Spectral Graph Wavelets on the Cortical Connectome and Regularization of the EEG Inverse Problem

Transcription

1 Spectral Graph Wavelets on the Cortical Connectome and Regularization of the EEG Inverse Problem David K Hammond University of Oregon / NeuroInformatics Center International Conference on Industrial and Applied Mathematics 11:30-11:55 July 22,

2 Overview Spectral Graph Wavelet Transform (SGWT) Motivations from Classical Wavelet Analysis Spectral Graph Theory SGWT construction EEG source estimation Electrical head modeling Cortical connectome graph construction Sparse representation with Cortical Spectral Graph Wavelets 2

3 SGWT : wavelets on weighted graphs Weighted graphs : N vertices Symmetric adjacency matrix A i,j = w i,j (w i,j 0) f(i) : value on i th vertex Wavelets on weighted graph: Linear, multiscale representation for functions on vertices f R N Why? ψ γ R N localized in space and frequency c γ = ψ γ,f Wavelets : very useful, but classically limited to Euclidean space Graphs : flexibly model complicated data domains 3

4 Preview : some SGWT graph wavelets 4

5 Classical wavelet analysis Signal f L 2 (R) mother wavelet ψ Analysis : wavelet coefficients W f (t, u) =ψ t,u,f translation & dilation ψ t,u (x) = 1 x u ψ t t Reconstruction (Orthogonal Wavelet case) f = n,j W f (2 j, 2 j n)ψ 2j,2 j n(x) Signal manipulation often easier in wavelet transform space 5

6 Towards Wavelets on Graphs Problem : dilation and translation on irregular Graph? Wavelet transform in Fourier domain : W f (t, x) =f ψ t W f (t, w) = ˆf(ω) ˆψ (tω) ψ t (x) = 1 t ψ( x t ) Individual wavelets : apply operator to δ u (x) = δ(x u) ψ t,u ( x) =W δu (t, x) = F 1 ˆδu (ω) ˆψ (tω) localization dilation Dilate operator ˆψ (tω) in frequency domain, then localize Analog of Fourier transform on weighted graphs => Spectral graph theory 6

7 Degree matrix N D i,i = k=1 A k,i Spectral Graph Theory Graph Laplacian L = D A Spectral decomposition of L Lχ l = λ l χ l {χ l } N l=1 0 = λ 1 λ 1... λ N orthonormal basis Graph Fourier transform ˆf(l) =χ l,f = n χ l (n)f(n) Graph inverse Fourier transform f = l ˆf(l)χ l 7

8 Spectral Graph Wavelets Wavelet kernel g : R + R + H., P. Vandergheynst, R. Gribonval 2011 Applied and Computational Harmonic Analysis g(t 1 λ) analogous to ˆψ (ω) Wavelet operator (at scale t j ) T t j g = g(t j L):R N R N Action defined on eigenvectors T t j g χ l = g(t j λ l )χ l λ λ N Wavelets ψ j,n = T t j g δ n SGWT coefficients at scale j T t j g f = l g(t j λ l ) ˆf(l)χ l 8

9 Spectral Graph Wavelets h(λ) g(t 1 λ) g(t 2 λ) g(t 3 λ) Scaling function band T h = h(l) φ n = T h δ n K-fold Overcomplete W : R N R KN λ K=4 λ N Fast SGWT Polynomial approximation of g(t j x) Avoids diagonalization of L 9

10 Application to EEG Source Estimation 10

11 Electroencephalography (EEG) J 2 J3 Φ 1 Φ 2 J 1 EEG signal : arises from dipole current sources + volume conduction Source estimation : Infer current sources J from electrode measurements Φ Access to subject specific brain anatomy : MRI : measure tissue geometry Diffusion tensor imaging (DTI) + tract tracing : measure brain connectivity How to exploit connectivity knowledge? Sparsity in cortical graph wavelet basis. 11

12 EEG forward problem Quasi-static Maxwell PDE φ : potential s : current sources σ(x, y, z) : tissue conductivity (σ φ) =s (σ φ) n =0 on boundary Finite-difference solver, on 1mm 3D grid Lead field matrix K e 1 e 2 e 3 d 1 n K ij : solution at e i with unit source at d j N d d 4 d 2 Φ=KJ N e K d 3 J R N d Φ R N e : source current amplitudes : electrode voltages 12

13 Cortical connectome graph Cortical grey matter (neuron cell bodies) connected by white matter fibers (axons) Diffusion along fiber directions diffusion ellipsoid λ1 v 1 λ2 v 2 r T D 1 r =1 Diffusion Tensor Imaging (DTI) measures water diffusion direction at every voxel Tractography dominant eigenvector v 1 reveals fiber orientation v 1 (x 1 ) v 1 (x 2 ) compute streamlines, from seed points ( a little more complicated, in practice... ) γ (t) =v 1 (γ(t)) using method developed by S Warfield, B Scherrer, Children s Hospital, Harvard 13

14 Cortical connectome graph seed in white matter, retain tracts connecting cortex to cortex global (tract-based) connectome A tr a tr i,j = k connecting i and j 1 length(k) local (adjacency-based) connectome A loc a loc i,j = length of boundary between patches i,j hybrid connectome A A = λ tr A tr + λ loc A loc 14 showing 12,124 out of 775,939 cortical-cortical tracts

15 Cortical connectome graph seed in white matter, retain tracts connecting cortex to cortex global (tract-based) connectome A tr a tr i,j = k connecting i and j 1 length(k) local (adjacency-based) connectome A loc a loc i,j = length of boundary between patches i,j hybrid connectome A A = λ tr A tr + λ loc A loc 15 showing 12,124 out of 775,939 cortical-cortical tracts

16 Cortical connectome graph seed in white matter, retain tracts connecting cortex to cortex global (tract-based) connectome A tr a tr i,j = k connecting i and j 1 length(k) local (adjacency-based) connectome A loc a loc i,j = length of boundary between patches i,j hybrid connectome A A = λ tr A tr + λ loc A loc 16 showing 12,124 out of 775,939 cortical-cortical tracts

17 cortex patches, showing patch centers 17

18 Global (tract-based) connectome graph 18

19 Local connectome graph 19

20 linear superposition : Φ=KJ Inverse problem : Given Φ, find J EEG inverse problem N e equations, N d unknowns infinitely many possibilities for J! Find J minimizing Φ KJ 2 + f(j) f(j) : small for good J large for bad J data fidelity prior how to build prior using connectivity? 20

21 Sparse representation with Cortical Graph Wavelets Construct SGWT from hybrid connectome A W : R N d R KN d expand J = i c i ψ i = W T c prior : 1 penalty on c c = argmin c J = W T c Φ KW T c λ c 1 (P) (P) is convex, L1-LS program solve with truncated Newton interior point method Kim, Koh, Lustig, Boyd, Gorinevsky

22 Motor Potential study Phan Luu (EGI) Validation : investigate source estimates for paradigm where we know where the sources should be Experimental paradigm: Press button (RT,LT,RP,LP) EEG recording setup : channel (257 electrodes) sensor net, 250 Hz average locked to button press onset (ERP) expect activation in contralateral motor cortex 22

23 subject 1470 Preliminary Results 40 ms before left thumb press minimum norm (l2 penalty) proposed (wavelet l1 penalty) dipole l1 penalty 23

24 subject ms before right thumb press minimum norm (l2 penalty) proposed (wavelet l1 penalty) dipole l1 penalty 24

25 Future Work Explore parameters (connectome graph construction, SGWT...) Sensible automatic selection of regularization parameters (L-curve) Spatiotemporal estimation via spatiotemporal graph More appropriate convex solver (path following for λ)? Alternative convex program formulations More subjects / different experimental paradigms 25

26 Acknowledgements Allen Malony Don Tucker Pierre Vandergheynst Rémi Gribonvale Adnan Salman Kai Li Simon Warfield Benoit Scherrer NIC, University of Oregon EGI / NIC, University of Oregon EPFL, Switzerland INRIA, France NIC, University of Oregon NIC, University of Oregon CRL, Children s Hospital, Harvard CRL, Children s Hospital, Harvard 26

27 FIN

28 Chebyshev polynomials for fast SGWT 1 g(t j x) p j (x) = M c j,k T k (x) k=0 0 0 λ 40 λ max Chebyshev approx for M=10 g(t j L)f p j (L)f = M k=0 c j,k (T k (L)f) T k (x) =2xT k 1 (x) T k 2 (x) T k (L)f =2L(T k 1 (L)f) T k 2 (L)f computing T k (L)f is fast if L is sparse 28

29 Some Cortical Graph Wavelets 29