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Bayesian fMRI analysis with Spatial Basis Function Priors

Guillaume Flandin & Will Penny. Bayesian fMRI analysis with Spatial Basis Function Priors. Variational Bayes scheme for voxel-specific GLM using wavelet-based spatial priors for the regression coefficients. SPM Homecoming, Nov. 11 2004. Spatial prior using a kernel.

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Bayesian fMRI analysis with Spatial Basis Function Priors

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  1. Guillaume Flandin & Will Penny Bayesian fMRI analysis with Spatial Basis Function Priors Variational Bayes scheme for voxel-specific GLM using wavelet-based spatial priors for the regression coefficients SPM Homecoming, Nov. 11 2004

  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 levelsFewer wavelet coefficients

  4. 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 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:

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