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