1. Multi-Scale Mapping Of fMRI Information On The Cortical Surface:
A Graph Wavelet Based Approach
Yi Chen, Radoslaw M. Cichy and John-Dylan Haynes
Bernstein Center for Computational Neuroscience Berlin & Charité – Universitätsmedizin Max-Planck-Institute for Human Cognitive and Brain Sciences, Leipzig
25/11/2011 Berlin

2. Multivariate Pattern Analysis of fMRI Signal?
Pattern Recognition
Haxby et al. Science, 2001

3. Spatial Range of MVPA Methods
Global
Local
Whole brain
ROI-based
Searchlight technique
 Searchlight technique affords unbiased, spatially localized information detection.
Haxby et al., Science, Haynes & Rees, Nature Rev. Neurosci, Kriegeskorte et al., PNAS, 2007

4. The fMRI Signal in Space
Information carried by the fMRI signals resides in the convolved cortical sheets.
The brain has a complex structure:
BOLD changes
 3D searchlight methods do not take this structural complexity of the brain into account.
3D searchlight
Jin & Kim, Neuroimage, 2008

5. Cortical Surface-based Searchlight
Searchlight on cortical surface mesh
3D searchlight neglects local geometry
Surface-based searchlight respects local geometry
Chen et al., NeuroImage, 2011

6. Application: Decoding Object Category
Object categories:
Trumpets vs Chairs vs Boats
Objects were rotating with randomly changing axis. Subjects were performing a Landolt-C task
Chen et al., NeuroImage, 2011

7. Surface-based vs 3D Method
Collateral sulcus
Fusiform gyrus
 Surface-based method observes local structure
Collateral sulcus
Fusiform gyrus
 3D method deteriorates spatial specificity
 Surface-based method localizes fMRI information more precisely
Chen et al., NeuroImage, 2011

8. Multiscale Organization of Brain Function
Hierarchical organization with increasing spatial scale
L R V1 V2 V3
Ocular dominance and orientation preference columns
Retinotopic maps
Object selective regions
 Knowing the spatial scale of patterns is crucial for understanding the brain’s functional organization
Yacoub et al., PNAS, Wandell, Encyclopedia Neurosci., 2007

9. Multiscale Analysis – Wavelet Transform
Wavelets, or “little waves”, are families of spatially local, band-passing filters:
Fine scale wavelet
Fine scale detail
Fine scale information
Transform
Output:
Large scale information
Large scale detail
Scale up
Large scale wavelet
 Information specific to different scales can be extracted with wavelets
Hackmack and Haynes, in prep.

10. Wavelets on Irregular Mesh
Wavelets on regular grid
Wavelets on irregular mesh
Translation invariant
Varies on translation
 On an irregular mesh, wavelet transform cannot be directly implemented

11. Another Way to Look at Discrete Fourier Transform
For a signal x defined on a one-dimensional, regular and circular field, we have:
Discrete Laplacian:
where K is a symmetric matrix, its eigenvectors, when sorted non-decreasingly w.r.t. eigenvalues:
Projecting a signal onto the space spanned by these eigenvectors is thus computationally equivalent to its Discrete Fourier Transform (DFT):
Manipulating the transform coefficients and exploiting the unitary property of U, we can implement filters on the frequency domain. The filtered signal is given by:
The diagonal matrix contains the Impulse Response function of the designed filter.
Taubin SIGGRAPH '95

12. Implementing Wavelets via Graph Laplacian
Generalized graph Laplacian H:
characterizes the geometric properties of the graph. The eigenvectors of graph Laplacian have a quasi-frequency property:
Wavelets on irregular mesh can then be defined on the eigenspectral domain:
Freq. Response
Biyikoglu et al., Laplacian Eigenvectors of Graphs, 2007 Hammond et al., Applied & Comp. Harmonic Analysis, 2009
Eigenspectrum

13. Multiscale Analysis on Irregular Mesh
Fine scale wavelet
Fine scale detail
Fine scale information
Transform
Outputs
Large scale detail
Large scale information
Large scale wavelet
 Spectral graph wavelets can be used to achieve multiscale analysis on irregular meshes

14. Anisotropic Filters on Cortical Surface
Vertical
Horizontal
Fine scale
Large scale
 Anisotropic filters are possible by using different geometric schemes for the graph Laplacian

15. Multiscale Analysis of Object Categories/Exemplars
Objects vs Scenes vs Body parts vs Faces
Exemplars:
Child vs Female vs Male
2-step procedure:
BOLD estimates were sampled onto the cortical surface & transformed with spectral graph wavelets
At each scale, the outputs from the filter banks were taken as feature vectors for classification
Cichy et al., Cerebral Cortex, 2011

16. Scale Differentiated Analysis of Exemplar and Category Encoding
Categories
Exemplars
Fine Scale
Large Scale
z-score
 Categories are preferentially encoded in large scale and exemplars in fine scale

17. Summary Cortical surface-based method
respects natural geometry of the brain
improves spatial specificity of MVPA
Multi-scale analysis on the cortical surface
can extract information from fMRI signals at different scales using spectral graph wavelets
shows that object categories and exemplars are encoded in different spatial scales in the ventral visual stream
 The combination of surface-based technique and multi- scale information mapping promises a better understanding of human brain function

18. Acknowledgements NEUROCURE John-Dylan Haynes Radoslaw M. Cichy
Jakob Heinzle
Kerstin Hackmack
Fernando Ramirez
NEUROCURE

19. Appendix

20. Spectral Graph Wavelets & Fast Algorithm
For filter with compact spatial support, its impulse response function defined on eigenspectral domain needs to be continuously differentiable.
Wavelet functions are defined by a family of dilated versions of a single function (mother wavelet).
Mother wavelet needs to meet the admissibility condition.
Fast algorithm is possible by approximating the wavelet function on eigenvalue domain with truncated orthogonal polynomials (e.g. Chebyshev polynomial), and calculating the eigenspace projection with recursive sparse matrix vector multiplications (Sect.6, Hammond et al., 2009).
Note, however, by adopting above fast algorithm, the dilation of mother wavelet is now carried on the eigenvalue domain, rather than the eigenvalue’s rank/index domain.
Hammond et al., Applied & Comp. Harmonic Analysis, 2009

21. Multiscale Analysis on Regular Grid
Fine scale wavelet
Fine scale detail
Fine scale detail
Large scale detail
Transform
Outputs
Large scale detail
Large scale wavelet