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1 Gaussian Random Fields for Statistical Characterization of Brain Submanifolds. Sarang Joshi Department of Radiation Oncology and Biomedical Enginereing University of North Carolina at Chapel Hill Sarang Joshi May 3,

Global Shape Models for Computational Anatomy.

: Collection of 0,1,2, and 3-dimensional compact sub-manifolds M of

H: Family of transformations of accommodating variability. h 2 H : 3.

2 Global Shape Models for Computational Anatomy. Homogeneous anatomy characterized by ( H I P). 1. : Collection of 0,1,2, and 3-dimensional compact sub-manifolds M of IR 3, = [ M : 2. H: Family of transformations of accommodating variability. h 2 H : 3. I: set of anatomical imagery fmri, CT, PET, CRYOSEC- TIONg. I 2 I :! IR N 4. P : probability measures on the space of transformations H. Sarang Joshi May 3,

3 Global Shape Models for Computational Anatomy. H constructed from group of dieomorphisms of coordinate system. h 2 H dened via vector elds of displacements. x = (x 1 x 2 x 3 ) 2 7! h(x) = x ; (u 1 (x) u 2 (x) u 3 (x)) I: Homogeneous space of the group H. 1. Two images I 1 I 2 2 I are topologically equivalent. 2. 9h 2 H such that I 1 = I 2 (h(x)) x h ;1 2 H such that I 2 = I 1 (h ;1 (x)) x2. 4. I: orbit of a single anatomy I 0, under the group action H: I H I 0 Anatomical variability understood via empirical construction of probability measures P on H. 1. Given family of anatomical images fi 0 I 1 I N g construct \registration" transformations fh i i = 1 Ng h i 2 H mapping provisory template I 0 to the family. 2. Given maps fh i i = 1 Ng estimate P. Sarang Joshi May 3,

Representation of Sub-Structures of the Brain.

4 Representation of Sub-Structures of the Brain. The surface M of neuro-anatomically signicant substructure is assumed to be a smooth two-dimensional C 2 sub-manifold of IR 3. Build local quadratic charts. T xi (M) n i h j x j x i b 1xi x p j Sarang Joshi May 3, b 2x i

5 Shape of 2-D Sub-Manifolds of the Brain: Hippocampus. The provisory template hippocampal surface M 0 is carried onto the family of targets: M 0 h 1 h ;! 2 h ; ;! M 1 M 0 ; ;! N M 2 M 0 ; h ;1 1 h ;1 2 h ;1 N M N : h 1 o M 0 h 1 h 2 h 2 o M 0 M 0 Template h N h N o M 0 Sarang Joshi May 3,

6 Shape of 2-D Sub-Manifolds of the Brain: Hippocampus. The mean transformation and the template representing the entire population: h = 1 N NX i=1 h i M temp = h M 0 : The mean hippocampus of the population of thirty subjects. Sarang Joshi May 3,

7 Shape of 2-D Sub-Manifolds of the Brain: Hippocampus. Mean hippocampus representing the control population: h control = 1 N control N control X i=1 h control i M control = h control M 0 : Mean hippocampus representing the Schizophrenic population: h schiz = 1 N schiz N schiz X i=1 h schiz i M schiz = h schiz M 0 : Control population Schizophrenic population Sarang Joshi May 3,

8 Gaussian Random Vector Fields on 2-D Sub-Manifolds. Hippocampi M i i= 1 N deformation of the mean M temp : M i : fyjy = x + u i (x) x2 M temp g u i (x) = h i (x) ; x x 2 M temp : Vector eld u i (x) shown in red. Construct Gaussian random vector elds over sub-manifolds. Sarang Joshi May 3,

9 Gaussian Random Vector Fields on 2-D Sub-Manifolds. Let H(M) be the Hilbert space of square integrable vector elds on M. Inner product on the Hilbert space H(M): hf gi = 3 X i=1 Z M f i (x)g i (x)d(x) where d is a measure on the oriented manifold M. Denition 1 The random eld fu(x) x 2 Mg is a Gaussian random eld on a manifold M with mean u 2 H(M) and covariance operator K u (x y) if 8f 2 H(M), hf i is normally distributed with mean m f 2 f = hk uf fi = h u fi and variance Gaussian eld is completely specied by it's mean u and the covariance operator K u (x y). Construct Gaussian random elds as a quadratic mean limit using a complete IR 3 -valued orthonormal basis f k k = 1 2 g h i j i = 0 i6= j Sarang Joshi May 3,

10 Isotropic Stochastic Process on The Sphere: Oboukhov expansion Let H(S) be the Hilbert space of square integrable functions on the sphere S. Inner product on the Hilbert space H(S): hf gi = Z S f( )g( )sin()dd where sin()dd is the measure on the sphere S. The Spherical Harmonics Y m n are a complete orthonormal basis of H(S) Dene process fu(p) p2 Sg u(p) = lim N!1 in q:m: NX nx Z nmyn m (p) n=0 m=;n where Z nm are zero mean independent Gaussian random variables with variance n with P n < 1. Theorem: The stochastic process fu(p) p 2 Sg constructed above is an isotropic zero mean q.m. continuous Gaussian process with covariance K(x y) = 1 X n=0 n P n (cos d(x y)) Sarang Joshi May 3,

11 Gaussian Random Vector Fields on 2-D Sub-Manifolds. Theorem 1 Let fu(x) x2 Mg be a Gaussian random vector eld with mean m U 2 H and covariance K U of nite trace. There exists a sequence of nite dimensional Gaussian random vector elds fu n (x)g such that where U(x) q.m. = lim n!1 U n (x) U n (x) = n X k=1 Z k (!) k (x) fz k (!) k = 1 g are independent Gaussian random variables with xed means EfZ k g = k and covariances EfjZ i j 2 g; EfZ i g 2 = 2 i = i P i i < 1 and (hk k ) are the eigen functions and the eigen values of the covariance operator K U : i i (x) = Z M K U(x y) i (y)d(y) where d is the measure on the manifold M. If d, the surface measure on M temp is atomic around the points x k then f i g satisfy the system of linear equations i i (x k ) = M X j=1 ^K U (x k y j ) i (y j )(y j ) i= 1 N where (y j ) is the surface measure around point y j. Sarang Joshi May 3,

12 Eigen Shapes of the Hippocampus. Assume that deformation elds fu i (x) i = 1 Ng are realizations from a Gaussian eld on the surface of the mean hippocampus M temp. Empirical estimate of the covariance operator given by ^K U (x y) = 1 N ; 1 NX i=1 u i (x)u i (y) T : Numerically compute the eigenfunctions and eigenvalues of using Singular Value Decomposition: 1. Let f (i) i = 1 Ng be vectors of length 3M with (i) k = i (x k ) 2. be a diagonal matrix of size 3M 3M with 3j 3j = 3j+1 3j+1 = 3j+2 3j+2 = (y j ) 3. ^KU be a 3M 3M symmetric matrix with ^K i j = ^K U (x i x j ): The system of linear equations in the above theorem becomes i (i) = ^K (i) i= 1 N: The basis vectors f (i) i = 1 Ng are generated by diagonalizing the matrix ^K. Sarang Joshi May 3,

13 Eigen Shapes of the Hippocampus. Eigen shapes E i i = 1 N dened as: E i = fx + ( i ) i (x) : x 2 M temp g : Eigen shapes completely characterize the variation of the submanifold in the population. Sarang Joshi May 3,

14 Statistical Signicance of Shape Dierence Between Populations. Assume that fu schiz j u control j g j = 1 15 are realizations from a Gaussian process with mean u schiz and u control and common covariance K U. Statistical hypothesis test on shape dierence: H 0 : u norm H 1 : u norm = u schiz 6= u schiz Expand the deformation elds in the eigen functions i : fz schiz j u schiz(j) N (x) = N X i=1 u control(j) N (x) = N X i=1 Z schiz(j) i i (x) Z control(j) i i (x) Zj control j = 1 15g Gaussian random vectors with means Z schiz and Z control and covariance. Hotelling's T 2 test: T 2 N = M 2 ( ^ Z norm ; ^ Z schiz ) T ^;1 ( ^ Z norm ; ^ Z schiz ) : N T 2 N p-value : P N (H 0 ) N: number of eigen functions. Sarang Joshi May 3,

15 Bayesian Classication on Hippocampus Shape Between Population. Bayesian log-likelihood ratio test: H 0 : normal hippocampus, H 1 : schizophrenic hippocampus. N = ;(Z ; ^ Z schiz ) y ^;1 (Z ; ^ Z schiz ) + (Z ; Z ^ norm ) y ^;1 (Z ; Z ^ H 0 < norm ) > 0 H 1 Use Jack Knife for estimating probability of classication: Log Likelihood Ratio Control Schiz Sarang Joshi May 3,

16 DISTRIBUTION FREE STATISTICAL TESTING. Use Fisher's method of randomization to empirically estimate the distribution of the test statistics with out the Gaussian assumption. Under the null hypothesis H 0 the expansion coecients (Z N i ) schiz Z N i ) control ) for i = 1 M are independent random samples from a single population. Each of the ( 2N N ) possible permutations of the data are equally likely and can be used for estimating the distribution of the test statistics under the hypothesis H 0. For N = 15, 1:551175e + 08 dierent combinations. Use monte carlo simulations for estimating the probability distribution by generating uniformly distributed random combinations. Sarang Joshi May 3,

17 DISTRIBUTION FREE STATISTICAL TESTING. Estimate probability distribution of the T 2 statistics under the hypothesis H 0 computed using monte carlo simulations Number of Simulations = 100,000 P = The signicance level or the p-value becomes: P = Z 1 T 2 ^F (f)df : Using 4 eigen shapes the p-value is estimated at 0:0232 Statistically signicant dierence in the populations. Sarang Joshi May 3,

A 4000 Hippocampus volume (mm 3 ) 3500 3000 2500 2000 1500 1000 L R L R L R Younger control CDR 0 CDR 0.

CDR 0.5 Consistent CDR 0.5 John G. Csernansky, Lei Wang, Sarang Joshi, J.

18 A 4000 Hippocampus volume (mm 3 ) L R L R L R Younger control CDR 0 CDR 0.5 B 4000 Hippocampus volume (mm 3 ) L R L R L R L R Younger control Consistent CDR 0 Inconsistent CDR 0.5 Consistent CDR 0.5 John G. Csernansky, Lei Wang, Sarang Joshi, J. Philip Miller, Mohktar Gado, Daniel Kido, Daniel McKeel, John C. Morris, Michael I. Miller, Hippocampal Deformities Distinguish Dementia of the Alzheimer Type from Healthy Aging, submitted to Science, 1999.

A Outward, p < 0.05 p > 0.05 Inward, p < 0.05 R L R L B Inconsistent CDR 0.5 vs. Consistent CDR 0 Outward, 3.3mm Consistent CDR 0.5 vs. Consistent CDR 0 Inward, 3.3mm R L R L C p =.015 p =.

19 A Outward, p < 0.05 p > 0.05 Inward, p < 0.05 R L R L B Inconsistent CDR 0.5 vs. Consistent CDR 0 Outward, 3.3mm Consistent CDR 0.5 vs. Consistent CDR 0 Inward, 3.3mm R L R L C p =.015 p =.0015 Log-likelihood ratio Log-likelihood ratio Consistent CDR 0 Inconsistent CDR 0.5 Consistent CDR 0 Consistent CDR 0.5 John G. Csernansky, Lei Wang, Sarang Joshi, J. Philip Miller, Mohktar Gado, Daniel Kido, Daniel McKeel, John C. Morris, Michael I. Miller, Hippocampal Deformities Distinguish Dementia of the Alzheimer Type from Healthy Aging, submitted to Science, 1999.

A Outward, p < 0.05 p > 0.05 Inward, p < 0.05 R L Consistent CDR 0 vs. Younger control B Outward, 3.3mm R L Inward, 3.3mm C p <.001 Log-likelihood ratio Younger control Consistent CDR 0 John G.

20 A Outward, p < 0.05 p > 0.05 Inward, p < 0.05 R L Consistent CDR 0 vs. Younger control B Outward, 3.3mm R L Inward, 3.3mm C p <.001 Log-likelihood ratio Younger control Consistent CDR 0 John G. Csernansky, Lei Wang, Sarang Joshi, J. Philip Miller, Mohktar Gado, Daniel Kido, Daniel McKeel, John C. Morris, Michael I. Miller, Hippocampal Deformities Distinguish Dementia of the Alzheimer Type from Healthy Aging, submitted to Science, 1999.

Statistics of Shape: Eigen Shapes PCA and PGA

Sarang Joshi 10/10/2001 #1 Statistics of Shape: Eigen Shapes PCA and PGA Sarang Joshi Departments of Radiation Oncology, Biomedical Engineering and Computer Science University of North Carolina at Chapel