rigid Deformation

2D Conformal MapDifferent Conformal Modules

Texture MapAngle Preserving

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• Idea: Perform detection and matching on conformal mapping images color encoded by mean curvature of 3D surface.– Method: 1) Graph Cut Segmentation and 2) Graph Matching methods

Internal Feature Detection and Matching

2D Conformal MapMean Curvature

Segmentation Haustral Folds

ExtractionFeature Points

MatchingFeature Correspondence

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Conformal Map - Matching Framework

3D Surface

2D Conformal Map

3D Surface

2D Conformal Map

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• Idea: Compute harmonic map between two 2D maps with feature correspondence constraints– One-to-one mapping– Linear computational complexity

Conformal Map Based Surface Matching

Supine =>Prone Deformed Supine Registration

Polyp on Prone

Polyp on Supine

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• Data– National Institute of Biomedical Imaging and Bioengineering (NIBIB) Image and

Clinical Data Repository, provided by the National Institute of Health (NIH) • Registration Accuracy

– Averaged distance error in R3 (mm)– Better than existing centerline-based methods, similar to [4]

• Advantage: One-to-one surface registration

Experiments

Table 1. Comparison of average millimeter distance error between existing methods.

Methods Distance Error

Our Conformal Geometry Based Method 7.85mm

Haustral fold registration [4] 5.03 mm

Centerline registration + statistical analysis [12] 12.66mm

Linear stretching / shrinking of centerline [1] 13.20mm

Centerline feature matching + lumen deformation [14] 13.77mm

Centerline point correlation [3] 20.00mm

Taenia coli correlation [10] 23.33mm

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Conclusion

• Conformal Geometry for Supine-Prone Registration– 3D problem => 2D matching problem– Internal feature correspondence based on 2D conformal

mapping images color encoded by mean curvature. – Surface registration by harmonic map with feature

correspondences, not only the feature points.

• Advantage– One-to-one and onto surface registration (diffeomorphism)– Efficiency: linear time complexity– Accuracy: low averaged distance error

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Questions?

Thanks!