Statistics of Shape: Eigen Shapes PCA and PGA

Transcription

1 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 Hill

2 Sarang Joshi 2/16/2003 #2 Study of the Shape of the Hippocampus Two approaches to study the shape variation of the hippocampus in populations: Statistics of deformation fields using Principal Components Analysis Statistics of medial descriptions using Lei Groups:- Principal Geodesic Analysis

3 Statistics of Deformation Fields Sarang Joshi 2/16/2003 #3

4 Sarang Joshi 2/16/2003 #4 Hippocampal Mapping Atlas Patients

5 Sarang Joshi 2/16/2003 #5 Hippocampal Mapping Atlas Subjects

6 ' 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, %

7 ' 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, %

8 ' 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, %

9 ' 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, %

10 ' 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, %

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. $ 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, %

13 ' 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, %

14 ' 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, %

15 Sarang Joshi 2/16/2003 #6 Statistics of Medial descriptions Each figure a quad mesh of medial atoms: { m m 0 i, j 0 i, j : = i = 1L N, j = 1L M ( x i, j, r, F, θ ) Medial atom parameters include angles and rotations. Medial atoms do not form a Hilbert Space Cannot use Eigen Shape for statistical characterization!! }

16 Sarang Joshi 2/16/2003 #7 Statistics of Medial descriptions Set of all Medial Atoms forms Lie-Group m = ( x i, j, r, F, θ ) m R 3 R + SO (3) SO (2) R 3 R + SO(3) SO(2) :Position x :Radius r :Frame :Object angle

17 Sarang Joshi 2/16/2003 #8 Lie Groups A Lie group is a group G which is also a differential manifold where the group operations are differential maps. R R µ : ( x, ι : x a y) x a 1 : G xy : a G G G a Both composition and the inverse are differential maps 3 + : µ ( x, : SO(3) SO(2) : y) x Multiplica tive : = + y, x 1 reals Orthogonal 2Orthogonal = x µ ( x, Matrix Matrix y) = G xy, Group Group x 1 = 1 x

18 Sarang Joshi 2/16/2003 #9 Lie Group Means Algebraic mean not defined on Lie Groups Use geometric definition: Remanian Distance well defined on a Manifold. Given N medial atoms m { : i = 1L N } m i the mean is defined as the group element that minimizes the average squared distance to the data. m = arg min 1 N N i= 1 d ( m, m i ) 2 No closed form solution need to use Lie-Group optimization techniques.

19 Sarang Joshi 2/16/2003 #10 Geodesic Curves Medial manifold is curved and hence no straight lines. Distance minimizing Geodesic curves are analogous to straight lines in Euclidian Space. Geodesics in Lie Groups are given by the exponent map: g ( t) = exp( ta) Geodesics are one parameter sub-groups analogous to 1-dimensional subspaces in N R

20 Sarang Joshi 2/16/2003 #11 Principal Geodesics Since the set of all medial atoms is a curved manifold linear PCA is not defined as well. Principal Geodesics are defined as the geodesics that minimize residual distance. No closed form solution: Needs non linear optimization.