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

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. 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 1 4

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

7. Brain structure in schizophrenia
Regions of significant but subtle brain atrophy in patients w/ 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 
Log-Jacobian
Residual

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

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
Non-uniqueness A B
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
(0,1,0) I1
(0,0,1)
(1,0,0) 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. K-NN classification and related techniques?
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]
Optimal [h*, R*]

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

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
Global PCA
Optimal, L1 norm
Local 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: W WTF = 0
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