Research Trajectory. Alexey A. Koloydenko. 25th January Department of Mathematics Royal Holloway, University of London

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1 Research Trajectory Alexey A. Koloydenko Department of Mathematics Royal Holloway, University of London 25th January 2016 Alexey A. Koloydenko Non-Euclidean analysis of SPD-valued data

2 Voronezh State University (RU) Norwich University (USA) University of Massachusetts Amherst (USA) University of Chicago (USA) Eurandom (NL) University of Nottingham Royal Holloway, University of London Alexey A. Koloydenko Non-Euclidean analysis of SPD-valued data

3 Natural Image Statistics Image Analysis of Speech Hidden Markov Models Biomedical Imaging Diffusion-Weighted MRI Non-Euclidean Statistics and patterns in high dimensional data Computer Aided Diagnosis and Treatment of Cancer Alexey A. Koloydenko Non-Euclidean analysis of SPD-valued data

4 Diffuion profiles - equiprobability contours [Alexander, 2006]

5 Diffusion Tensor A standard ( single compartment ) model for diffusion X N(0, 2tD), t - diffusion time, D - diffusion tensor Multicompartment models - mixtures of Gaussians K p(x) = a k f (x; 0, 2tD k ), k=1 Kk=1 a k = 1 a 1,..., a K > 0 Diffusion tensors are symmetric positive semi-definite matrices, λ represented by ellipsoids via D = V 0 λ 2 0 VT, V O(3). 0 0 λ 3

6 DT-based Measures of Connectivity/Anisotropy D = Eigenvectors e 1 = (1, 0, 0) T e 2 = (0, 1, 0) T e 3 = (0, 0, 1) T V = e 1 e 2 e 3 = I Invariant (shape) characteristics Mean diffusivity MD (Isotropic) Apparent Diffusion Coefficient ADC Volume ratio VR = λ = tr(d) 3 = 2 = (D xx + D yy + D zz )/3 =MD. = det(d) MD 3 = 3/4 1 Fractional anisotropy FA = (λ j λ) 2 j=1 3 λ 2 j j= < 1

7 Averaging, interpolation, extrapolation, regularisation, smoothing, registration and inference Need for interpolation and smoothing In group comparison studies, arrays of FA or ADC values within a certain region of interest (ROI) are compared. Multiple instances of ROI need to be co-registered. DWMRI resolution is relatively low, voxel= 1mm 1mm 2mm Solutions: Interpolation carried out directly on FA or ADC maps ignores important information (orientation, size), interpolation carried out on raw data could be unnecessarily difficult. Operate on diffusion tensor estimates D Significance: Conclusions depend on the method used [Pasternak & Basser, 2012].

8 How to average and interpolate Ds? Ω is the set of 3 3 non-negative definite symmetric matrices D. If D 1, D 2..., D N Ω and w 1, w 2,..., w N are non-negative weights with w i = 1, then D E (w 1, w 2,..., w N ) = N w i D i Ω. i=1 tr D E (w 1, w 2,..., w N ) = N w i tr D i. i=1 tr D 1 = tr D 2 tr D E (w 1, w 2 ) = tr D 1 = tr D 2 for all weights with w 1 + w 2 = 1. [Pasternak & Basser, 2012] argue preserving MD/ADC is desirable.

9 Alternatives Negative weigths (extrapolation) lead out of space D E (w 1, w 2 ) Ω. Numerical estimation algorithms require explicit constraints to prevent this. [Pennec, 2009, X. Pennec & Ayache, 2006] argue preservation of volume (determinant) is more important: (1 α) confidence ellipsoid has volume 4 3 π tc α det D. However, Euclidian averages are prone to swelling: det D E (w 1, w 2 ) det D i.

10 Swelling D E (w 1, w 2 ) = w 1 D 1 + w 2 D 2, w i 0 The Minkowski determinant Theorem [Marcus & Minc, 1992] gives (det D E (w 1, w 2 )) 1 3 w1 (det D 1 ) w2 (det D 2 ) 1 3 with the equality if and only if D 2 = cd 1 for some c 0. Thus, w 1 + w 2 = 1 det D E (w 1, w 2 ) min{det D 1, det D 2 }, and if det D 1 = det D 2 but the tensors are distinct, swelling is inevitable as for positive weights w 1, w 2. det D E (w 1, w 2 ) > det D 1 = det D 2

11 Metrics Given a metric d and probability measure ν on Ω ED = arg inf d(d, D ) 2 ν(dd) D Ω is known as the Fréchet, or Karcher, mean. For finite samples, weighted Fréchet sample average D d (w 1, w 2,..., w N ) = arg min D Ω The Euclidean distance N w i d(d i, D ) 2. d E (D 1, D 2 ) = D 1 D 2 = tr((d 1 D 2 ) T (D 1 D 2 )) gives the Euclidean (weighted) Fréchet sample average N i=1 w i D i = arg min D Ω i=1 N w i d E (D i, D ) 2. i=1

12 Relevant non-euclidian Riemannian distances Affine invariant Riemannian metric [Batchelor et al., 2005, Fletcher & Joshi, 2007, Moakher, 2005, Pennec, 2009, Pennec et al., 2006, Schwartzman et al., 2008, X. Pennec & Ayache, 2006] on Ω + : d R (D 1, D 2 ) = log ( ) D 1/2 1 D 2 D 1/2 1 = d R (I, D 1/2 1 D 2 D 1/2 1 ), gives a geometric weighted average w 1 +w 2 ( ) 2 D R (w 1, w 2 ) = D1 D D 2D 1 w2 w 1 +w D 2 1, but no closed form solution for N > 2, uses a numerical algorithm. det D R (w 1, w 2 ) = (det D 1 ) w 1 det D 2 ) w 2 det D 1 = det D 2 det D R (w 1, w 2 ) = det D i

13 Log-Euclidian [Arsigny et al., 2007, Pennec, 2009, X. Pennec & Ayache, 2006] d L (D 1, D 2 ) = log(d 1 ) log(d 2 ) Gives another geometric weighted average D L (w 1, w 2,..., w N ) = exp ( Ni=1 w i log(d i ) ), efficiently computed. Otherwise, generally very similar to d R : det D L (w 1, w 2 ) = det D R (w 1, w 2 ) det D 1 = det D 2 det D L (w 1, w 2 ) = det D i. FA(D R (w 1, w 2 )) FA(D L (w 1, w 2 )).

14 Power-Euclidean metric d A (D 1, D 2 a) = 1 a Da 1 Da 2 a 0 + d L(D 1, D 2 ) gives the power Euclidean weighted average 1a N D A (w 1, w 2,..., w N a) = w i D a i. tr D a 1 = tr Da 2 tr D A(w 1, w 2 a) a = tr D a i. i=1 FA(D a ) is increasing in a > 0 [Zhou et al., 2015].

15 Square root-based distance [Dryden et al., 2009] d H (D 1, D 2 ) = 2d A (D 1, D 2 0.5) = D 1/2 1 D 1/2 2. N D H (w 1, w 2,..., w N ) = i=1 w i D 1 2 i 2. tr D = tr D tr D H (w 1, w 2 ) 1 2 = tr D 1 2 i. PA(D) =FA(D 1 2 ) FA(D).

16 Procrustes size-and-shape [Dryden et al., 2009], [Le, 1988], [Kendall, 1989] where Q i Q T i = D i. d S (D 1, D 2 ) = min R O(3) Q 1 Q 2 R, The minimizer ˆR = UV T, where U, V are as in SVD Q T 1 Q 2 = V U T. Q i = D 1 2 i ˆR S O(3). D S (w 1, w 2 ) = (w 1 Q 1 + w 2 Q 2 ˆR)(w 1 Q 1 + w 2 Q 2 ˆR) T, for N > 2 need an iterative algorithm to compute D S (w 1, w 2,..., w N ). Weighted Generalized Procrustes Algorithm [Zhou et al., 2015].

17 What does d S preserve? rank D 1 = rank D 2 rank D S (w 1, w 2 ) = rank D i [Zhou et al., 2015]. Thus, planar (linear) diffusion remains planar (linear) in interpolation and extrapolation. General statistical inference: Confidence bounds can be computed and effects tested by mapping the data to the tangent space of Ω at D S (w 1, w 2,..., w N ) (typically with w i = 1/N) [Dryden et al., 2009].

18 A key relationship between d H and d S Theorem 1 Let D 1, D 2 Ω. Then, 0.5dH (D 1, D 2 ) d S (D 1, D 2 ) d H (D 1, D 2 ). Moreover, if D 1 D 2, and D 1 and D 2 are of rank 1, then 0.5dH (D 1, D 2 ) < d S (D 1, D 2 ) and d S (D 1, D 2 )/d H (D 1, D 2 ) 0.5 as d(d 1, D 2 ) 0 in any metric d.

19 Trace and Determinant Inequalities: Theorem 2 Let w 1, w 2 be probability weights, and let D 1 and D 2 be n n positive definite symmetric matrices, and let D L,R (w 1, w 2 ), D H (w 1, w 2 ), and D E (w 1, w 2 ) be their Riemannian (Log-Euclidean or affine), root-euclidean, and Euclidean weighted averages, respectively, where w 1, w 2 0. Then we have det D L,R det D S det D H det D E. tr D R tr D L tr D H tr D S tr D E. tr D L (w 1, w 2,..., w N ) tr D A (w 1, w 2,..., w N a) tr D E (w 1, w 2,..., w N ) for all a (0, 1), and in particular for a = 0.5, which corresponds to the square root Euclidean d H averaging.

20 References I Alexander, D Visualization and image processing of tensor fields. Springer. Chap. An introduction to computational diffusion MRI: the diffusion tensor and beyond. Arsigny, Vincent, Fillard, Pierre, Pennec, Xavier, & Ayache, Nicholas Geometric means in a novel vector space structure on symmetric positive-definite matrices. Siam journal on matrix analysis and applications, 29(1), Batchelor, P. G., Moakher, M., Atkinson, D., Calamante, F., & Connelly, A A rigorous framework for diffusion tensor calculus. Magnetic resonance in medicine, 53(1), Bhatia, Rajendra, & Grover, Priyanka Norm inequalities related to the matrix geometric mean. Linear algebra and its applications, 437(2), Bhatia, Rajendra, & Kittaneh, Fuad Notes on matrix arithmeticâgeometric mean inequalities. Linear algebra and its applications, 308(1-3),

21 References II Dryden, I. L., Koloydenko, A. A., & Zhou, D Non-euclidean statistics for covariance matrices, with applications to diffusion tensor imaging. Ann. appl. statist., 3(3), Fletcher, P.T., & Joshi, S Riemannian geometry for the statistical analysis of diffusion tensor data. Signal processing, 87(2), Marcus, Marvin, & Minc, Henryk A survey of matrix theory and matrix inequalities. Dover Publications, Inc., New York. (English summary) Reprint of the 1969 edition. Moakher, Maher A differential geometric approach to the geometric mean of symmetric positive-definite matrices. Siam journal on matrix analysis and applications, 26(3), Pasternak, O., Sochen-N., & Basser, P.J New developments in the Visualization and Processing of Tensor Fields. Berlin: Springer Berlin Heidelberg. Chap. Metric Selection and Diffusion Tensor Swelling, pages

22 References III Pennec, X Statistical computing on manifolds: From riemannian geometry to computational anatomy. Pages of: Proceedings etvc, vol. 5416Springer Berlin Heidelberg, for LNCS. Pennec, Xavier, Fillard, Pierre, & Ayache, Nicholas A Riemannian framework for tensor computing. International journal of computer vision, 66, Rodrigo de Luis-García, Carlos Alberola-L opez, & Westin, Carl-Fredrik New developments in the Visualization and Processing of Tensor Fields. Berlin: Springer Berlin Heidelberg. Chap. On the Choice of a Tensor Distance for DTI White Matter Segmentation, pages Schwartzman, Armin, Dougherty, Robert F., & Taylor, Jonathan E False discovery rate analysis of brain diffusion direction maps. Ann. appl. stat, 2(1),

23 References IV X. Pennec, P. Fillard, & Ayache, N A Riemannian framework for tensor computing. International journal of computer vision, 66, Zhou, D., Dryden, I.L., Koloydenko, A.A., Audenaert, K.M.R., & Li, B Regularisation, interpolation and visualisation of diffusion tensor images using non-euclidean statistics. Journal of Applied Statistics.

24 Current work and future outlook Non-Euclidean Statistics: Ranks are preserved for N = 2, N > 2 is an open problem Covariance operators and n = Powers other than a = 1 2 and Procrustes optimisation and more! HMM: Bayesian extensions, Pairwise Markov Chain models. Raman Spectroscopy for Skin Cancer diagnosis and treatment: Explicit spatial dependence, Hidden Markov Random Fields and Conditional Random Fields. Alexey A. Koloydenko Non-Euclidean analysis of SPD-valued data