Bayesian fMRI analysis with Spatial Basis Function Priors

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Bayesian fMRI analysis with Spatial Basis
Function Priors
Variational Bayes scheme for voxel-specific GLM
using wavelet-based spatial priors for the
regression coefficients

• Guillaume Flandin Will Penny

SPM Homecoming, Nov. 11 2004
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Spatial prior using a kernel

• Spatial prior over regression and AR coefficients
• Data-driven estimation of the amount of smoothing
  (different for each regressor)
• Does not handle spatial variations in
  smoothness? spatial basis set prior

Penny et al, NeuroImage, 2004
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Orthonormal Discrete Wavelet Basis Set

• Decomposition of time series/spatial processes on
  an orthonormal basis set with
• Multiresolution time-frequency/scale-space
  properties
• Natural adaptivity to local or nonstationary
  features

• Good properties
• Decorrelation / Whitening,
• Sparseness / Compaction,
• Fast implementation with a pyramidal algorithm
  in O(N) complexity

Increased levelsFewer wavelet coefficients
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Orthonormal Discrete Wavelet Transform (DWT)

• Wavelet transform

Wavelet coefficients Nx1
Data Nx1
Set of wavelet basis functions NxN

• Inverse transform
• Multidimensional transform
• No need to build V in practice, thanks to
  Mallats pyramidal algorithm.

Daubechies Wavelet Filter Coefficients
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Wavelet shrinkage or nonparametric regression

• Signal denoising technique based on the idea of
  thresholding wavelet coefficients.

DWT
IDWT
Thresh.
Nonlinear operator ?
DWT
gt Threshold ?
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3D denoising of a regression coefficient map
Histogram of the wavelet coefficients
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Bayesian Wavelet Shrinkage

• Wavelet coefficients are a priori independent,
• The prior density of each coefficient is given by
  a mixture of two zero-mean Gaussian.

• Consider each level separately
• Applied only to detail levels

Negligible coeffs.
Significant coeffs.

• Estimation of the parameters via an Empirical
  Bayes algorithm

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Generative model
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Variational Bayes
Approximate posteriors

• Iteratively updating Summary Statistics to
  maximise a lower bound on evidence

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Summary / Future

• Variational Bayes scheme for voxel-specific GLM
  using wavelet-based spatial priors for the
  regression coefficients
• Replace the mono scale Gaussian filtering (gt
  anisotropic smoothing amount of smoothness
  estimated from data)
• Lower the quantity of data to deal with in the
  iterative algorithm

• Implementation gt spm_vb_(2D vs. 3D,
  level-dependent parameters, Gibbs-like
  oscillations, )
• General framework which allows lots of
  adaptations and improvements

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Wavelet denoising

• Signal denoising technique based on the idea of
  thresholding wavelet coefficients