Table 1.

Summary of the main steps of the algorithm for multiple hypothesis testing in the spatial domain after denoising in the wavelet domain; see also Fig 1.

Compute the 2D DWT cofficients of a set of 10 permuted parametric maps derived from repeated wavelet resampling of a functional MRI dataset

Combine the coefficients corresponding to the same level and orientation of the 2D DWT and store these as the empirical null data.

Extract the coefficients corresponding to a specific level and orientation from the empirical null data. Estimate the corresponding variance and designate it as the noise variance, σ_n(l_i),corresponding to level l_i. Repeat the procedure for every level and orientation of the decomposition.

BaybiShrink Denoising Algorithm : For each 2D DWT coefficient of the observed map,

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Estimate the noisy signal variance σ̂_y using neighboring coefficients

{\hat{σ}}_{y}^{2} = \frac{1}{M} \sum_{y_{ι} \in N (k)} y_{ι}^{2}

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Estimate the signal variance σ̂ using

\hat{σ} = \sqrt{{({\hat{σ}}_{y}^{2} - {\hat{σ}}_{n}^{2} (l_{ι}))}_{+},}

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Estimate the threshold value using

τ_{w} = {\hat{σ}}_{n}^{2} (l_{i}) / \hat{σ}

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Estimate each coefficient using Eqn. (1) with the estimated threshold value.

Invert the denoised coefficients to obtain the denoised observed maps.

Generate a new set of permuted parametric maps by fMRI time series resampling, repeat steps (1)–(4) and invert the DWT. Select a percentile value and compute the threshold value τ_s.

Define a subset of voxels for significance testing by thresholding the denoised observed maps with the threshold value, τ_s, computed in the previous step.

Compute the P-values for each of the voxels surviving step (7) using a permutation distribution of voxel statistics derived from the second set of permuted maps. Select an appropriate false discovery rate, α_FDR, and apply the FDR procedure to control Type I error in observed maps.