1. An Integrated Pose and Correspondence Approach to Image Matching Anand Rangarajan Image Processing and Analysis Group Departments of Electrical Engineering and Diagnostic Radiology Yale University

2. Motivation I Human Brain Mapping: –Different subjects. Statistical analysis. Normal vs. abnormal. –Different times. Detect significant change, help diagnosis. –Different modalities. Combine complementary information.

3. Motivation II Difficulty : –Variability in pose, size, shape and acquisition. Brain registration : –Common coordinate frame. –Data comparable. –Quantitative analysis.

4. Results Interactive 3D Sulcal Tracing

5. Overview Extract features: –Sulcal traces represented as point sets. –Labeling, ordering information [optional]. Jointly solve feature correspondence and spatial mapping.

6. Overview II Part II: Information Analysis: –Measurements. –Learn from the data, construct statistical models. e.g., probabilistic atlas for structures / functions. –Make inference for new data based on the learned models. e.g., automated sulcal labeling, segmentation, computer aided diagnosis.

7. Outline Related work. The approach. –Point-based representation of sulci. –Robust point matching algorithm. Results and examples. Future work.

8. Other Work in Brain Registration Voxel-based methods: –Volumetric Warping: Christensen et al., Gee et al., Collins et al. Feature-based methods: –Landmarks: Bookstein. –Curves: Sandor and Leahy, Collins et al. –Surfaces: Thompson et al., Davatzikos et al. –Sulcal Graphs: Lohmann and von Cramon.

9. Approach Rationale Voxel intensity matching does not ensure that corresponding sulci indeed match. Landmarks hard to define. Extraction, representation and matching of cortical curves / surfaces / graphs is difficult.

10. Our Approach Point-based Representation Hundreds of points, statistically more robust than just a few landmarks. Additional information can be used: –Major sulcal labels. Further analyses made easy: –Procrustes mean. –Eigen-analysis of the error covariance matrix.

11. Our Approach Robust Point Matching (RPM) Estimation : –Correspondence and spatial mapping. Softassign: –Soft correspondence. –Allows partial matching, noise. –Less sensitive to local minima. Handles outliers.

12. Robust Point Matching Alternating Optimization When correspondence M is known, standard least squares solution for spatial mapping A. When spatial mapping A is fixed, assignment solution for correspondence M. –Softassign - soft correspondence. –Deterministic Annealing - temperature T.

13. Robust Point Matching Energy Function

14. Robust Point Matching Step I. Solve Spatial Mapping Given correspondence M, find the optimal spatial mapping A (affine): Standard least-squares solution. Gradually relaxed regularization on 

15. Robust Point Matching Part II. Softassign Given spatial mapping A, solve the Linear Assignment Problem: subject to

16. Robust Point Matching Step II. Softassign Two-way constraints M ij M M  i Row Normalization M ij M M  j Col. Normalization Positivity =exp( )Q ij  M Step I: M ij = exp ( - Q ij /T). Step II: Double Normalization. Sinkhorn’s Algorithm. Outlier rejection using slack variables.

17. Robust Point Matching Part II. Softassign Deterministic Annealing : –T as an extra parameter. –F = E assign - TS = Gibbs Distribution : –Positivity ganranteed. –High T, insensitive to Q, uniform M. –Low T, sensitive to Q, binary M.

18. Robust Point Matching Algorithm Summary Start: uniform M, high temperature T. Do until final temperature is reached. –Given M, solve for spatial mapping A. –Given A, use Softassign to update M. Decrease temperature.

19. Experiment on Brain Sections

20. Results of Method

21. Results Interactive 3D Sulcal Tracing

22. Results RPM Example Two labeled sulcal point sets, initial position.

23. RPM without label information

24. Results Visual Matching Comparison

26. Quantitative Comparison

28. Future Work Error measure on the entire volume. Fully non-rigid 3D spatial mapping. –Thin-plate spline and correspondence. Automated sulcal extraction, Zeng et al. Investigate partially labeled case. Automated labeling. Atlas construction.

29. The End

30. Thin-plate-spline Implementation

32. Results Visual Matching Comparison TPS