Wei Zeng Joseph Marino Xianfeng Gu Arie Kaufman Stony Brook University, New York, USA

1. Conformal Geometry Based Supine and Prone Colon Registration Wei Zeng Joseph Marino Xianfeng GuArie Kaufman Stony Brook University, New York, USA The MICCAI 2010 Workshop on Virtual Colonoscopy and Abdominal Imaging 2010-09-20

2. Overview • 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

3. Algorithm Supine & Prone ColonSurfaces (S1, S2) Anatomical Landmark Extraction Conformal Mapping(φ1, φ2) Holomorphic Differentials Internal Feature Detection & Matching Constraints: FeatureCorrespondence of (S1, S2) Harmonic Map Registration Harmonic Energy Linear System Optimization

4. Anatomical Landmarks Extraction • Idea: Extract anatomical landmarks using existing methods • Taenia coli – Slicing the colon surface open • Flexures – Dividing the colon to 5 segments Taenia Coli Flexures

5. Conformal Map - Holomorphic Differentials • Idea:Solve harmonic functions with Dirichlet boundary conditions. • Colon segment: topological cylinder, denoted as triangular mesh 3D SurfaceNon-rigid Deformation 2D Conformal MapDifferent Conformal Modules Texture MapAngle Preserving

6. Internal Feature Detection and Matching • 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 SegmentationHaustral Folds ExtractionFeature Points MatchingFeature Correspondence 2D Conformal MapMean Curvature

7. Conformal Map - Matching Framework 3D Surface 3D Surface 2D Conformal Map 2D Conformal Map

8. Conformal Map Based Surface Matching • Idea: Compute harmonic map between two 2D maps with feature correspondence constraints • One-to-one mapping • Linear computational complexity Polyp on Supine Polyp on Prone Supine => Prone Deformed Supine Registration

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

10. Conclusion • Conformal GeometryforSupine-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

11. Questions? Thanks!