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A subspace approach to blind coil sensitivity estimation in parallel MRI


In parallel MRI, subsampled k-space data are simultaneously collected by multiple coils. Each coil introduces a sensitivity map (CSM) that is multiplied pointwise with the single image to be reconstructed. In ESPIRiT [1], for each pixel location in each coil, an eigen-decomposition is applied to small matrices to obtain CSMs. However, this approach can be time-consuming for larger imaging problems. Here, we exploit smoothness of the coil sensitivities in the image domain to model them as small finite impulse response (FIR) filters in k-space as in PRUNO [2]. Since pointwise-multiplication in image domain corresponds to convolution in k-space, parallel MRI problem can be expressed as a blind image deconvolution problem; consequently, a subspace approach [3] can be used to estimate the k-space coefficients of the CSMs.


If yi, x and hi represent fully sampled k-space data, true image, and k-space coefficients of the ith CSM, then the problem can be written as yi = x*hi = Xhi. Further, the multichannel convolution can be written as Y = XH. Thus, if x has full rank, then the null-space of Y is equivalent to the null-space of H. As a result, the null-space (equivalently row-space) vectors used to reconstruct yi from subsampled data in PRUNO, can be used for the estimation of k-space coefficients of CSMs efficiently using the following optimization problem: h = argmaxh ||Vh||2+μ||Rh||2, where R represents a low-pass filter, and V involves convolution matrices of filters obtained from rowspace vectors. For validation, real-time, free-breathing 3-fold cine data were collected on a 3T Siemens scanner with matrix size 161 × 144 × 12 × 48. For CSM estimation, a 161 × 24 fully sampled k-space was obtained from 3 consecutive time frames. Among 432 singular values, the largest 70 were used as row-space vectors. A Gaussian function was selected for the low-pass R. The eigenvector associated with the largest eigenvalue of VHV+μ RHR was calculated to yield the 8 × 8 estimated k-space coefficients of the CSMs for μ = 5. Finally, the sensitivities were normalized with their square-root sum-of-squares (SoS) image.


Estimated CSM for one coil and its SoS-normalized version are demonstrated in Figure 1. SENSE[4] reconstructions for one of the frames are given in Figure 2 for the estimated CSMs and their SoS-normalized versions. As seen, inhomogeneity and artifacts existing in SENSE reconstruction is significantly reduced with the normalized CSMs. Compared to the image domain processing, the proposed k-space estimation of CSM was 10 times faster.

Figure 1
figure 1

Left: estimated coil sensitivity for one of 12 coils; Right: estimate normalized by the sum-of-squares of the estimated sensitivities.

Figure 2
figure 2

SENSE reconstructions from the estimated sensitivities on the left and their SoS normalized versions on the right. Red arrow points to a strong artifact.


The proposed k-space approach for CSM estimation using subspace methods and a simple normalization provides both low computational complexity and the flexibility to incorporate both regularization and a low-dimensional parameterization of the smooth CSMs.


This work was supported by DARPA/ONR grant N66001-10-1-4090.


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This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The Creative Commons Public Domain Dedication waiver ( applies to the data made available in this article, unless otherwise stated.

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Gungor, D.G., Ahmad, R. & Potter, L.C. A subspace approach to blind coil sensitivity estimation in parallel MRI. J Cardiovasc Magn Reson 16 (Suppl 1), W1 (2014).

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