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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 - PowerPoint PPT Presentation
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Christos Davatzikos
Director, Section of Biomedical Image AnalysisDepartment of Radiology
Joint Affilliations: Electrical + Systems Engineering Bioengineering
University of Pennsylvaniahttp://www.rad.upenn.edu/sbia
Morphological Appearance Manifolds for Computational Anatomy: Group-wise
Registration and Morphological Analysis
T S
h(.)
Template Subject
≈Template MR image Warped template
Shape A
Shape B
Elastic or fluid transformation
(a<1) * Identity transformation + Residual
Det J (.) < 1
The diffeomorphism is not the best way to describe these shape differences: the residual, after a “reasonable” alignment, is better
Earlier attempts to include residuals
1 1
1 14
• Tissue-preserving shape transformations (RAVENS maps) (Davatzikos et.al., 1998, 2001)
• “modulated” VBM, Ashburner et.al., 2001
RAVENS mapOriginal shape
A variety of studies of aging, AD , schizophrenia, …
Regions of longitudinal decrease of RAVENS maps in healthy elderly
Alzheimer’s Disease
• Brain structure in schizophrenia
Regions of significant but Regions of significant but subtle brain atrophy in subtle brain atrophy in patients w/ schizophreniapatients w/ schizophrenia
T-statistic
T-statistic
Machine learning tools foridentification of spatial patternsof brain structure
Davatzikos et.al., Arch. of Gen. Psych.
Extended Formulation for Computational Anatomy: Lossless representation
TemplateAverage
Registration of 158 brains of older adults
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)
Synthesized Atrophy (thinning)
Shapes w/o thinning Shapes with thinning
Statistical test (VBM, DBM, TBM, …)
Voxel-based statistical analysis
(Image/Feature Matching) + λ (Regularization)
Registration algorithm:
Log-Jacobian Residual
Detected atrophy: p-values of group differences for different and
Log-Jacobian Residual
Detected atrophy: p-values of group differences for different and
M = [h, Ri] or [log det(J), Ri] as morphological descriptor
(Image/Feature Matching) + λ (Regularization)
Small λ Small Residual RLarge λ Large Residual R
Non-uniqueness: a problem
Non-uniquenessBA
Template 1
Template 2
Inter-individual and group comparisons depend on the template
Group average templates alleviate this problem to some extent, but still they are single templates
Anatomical Equivalence Classes formed by varying θ
Related work in Computer Vision: Image Appearance Manifolds
• Variations in lighting conditions
• Pose differences
Image appearance manifolds: Facial expression
…. Morphological Appearance Manifolds
Problem: Non-differentiability of IAM
• Spatial smoothing of images Scale-space approximations of IAM
• Smoothing of the manifold via local PCA or other method
(1,0,0)
(0,1,0)
(0,0,1)
I3
I2
I1
From Wakin, Donoho, et.al.
Some things that can be done with non-unique representations:
K-NN classification and related techniques?
Non-metric distance Not appropriate for analysis
Find the points on these manifolds that minimize variance
• Unique morphological descriptor
• Group-wise registration
Initial Linear Approximation of the Manifolds: PCA
Results from synthesized atrophy detection
Log-Jacobian has much poorer detection sensitivity
Optimal (min variance) Representation
Best result obtained for the un-optimized [h,R]
T1 T2 T3
Optimal [h*, R*]
Minimum p-values
• Jacobian is highly insufficient and dependent on regularization
• Excellent detection of group difference and stability for the optimal descriptor
Detected atrophy agrees with the simulated atrophy
Best [h, R] ( = 7) Optimal [h*, R*]
• Longitudinal atrophy was simulated in 12 MRI scans
• Plots of estimated atrophy were examined for un-optimized and optimized descriptors
Time-point 1
Time-point 2
Time-point L
Robust measurement of change in serial scans
Regions With Simulated Atrophy
Linear MAM approximation
iQ̂
Global PCA
where is the mean of AEC and Vij is the eigen vectors
Limitation: cannot capture the nonlinearity of AEC
Locally-linear MAM approximation
Experimental results Shifted 2D subjects
Shift the 2D subject randomly.
Healthy subjects Patient subjectswith atrophy
Experimental results Shifted 2D
subjects
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
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
Limitation: L1 distance criterion is non-differentiable. Method: Convex programming (
S. Boyd and L. Vandenberghe, 2004)
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
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.
•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, …)
2
,
min
[ ] ,[ ] 0
FF G
ij ijF G
X FG
Non-negative matrix factorization (NMF): We can assume sample can be represented as multiplication of low rank positive matrices
Sparse Image Representations
Curse of Dimensionality in High-D Classification
Optimal NMF decomposition in Alzheimer’s Disease
2
,
min ( , )
, Feasible set
FF G
J F G
F G
X FG
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)
W
WTF = 0
MAM1
MAML
MAM2
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
Thanks to …
• Sokratis Makrogiannis• Sajjad Baloch• Naixiang Lian• Kayhan Batmanghelich