Christos Davatzikos Director, Section of Biomedical Image Analysis Department of Radiology

1. Morphological Appearance Manifolds for Computational Anatomy: Group-wise Registration and Morphological Analysis Christos Davatzikos Director, Section of Biomedical Image Analysis Department of Radiology Joint Affilliations: Electrical + Systems Engineering Bioengineering University of Pennsylvania http://www.rad.upenn.edu/sbia

2. h(.)

3. ≈ MR image Warped template Template Template Subject

4. Det J (.) < 1 Elastic or fluid transformation Shape A (a<1) * Identity transformation + Residual Shape B The diffeomorphism is not the best way to describe these shape differences: the residual, after a “reasonable” alignment, is better

5. 1 1 4 1 1 Earlier attempts to include residuals • Tissue-preserving shape transformations (RAVENS maps) (Davatzikos et.al., 1998, 2001) • “modulated” VBM, Ashburner et.al., 2001 RAVENS map Original shape

6. A variety of studies of aging, AD , schizophrenia, … Regions of longitudinal decrease of RAVENS maps in healthy elderly Alzheimer’s Disease

7. Regions of significant but subtle brain atrophy in patients w/ schizophrenia • Brain structure in schizophrenia T-statistic Machine learning tools for identification of spatial patterns of brain structure Davatzikos et.al., Arch. of Gen. Psych.

8. Extended Formulation for Computational Anatomy: Lossless representation

9. HAMMER: Deformable registration • Each voxel has an attribute vector used as “morphological signature” in matching template to target • Hierarchical matching: from high-confidence correspondence to lower-confidence correspondence • (Shen and Davatzikos, 2002) Average Template Registration of 158 brains of older adults

10. Synthesized Atrophy (thinning) Shapes with thinning Shapes w/o thinning

11. Voxel-based statistical analysis  Statistical test (VBM, DBM, TBM, …) Registration algorithm: (Image/Feature Matching) + λ (Regularization)

12. Detected atrophy: p-values of group differences for different  and  Residual Log-Jacobian

13. Detected atrophy: p-values of group differences for different  and  Residual Log-Jacobian

14. M = [h, Ri] or [log det(J), Ri] as morphological descriptor (Image/Feature Matching) + λ (Regularization) Small λ Small Residual R Large λ Large Residual R Non-uniqueness: a problem

15.   Inter-individual and group comparisons depend on the template Template 1 Non-uniqueness B A Template 2 Group average templates alleviate this problem to some extent, but still they are single templates

17. Anatomical Equivalence Classes formed by varying θ

18. Related work in Computer Vision: Image Appearance Manifolds • Variations in lighting conditions • Pose differences

19. Image appearance manifolds: Facial expression

20. …. Morphological Appearance Manifolds

21. Problem: Non-differentiability of IAM I2 (0,1,0) I1 (1,0,0) (0,0,1) I3 • Spatial smoothing of images  Scale-space approximations of IAM • Smoothing of the manifold via local PCA or other method

22. From Wakin, Donoho, et.al.

23. Some things that can be done with non-unique representations: K-NN classification and related techniques? Not appropriate for analysis Non-metric distance

24. Find the points on these manifolds that minimize variance

25. Unique morphological descriptor • Group-wise registration

26. Initial Linear Approximation of the Manifolds: PCA

27. Results from synthesized atrophy detection Optimal (min variance) Representation Log-Jacobian has much poorer detection sensitivity

28. Best result obtained for the un-optimized [h,R] T2 T3 T1 Optimal [h*, R*]

30. Jacobian is highly insufficient and dependent on regularization • Excellent detection of group difference and stability for the optimal descriptor Minimum p-values

31. Best [h, R] ( = 7) Optimal [h*, R*] Detected atrophy agrees with the simulated atrophy

32. Robust measurement of change in serial scans Time-point 1 • Longitudinal atrophy was simulated in 12 MRI scans • Plots of estimated atrophy were examined for un-optimized and optimized descriptors Time-point 2 Time-point L

33. Regions With Simulated Atrophy

34. Linear MAM approximation • Global PCA where is the mean of AEC and Vij is the eigen vectors • Limitation: cannot capture the nonlinearity of AEC

35. Locally-linear MAM approximation

36. Experimental results • Shifted 2D subjects • Shift the 2D subject randomly. Healthy subjects Patient subjects with atrophy

37. Experimental results • Shifted 2D subjects

38. Experimental results • Shifted 2D subjects Determinant of Jacobian RAVENS map (smaller ) RAVENS map (Larger ) Optimal, L2 norm Global PCA Optimal, L1 norm Local PCA Optimal, L1 norm Global PCA

39. Some of the findings using nonlinear MAM approximation • Nonlinear approximations don’t necessarily improve the results, and are certainly more vulnerable to local minima • (smoothness or local minima might be the reasons) • L1-norm is a better criterion of image similarity than L2-norm

40. Limitation: L1 distance criterion is non-differentiable. • Method: Convex programming (S. Boyd and L. Vandenberghe, 2004)

41. Optimization Criterion • L1 distance criterion • Based on PCA representation: rewrite the difference of the ith and jth subjects as where , and • To simplify the expression, set , , , and then

42. Optimization Criterion • L1 distance criterion and convex programming • L1 distance criterion: • Let , and . Then L1 distance criterion becomes: We can use convex programming to optimize the cost function.

43. Sparse Image Representations Curse of Dimensionality in High-D Classification Non-negative matrix factorization (NMF): We can assume sample can be represented as multiplication of low rank positive matrices • It is experimentally (and under some conditions mathematically) that it leads to part-based representation of image • non-negativity yields sparsity? Not necessarily, many revision has been proposed (Orthogonality while keeping positivity, …)

44. Optimal NMF decomposition in Alzheimer’s Disease

45. Extension of NMF: • Find directions that form good discriminants between two groups (e.g. patients and controls) • Prefer certain directions (prior knowledge) • Avoid certain directions (e.g. directions along MAM’s) MAM1 W MAM2 MAML WTF = 0

46. Conclusion • The conventional computational anatomy framework can be insufficient • is a complete (lossless) morphological descriptor • Non-uniqueness is resolved by solving a minimum-variance optimization problem • Robust anatomical features can potentially be extracted by seeking directions that are orthogonal to MAMs

47. Thanks to … • Sokratis Makrogiannis • Sajjad Baloch • Naixiang Lian • Kayhan Batmanghelich