For a real object k-space is Hermitian symmetric, so only half of k-space is required to reconstruct images, that is, sparse sampling can be used. Half k-space reconstruction requires a few additional k-space lines to overcome phase shifts using homodyne methods¹. However, the existing half k-space reconstruction is vulnerable to off-resonance, which is important for functional MRI. This method can result in loss of signal when B₀ gradients cause the center of k-space to be unsampled. In this study, we introduce a new method for half k-space acquisition that is more robust to off-resonance.

Data Acquisition: With IRB approval, we scanned a human brain using a 2D Cartesian sequence. Data were acquired using a 3T GE whole-body MRI scanner equipped with a single-channel RF receive coil and a single-shot gradient-echo sequence with TE/TR=25/2000 ms, 3.4 mm x 3.4 mm x 4 mm voxels, FOV=22 cm×22 cm, 10 slices, BW=500 Hz/pixel, flip angle=80 degrees, and scan time=2 min. To see how a linear gradient affects data, we acquired reference data (no gradient applied) and data with a field gradient applied using a linear shim in the PE direction. To calculate the amount of shift in k-space from the phase map, the gradient applied image is divided by the reference image. Image Reconstruction: For homodyne reconstruction, half k-space lines and additional lines are acquired (Fig. 1 (a)) and for the new method (even/odd), even lines of the left half, odd lines of the right half and additional full k-space lines at the center (same number of lines as those of homodyne) are acquired (Fig. 1 (b)). Fig. 1 (c) and (d) show k-space data from both methods. We are able to reconstruct the image with those half k-space lines using conjugate symmetry (1). Here x denotes PE direction, and the corresponding suffix is dropped on k. $$ \begin{align*} f(x) &=\int_{-\infty}^{+\infty} F_{e}(k)\cdot e^{ikx}dk +\int_{-\infty}^{+\infty} F_{e}(k)\cdot e^{ikx}dk \\ &=\int_{-\infty}^{0} F_{e}(k)\cdot e^{ikx}dk+\int_{-\infty}^{0} F_{e}^{*}(-k)\cdot e^{ikx}dk+\int_{0}^{+\infty} F_{o}^{*}(-k)\cdot e^{ikx}dk+\int_{0}^{+\infty} F_{o}(k)\cdot e^{ikx}dk \\&=\ 2Re(f_{e}+f_{o}) \ \cdot\cdot\cdot (1) \end{align*} $$

Conjugate symmetry is satisfied only when the signal is real: $$$ F_{e}(k)=F_{e}^{*}(-k), \ F_{o}(k)=F_{o}^{*}(-k) $$$, however, the MR image is not purely real in practice. Therefore, the phase of the image is unwound by phase correction as in homodyne reconstruction. A phase map is reconstructed from the fully sampled low frequency data, and its conjugate used to correct the half-sampled data. 1) Simulation with reference data: To study how linear gradients affect to the images, we simulated different amount of shifts in k-space to the reference data and reconstructed them using both methods. 2) Reconstruction of gradient applied data: In order to guarantee conjugate symmetry while avoiding effects of off-resonance, we corrected the shifted k-space center by resampling. The amount of shift is calculated from the phase map (3). We applied both half k-space reconstruction methods to the shift corrected data and compared them. $$ \begin{align} \delta_{offset}&=\gamma\int_{0}^{TE} {G(t)\ dt} = \gamma\cdot G\cdot TE, \ where \ G=\frac{{18.76 \ [Hz]}\ /\ {\gamma}}{resoultion \times pixel \ distance} = 0.0044 \ [G/cm] \ \cdot\cdot\cdot (3) \end{align} $$

1) Simulation with reference data: For reference data, the reconstructed image using homodyne is greatly attenuated as the amount of shift increases, however, the reconstructed images using even/odd method is not attenuated as much as those from homodyne. Since most of the image energy is at the center of k-space, the reconstructed image will be highly attenuated if we do not acquire data at the center. If there is a large amount of shift, the data at the center will be lost with homodyne reconstruction (Fig. 2 (a)). However, even/odd reconstruction does not lose the data at the center even though there is a large amount of shift (Fig. 2 (b)). 2) Reconstruction of gradient applied data: For gradient applied image, there is a 15.0 sample shift in PE direction from calculation described in method and we corrected them so that conjugate symmetry can be applied. The reconstructed images using both reconstruction methods are shown in Fig. 3. Without gradient applied, homodyne acquisition shows signal loss at the center and frontal part of the brain (Fig.3 (a)), however, even/odd acquisition results in successful reconstruction (Fig. 3 (b)). The gradient applied images are reconstructed after shift correction (Fig. 3 (c) and (d)). Both images are comparable to the reference images (Fig. 3 (a) and (b)). As with the reference images, homodyne acquisition shows signal loss at the center and frontal part of the brain, however, even/odd acquisition shows successful reconstruction (Fig. 3 (c) and (d)).