Steven P Allen^{1}, Xue Feng^{1}, Samuel Fielden^{2}, and Craig H Meyer^{1}

We introduce a novel objective function for automatic deblurring of images acquired with non-2DFT trajectories. When paired with the recently introduced retraced, spiral-in-out trajectory, this objective function provides two advantages over previously established functions: it is invariant with incidental phase and is less susceptible to spurious extrema. These advantages lead to effective deblurring over a larger range of off resonance conditions and readout durations. Here, using simulations and phantom studies, we compare the sensitivity of this objective function to spurious extrema to a previously proposed function.

Spiral
acquisition techniques have desirable properties including efficient data
acquisition and resistance to motion. However, image blurring due to field inhomogeneities remains a major barrier to their widespread adoption.
The “automatic” method^{1–3} can be employed to
remove inhomogeneity blurring, especially when other methods—such as a field map,
linear corrections^{4,5}, and KESA^{6}—fail. However, when the readout duration is long or the incidental phase has not been removed, this method's objective
function becomes susceptible to spurious minima, producing corrupted images.

Here, we present an automatic deblurring objective
function suitable for a retraced in-out spiral acquisition scheme (RIO)^{7} with long readouts. This objective function is both invariant with incidental phase and insensitive to spurious extrema. We assess the suitability of this objective function for a
RIO acquisition strategy and compare its performance to that proposed by Lee et
al^{3}.

When using a RIO acquisition scheme, k-space is sampled by two trajectories that are both symmetric and
mirror images of each other^{7}. Multiplying the two sets of acquired raw data by a conjugate phase term representing some
estimate of the off resonant frequency, $$$\theta$$$, and then averaging the results together yields the following signal equation: $$ s(t,\theta)=\int m(r)e^{-i2\pi k(t)r}cos[t(\omega(r)-\theta)]dr $$ where $$$\omega(r)$$$ is the true off resonant frequency at location $$$r$$$. Under this scheme, the pathologic phase modulation commonly introduced to the object's k-domain representation by field inhomogeneities is transformed into a more benign cosine modulation. For a
point object, $$$s(t)$$$ is
real valued. Further, Parseval's theorem indicates that the energy of the reconstructed point spread function is maximized when $$$\theta=\omega(r)$$$. Thus, one can estimate the field map, $$$\theta(r)$$$, by solving
$$\theta(r)=arg min_{\theta(r)}[\int_{r\in A}I(r,\theta)I(r,\theta)^*dr +\lambda *var_{r\in B}(I(r,\theta)I(r,\theta)^*)]$$ where $$$I(r,\theta)^*$$$ is the complex conjugate of the image, $$$I(r,\theta)$$$, reconstructed from $$$s(t,\theta)$$$, and regularizing term $$$var_{r\in B}(*)$$$ is the variance operator over the area $$$B$$$ and whose influence is modulated by $$$\lambda$$$. That is, the local energy and variance of the the image are maximized when $$$\theta$$$ is close to the local off resonant frequency.

The proposed objective function was used to deblur phantom images acquired in a 3T GE MR750 scanner using RTHawk (www.heartvista.com) and a RIO aquisition with a 56 ms readout duration (28 ms spiral-in and
spiral-out trajectories, respectively; TR:100 ms; TE:35 ms; Interleaves:4; FOV:16 cm; matrix:178x178; $$$\lambda$$$=1; A=49 pixels^{2};^{ }B=441pixels^{2}). A linear gradient introduced a bandwidth of ~400 Hz across the object. To provide a direct
comparison, field maps were estimated using the proposed objective function and
that of Lee et al. both prior to and after summation of the retraced acquisition
data. These functions were evaluated over a 400 Hz bandwidth at 1 Hz intervals.

A simulation of the two compared objective functions for a point object is shown in Fig. 1.A. The original, blurred spiral-in, spiral-out, and RIO summation images of the phantom are shown in Fig. 1.B while the deblurred images are shown in Fig. 1.C and objective functions evaluated at locations indicated by the yellow arrows are displayed in Fig. 1.D.

The proposed objective function contains two improvements: it is invariant with image phase and, in the case of a point object, is not susceptible to spurious extrema (see Fig 1.A). Spurious extrema are only encountered as a function of the specific structure of the imaged object. Thus, in Fig 1.C, the proposed function provides robust deblurring in nearly all parts of the image compared to that of Lee et al. A small artifact (star) in Fig 1.C persists because, at that location, the local magnetic field varies significantly over the areas of evaluation: $$$A$$$ and $$$B$$$. This artifact disappears with a less extreme field accross the object. Image noise persists in the phantom's voids (circle) because the objective functions evaluated in these regions were maximized by blurring signal from distant sources into the areas of evaluation. This artifact can be mitigated by altering the terms $$$A$$$ and $$$B$$$.

Lee et al. indicated that spurious minima corrupt the deblurring of a point object when approximately four cycles of off resonant phase can accrue during readout^{3}. In practice, this constraint can be even more strict. Our results here show that pairing the proposed objective function with a RIO acquisition provides effective deblurring even when more than ten cycles of phase accrue during readout. This combined scheme can enable deblurring of spiral acquistions over a much wider variety of readout lengths than previously possible.

1. Noll DC, Pauly JM, Meyer CH, Nishimura DG, Macovskj A. Deblurring for nonā2D fourier transform magnetic resonance imaging. Magn Reson Med. 1992;25(2):319-333.

2. Man LC, Pauly JM, Macovski A. Improved automatic off-resonance correction without a field map in spiral imaging. Magn Reson Med. 1997;37(6):906-913.

3. Lee D, Nayak KS, Pauly J. Reducing Spurious Minima in Automatic Off-Resonance Correction for Spiral Imaging. Proc Int Soc Magn Reson Med Kyoto, Japan, 2004. 2004;11:2678.

4. Chen W, Meyer CH. Fast automatic linear off-resonance correction method for spiral imaging. Magn Reson Med. 2006;56(2):457-462.

5. Smith TB, Nayak KS. Automatic off-resonance correction in spiral imaging with piecewise linear autofocus. Magn Reson Med. 2013;69(1):82-90.

6. Truong T-K, Chen N-K, Song AW. Application of k-space energy spectrum analysis for inherent and dynamic B0 mapping and deblurring in spiral imaging. Magn Reson Med. 2010;64(4):1121-1127.

7. Fielden SW, Meyer CH. A simple acquisition strategy to avoid off-resonance blurring in spiral imaging with redundant spiral-in/out k-space trajectories. Magn Reson Med. 2015;73(2):704-710.

Figure 1: A)
Simulated cost functions for a point object. B) Acquired spiral in,
spiral out, and RIO summation images before automatic deblurring. C) Deblurred
images using either the objective function provided by Lee et al. or the
proposed fuction and applied either before summation (Deblur-> Sum) or after summation (Sum->Deblur). The star and circle indicate features specific to the proposed objective function. D) The observed objective functions at pixels highlighted by the yellow
arrows in C. A spurious minimum exists at -140 Hz.