Wavelet domain de-noising of time-courses in MR image sequences

Magnetic resonance images acquired with high temporal resolution often exhibit large noise artifacts, which arise from physiological sources as well as from the acquisition hardware. These artifacts can be detrimental to the quality and interpretation of the time-course data in functional MRI studies. A class of wavelet-domain de-noising algorithms estimates the underlying, noise-free signal by thresholding (or 'shrinking') the wavelet coefficients, assuming the underlying temporal noise of each pixel is uncorrelated and Gaussian. A Wiener-type shrinkage algorithm is developed in this paper, for de-noising either complex- or magnitude-valued image data sequences. Using the de-correlation properties of the wavelet transform, as elucidated by Johnstone and Silverman, the assumption of i.i.d. Gaussian noise can be abandoned, opening up the possibility of removing colored noise. Both wavelet- and wavelet-packet based algorithms are developed, and the Wiener method is compared to the traditional Hard and Soft wavelet thresholding methods of Donoho and Johnstone. The methods are applied to two types of data sets. In the first, an artificial set of complex-valued images was constructed, in which each pixel has a simulated bimodal time-course. Gaussian noise was added to each of the real and imaginary channels, and the noise removed from the complex image sequence as well as the magnitude image sequence (where the noise is Rician). The bias and variance between the original and restored paradigms was estimated for each method. It was found that the Wiener method gives better balance in bias and variance than either Hard or Soft methods. Furthermore, de-noising magnitude data provides comparable accuracy of the restored images to that obtained from de-noising complex data. In the second data set, an actual in vivo complex image sequence containing unknown physiological and instrumental noise was used. The same bimodal paradigm as in the first data set was added to pixels in a small localized region of interest. For the paradigm investigated here, the smooth Daubechies wavelets provide better de-noising characteristics than the discontinuous Haar wavelets. Also, it was found that wavelet packet de-noising offers no significant improvement over the computationally more efficient wavelet de-noising methods. For the in vivo data, it is desirable that the groups of "activated" time-courses are homogeneous. It was found that the internal homogeneity of the group of time-courses increases when de-noising is applied. This suggests using de-noising as a pre-processing tool for both exploratory and inferential data analysis methods in fMRI.