1. 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

2. 2 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

3. 3 Algorithm Anatomical Landmark Extraction Constraints: Feature Correspondence of (S 1, S 2 ) Harmonic Energy Linear System Optimization Conformal Mapping (φ 1, φ 2 ) Supine & Prone Colon Surfaces (S 1, S 2 ) Internal Feature Detection & Matching Harmonic Map Registration Holomorphic Differentials

4. 4 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 ColiFlexures

5. 5 Idea: Solve harmonic functions with Dirichlet boundary conditions. –Colon segment: topological cylinder, denoted as triangular mesh Conformal Map - Holomorphic Differentials 3D Surface Non-rigid Deformation 2D Conformal Map Different Conformal Modules Texture Map Angle Preserving

6. 6 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 Map Mean Curvature Segmentation Haustral Folds Extraction Feature Points Matching Feature Correspondence

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

8. 8 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 =>ProneDeformed SupineRegistration Polyp on Prone Polyp on Supine

9. 9 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 R 3 (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. MethodsDistance Error Our Conformal Geometry Based Method7.85mm 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

10. 10 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

11. 11 Questions? Thanks!