1. 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

2. 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

3. 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 levels Fewer wavelet coefficients

4. Orthonormal Discrete Wavelet Transform (DWT)
Data [Nx1]
Wavelet coefficients [Nx1]
Set of wavelet basis functions [NxN]
Inverse transform:
Multidimensional transform
No need to build V in practice, thanks to Mallat’s pyramidal algorithm.
Daubechies Wavelet Filter Coefficients

5. Wavelet shrinkage or nonparametric regression
Signal denoising technique based on the idea of thresholding wavelet coefficients.
DWT
Thresh.
IDWT
Nonlinear operator 
DWT
=> Threshold 

6. 3D denoising of a regression coefficient map
Histogram of the wavelet coefficients

7. 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

8. Generative model

9. Approximate posteriors
Variational Bayes
Iteratively updating Summary Statistics to maximise a lower bound on evidence

10. Summary / Future
Variational Bayes scheme for voxel-specific GLM using wavelet-based spatial priors for the regression coefficients
Replace the mono scale Gaussian filtering (=> anisotropic smoothing + amount of smoothness estimated from data)
Lower the quantity of data to deal with in the iterative algorithm
Implementation => spm_vb_* (2D vs. 3D, level-dependent parameters, Gibbs-like oscillations, …)
General framework which allows lots of adaptations and improvements…

11. Wavelet denoising
Signal denoising technique based on the idea of thresholding wavelet coefficients: