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
Multivariate Pattern Analysis of fMRI Signal
?
Pattern
Recognition
Haxby et al. Science, 2001
Spatial Range of MVPA Methods
Global
Whole brain
Local
ROI-based
Searchlight technique
 Searchlight
technique affords unbiased, spatially localized
information detection.
•
Haxby et al., Science, 2001
Haynes & Rees, Nature Rev. Neurosci, 2006
Kriegeskorte et al., PNAS, 2007
The fMRI Signal in Space
Information
carried
by the structure:
fMRI signals resides in the convolved cortical
The brain has
a complex
sheets.
BOLD changes
 3D
searchlight methods do not take this
structural complexity of the brain into
account.
3D searchlight
•
Jin & Kim, Neuroimage, 2008
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
Application: Decoding Object Category
Object categories:
Trumpets vs Chairs vs Boats
Chen et al., NeuroImage, 2011
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
Multiscale Organization of Brain Function
Hierarchical organization with increasing spatial scale
V2
R
V3
V1
V2
L
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, 2008
Wandell, Encyclopedia Neurosci., 2007
Multiscale Analysis – Wavelet Transform
Wavelets, or “little waves”, are families of spatially local, band-passing
Fine scale wavelet
filters:
Fine scale detail
Fine scale information
Transform
Output:
Large scale information
Scale up
Large scale wavelet
Large scale detail
 Information specific to different scales can be extracted
with wavelets
Hackmack and Haynes, in prep.
Wavelets on Irregular Mesh
Wavelets on regular grid
Translation invariant
Wavelets on irregular mesh
Varies on translation
 On an irregular mesh, wavelet transform cannot be
directly implemented
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:
1
1
xi  ( xi 1  xi )  ( xi 1  xi )
2
2
x  K x where K is a symmetric matrix, its eigenvectors,
when sorted non-decreasingly w.r.t. eigenvalues:

1/ n
if j  1

(u j )h   1 / n sin( 2 h  j / 2 / n) if j is even
 1 / n cos( 2 h j / 2 / n) if j is odd
 

Projecting a signal onto the space spanned by these eigenvectors is thus computationally
equivalent to its Discrete Fourier Transform (DFT):
xˆ  U x
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:
~
x  U T D f xˆ , where D f is a diagonal matrix
The diagonal matrix D f contains the Impulse Response function of the designed filter.
Taubin SIGGRAPH '95
Implementing Wavelets via Graph Laplacian
 Generalized graph Laplacian H:
H f ( x)   wxy  f ( x)  f ( y )
y~ x
Wxy characterizes the geometric
properties of the graph. The
eigenvectors of graph Laplacian
have a quasi-frequency property:
Freq. Response
 Wavelets on irregular mesh can then be
defined on the eigenspectral domain:
Eigenspectrum
Biyikoglu et al., Laplacian Eigenvectors of Graphs, 2007
Hammond et al., Applied & Comp. Harmonic Analysis, 2009
Multiscale Analysis on Irregular Mesh
Fine scale wavelet
Fine scale detail
Fine scale information
Transform
Outputs
Large scale information
Large scale wavelet
 Spectral
Large scale detail
graph wavelets can be used to achieve multiscale
analysis on irregular meshes
Anisotropic Filters on Cortical Surface
Vertical
Horizontal
Fine scale
Large scale
 Anisotropic
filters are possible by using different geometric
schemes for the graph Laplacian
Multiscale Analysis of Object Categories/Exemplars
Categories:
Objects vs Scenes vs Body parts vs
Faces
2-step procedure:
Exemplars:
Child vs Female vs Male
 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
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

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 multiscale information mapping promises a better understanding
of human brain function
Acknowledgements
John-Dylan
Haynes
Radoslaw M.
Cichy
Jakob
Heinzle
Kerstin
Hackmack
Fernando
Ramirez
NEUROCURE
Appendix
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.
g 2 ( x)
0 x dx  Cg  , g (0)  0
 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).

g (t )  Tt ( g )  pt ( )  Wt ( f )  TL,t  f  pL,t ( f )
 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
Multiscale Analysis on Regular Grid
Fine scale wavelet
Fine scale detail
Fine scale detail
Transform
Outputs
Large scale detail
Large scale wavelet
Large scale detail