fMRI Signal Restoration Using a Spatio-Temporal Markov Random Field Preserving Transitions Xavier Descombes, Frithjof Kruggel, and D. Yves von Cramon Max Planck Institute of Cognitive Neuroscience, 22-26 Inselstrasse, 04103, Leipzig, Germany Received November 14, 1997 In fMRI studies, Gaussian filtering is usually applied to improve the detection of activated areas. Such lowpass filtering enhances the signal to noise ratio. However, undesirable secondary effects are a bias on the signal shape and a blurring in the spatial domain. Neighboring activated areas may be merged and the high resolution of the fMRI data compromised. In the temporal domain, activation and deactivation slopes are also blurred. We propose an alternative to Gauss- ian filtering by restoring the signal using a spatiotem- poral Markov Random Field which preserves the shape of the transitions. We define some interaction between neighboring voxels which allows us to reduce the noise while preserving the signal characteristics. An energy function is defined as the sum of the interaction poten- tials and is minimized using a simulated annealing algorithm. The shape of the hemodynamic response is preserved leading to a better characterization of its properties. We demonstrate the use of this approach by applying it to simulated data and to data obtained from a typical fMRI study.  1998 Academic Press INTRODUCTION Signal detection in the presence of random noise plays a crucial role in functional imaging as it repre- sents the first step of the experimental analysis. With efficient signal detection techniques it is possible to diminish the number of task repetitions and to better describe low-level activations. Both points are of great importance when studying cognitive processes by fMRI. The analysis of fMRI data is usually achieved by performing a voxelwise (parametric or nonparametric) statistical test in the time-series to detect areas with signal changes related to the experimental design. The statistical measure (say, a z-score) is then compiled in a statistical parameter map (SPM). Significantly acti- vated areas are found by thresholding the SPM (Fris- ton et al., 1994; Worsley and Friston, 1995; Xiong et al., 1996). The threshold is established by testing the null-hypothesis (H 0 -hypothesis). A P value which pre- vents false alarms is set to a given value (usually 0.05) and the threshold is derived by limiting the probability of false alarms to P. In such a binary decision process we can have four different outcomes: true positive (P(1 0 H 1 )), false positive (P(1 0 H 0 )), true negative (P(0 0 H 0 )), false negative (P(0 0 H 1 )). A spatial Gaussian filter is often used in fMRI studies to reduce the noise before computing the SPM and thus to prevent false alarms. The introduced smoothness is then taken into account when computing the P value (Friston et al., 1995; Forman et al., 1995). However, the signal itself is markedly affected by this lowpass filter, yielding blur- ring and possible displacement of activated areas. Moreover, signals of low amplitude are completely removed. Thus, the detection of activation is improved but the characteristics of the signal and the accuracy of the activated areas localization are lost. We propose to overcome this problem by restoring the data instead of filtering them. The restoration process smooths the noise but at the same time preserves the signal shape. Restoration has been widely investigated in signal and image processing (Andrews and Hunt, 1977). Here, we focus only on the problem of noise reduction. Filter- ing techniques and in particular the application of adaptive filters have provided solutions to the restora- tion problem. However, if noise reduction is considered as an inverse problem, the regularization theory which takes into account some a priori knowledge on the solution has been demonstrated to be more powerful. The Bayesian approach plays a key role in the regular- ization theory for its ability to integrate both a data attachment term (goodness of fit with respect to data) and some general and flexible priors (probability distri- bution modeling the expected signal/image). A popular approach is to use Markov random fields (MRFs) (Besag, 1974; Geman and Geman, 1984) to define the prior. The a priori knowledge induced by MRFs is general enough not to overly constrain the solution but providing interesting regularizing properties. MRFs were introduced in the engineering sciences by Besag (1974). Since then, they have been widely used, espe- cially in image processing, for various problems such as NEUROIMAGE 8, 340–349 (1998) ARTICLE NO. NI980372 340 1053-8119/98 $25.00 Copyright  1998 by Academic Press All rights of reproduction in any form reserved.