Addendum:

Today's release of the AFNI Permutation Test plug-in includes an option to model autoregressive noise. Steps added for this autoregressive model are indiated in red in this revised version of Figure 1:

Autoregressive modelling was suggested early on as a method of correcting for coloured noise as a preparatory step to permutation testing of fMRI time series [1]. Although methods based around permuting in a wavelet domain leave residuals that are more temporally independent in comparison to those left by autoregressive methods [2], it's uclear how much of this residual temporal dependence arises from coloured noise versus how much may reflect the presence of (subthreshold) signal. Recently it's been shown that wavelet methods, like Fourier methods, may tend to underestimate thresholds for statistical significance (and therefore risk producing false positive results) when the permuted series include frequency-localised BOLD signal as well as noise [3] – a condition that applies especially in block designs. For these reasons as well as for simplicity of implementation, I've chosen to implement here autoregressive rather than wavelet whitening.

1. JJ Locascio, PJ Jennings, CI Moore, S Corkin. Time Series Analysis in the Time Domain and Resampling Methods for Studies of Functional Magnetic Resonance Brain Imaging. Human Brain Mapping 5(3):168-193 (1997).
2. ET Bullmore, C Long, J Suckling, J Fadili, G Calvert, F Zelaya, TA Carpenter, MJ Brammer. Colored Noise and Computational Inference in Neurophysiological (fMRI) Time Series Analysis: Resampling Methods in Time and Wavelet Domains. Human Brain Mapping 12(2):61-78 (February 2001).
3. O Friman, C-F Westin. Resampling fMRI Time Series. NeuroImage 25(3):859-867 (15 April 2005).