ESPIRiT is a robust, auto-calibrating approach to parallel MR image reconstruction that estimates the subspace of coil sensitivity maps using an eigenvalue-based method. While it is robust to a range of parameter choices, having parameters that result in a tight subspace yields the best performance. We propose an automatic, parameter free method that appropriately weights the subspace using a shrinkage operator derived from Stein's Unbiased Risk Estimate. We demonstrate the efficacy of our technique by showing superior map estimation without user intervention in simulation and in-vivo data compared to the current default method of subspace estimation.

In ESPIRiT, a calibration matrix $$$(A)$$$ is constructed from auto-calibration signal (ACS) data. Its singular value decomposition (SVD) is used to characterize the signal subspace $$$(V_{||})$$$ and the noise subspace $$$(V_\perp)$$$.¹

$$\text{SVD}(A) = USV^*, \text{ where } V = \left[V_{||},V_{\perp}\right]$$

A self-consistency operator $$$\mathcal{G}_q$$$ is derived from the signal subspace. Self consistency implies $$$\mathcal{G}_q \, x = x$$$ where $$$x$$$ is the image. The point-wise eigenvalue decomposition of $$$\mathcal{G}_q$$$ is done in the image domain and eigenvectors with eigenvalues close to one with a tolerance threshold $$$(\gamma)$$$ are said to span sensitivity maps.¹

A hard threshold ($$$\lambda$$$) on the singular values is used to estimate the signal subspace $$$(V_{||})$$$. If $$$s_1 \geq s_2 \geq \dots \geq s_{\min \dim A}$$$ are the singular values of $$$A$$$,

$$V_{||} = VW, \text{ where } W = \text{diag}\left[\frac{\mathbb{1}(s_i > \lambda s_1)}{s_i}\right]$$

While the hard threshold works well in most cases, it can be sensitive to the choice of $$$\lambda$$$. It might include vectors from the noise subspace or null vectors from the signal subspace. Finding the right $$$\lambda$$$ is therefore crucial, but this problem is not convex. To overcome this, we replace the hard threshold with a soft threshold that weighs down the noise subspace. Soft thresholding is more robust to choice of threshold and it is possible to determine the optimal soft threshold $$$(\hat \lambda)$$$ using Stein's Unbiased Risk Estimate.²

Once the optimal soft threshold $$$(\hat \lambda)$$$ is determined, we define a new weighted subspace estimate.

$$\hat V_{||} =V\hat W, \text{ where } \hat W = \text{diag}\left[\frac{(s_i - \hat \lambda)_+}{s_i}\right]$$

This weighting operator $$$\hat W$$$ appropriately shrinks the norms of the singular vectors. This weighting of $$$V$$$ to estimate $$$V_{||}$$$ results in $$$\hat{\mathcal{G}}_q$$$ being related to $$$\mathcal{G}_q$$$ as follows,

$$\hat{\mathcal{G}}_q = W^* \mathcal{G}_q W$$

We assume the eigenvectors of $$$\hat{\mathcal{G}}_q$$$ span "weighted" sensitivity maps and hence the eigenvalues do not necessarily meet the "=1" eigenvalue condition.¹ To accommodate this, the second threshold $$$(\gamma)$$$ is set to $$$0.98$$$ times the maximum eigenvalue of the eigenvalue decomposition of $$$\hat{\mathcal{G}}_q$$$.

There are different ways to estimate the noise standard deviation. Since we hypothesize that $$$A$$$ is low-rank, we expect that the last few singular values should be contributions from noise only. In order to estimate noise standard deviation ($$$\sigma$$$), we generate a zero-mean, unit-variance Gaussian noise calibration matrix and fit its last one-fourth singular values to the last one-fourth singular values of $$$A$$$.

We look at the ESPIRiT maps generated from the two methods and qualitatively compare them. We use two fully sampled knee-datasets acquired from a Discovery MR 750 GE Scanner.³ A kernel of dimension $$$[6 \times 6]$$$ and a calibration region of dimension $$$[24 \times 24]$$$ are used. We use ESPIRiT's default threshold parameters $$$(\lambda = 0.001, \gamma = 0.8)$$$ to generate one set of ESPIRiT maps. The weighting method described above is used to generate the other set. We project the data on the above two sets of maps and compare the results and their difference images. (The difference image is the null projection. It should have only noise and no signal.)

Figures 1, 2, 3 and 4 exemplify our results. Even with data being acquired from the same scanner, we see that the default parameters can result in loose sensitivity maps as in the Figure 2 and tight maps as in Figure 4. Contrast this with the soft-weighting method which results in tight sensitivity maps in both cases. Looking at Figure 1 specifically, we see the tight sensitivity maps result in more noise in the difference image (1e) compared to difference image (1c), which is desirable. This is especially noticeable in regions surrounding the knee, where we expect no signal and only noise.

The effects of weighting the signal subspace on the second threshold need to be further studied. Nevertheless, the soft-weighting of singular vectors is a promising step towards having an ESPIRiT implementation that is both robust to noise and completely auto-calibrating.