Statistical noise analysis in GRAPPA using a parametrized noncentral Chi approximation model

Abstract

The characterization of the distribution of noise in the magnitude MR image is a very important problem within image processing algorithms. The Rician noise assumed in single-coil acquisitions has been the keystone for signal-to-noise ratio estimation, image filtering, or diffusion tensor estimation for years. With the advent of parallel protocols such as sensitivity encoding or Generalized Autocalibrated Partially Parallel Acquisitions that allow accelerated acquisitions, this noise model no longer holds. Since Generalized Autocalibrated Partially Parallel Acquisitions reconstructions yield the combination of the squared signals recovered at each receiving coil, noncentral Chi statistics have been previously proposed to model the distribution of noise. However, we prove in this article that this is a weak model due to several artifacts in the acquisition scheme, mainly the correlation existing between the signals obtained at each coil. Alternatively, we propose to model such correlations with a reduction in the number of degrees of freedom of the signal, which translates in an equivalent nonaccelerated system with a minor number of independent receiving coils and, consequently, a lower signal-to-noise ratio. With this model, a noncentral Chi distribution can be assumed for all pixels in the image, whose effective number of coils and effective variance of noise can be explicitly computed in a closed form from the Generalized Autocalibrated Partially Parallel Acquisitions interpolation coefficients. Extensive experiments over both synthetic and in vivo data sets have been performed to show the goodness of fit of out model.

Denoising and fast diffusion imaging with physically constrained sparse dictionary learning

“…These techniques were then extended to take into account the Rician noise nature of the DWI signal (Coupé et al, 2010;Descoteaux et al, 2008;Aja-Fernàndez et al, 2008;Tristán-Vega and Aja-Fernández, 2010;Aja-Fernández et al, 2011;Brion et al, 2011) and, recently, the non-central Chi-squared distribution in the case where parallel imaging is used Brion et al, 2011). However, only the technique of (Tristán-Vega and Aja-Fernández, 2010) performs denoising across the DWI channels, i.e.…”

Parallel MRI Noise Correction: An Extension of the LMMSE to Non Central χ Distributions

“…Whereas the Rice LMMSE oversmoothes the images, the nc-χ LMMSE performs a clearly visible correction with respect to the details, even in very defavorable conditions as at b = 4500s.mm −2 . The real data we used were obtained with GRAPPA reconstruction, and yet, it has been recently discussed in [3] that an effective number of channels, as well as an effective variance of noise, must be calculated to get a nc-χ noise model that better fits the data. Both effective calculated parameters are not stationnary and should be evaluated at each voxel of the image.…”

“…In case of a multiple-channel acquisition, with a Sum-of-squares (SoS) reconstruction, the noisy magnitude follows a non-central χ (nc-χ) distribution [2]. In the case of Generalized Autocalibrating Partially Parallel Acquisition (GRAPPA) reconstruction, [3] reminds that the noise is non-stationnary, so in this case, the nc-χ hypothesis becomes a good approximation. A particular case of the nc-χ distribution, namely the Rician distribution, appears in case of a single-channel acquisition, or when using the Sensitivity Encoding for Fast MRI (SENSE) algorithm [4].…”

Local estimation of the noise level in MRI using structural adaptation

“…SENSE (Pruessmann et al, 1999) in general leads to the special case of a Rician distribution L ¼ 1, with spatially varying scale parameter. With other parallel imaging methods like GRAPPA a non-central v distribution with adjusted, location dependent distribution parameters serves as a valid approximation of the true data distribution (Aja-Fernández et al, 2011). For further reading we refer, e.g., to Aja-Fernandez et al (2014).…”

“…For most parallel imaging methods the distribution depends on the reconstruction algorithm but is usually approximated by a (scaled) non-central v-distribution (Aja-Fernández et al, 2013) which includes the Rician distribution as a special case. The scale parameter r of these distributions is determined by the noise standard deviation of the complex valued noise in k-space, the local coil sensitivities and signal correlations between coils (Aja-Fernández and Tristán-Vega, 2012;Aja-Fernández et al, 2011). Accordingly, the noise level is generally not a global quantity over the MR image, but varies locally.…”