Kamlesh Pawar^{1,2}, Zhaolin Chen^{1,3}, Jingxin Zhang^{3,4}, N Jon Shah^{1,5}, and Gary F Egan^{1,2}

Non-Fourier encoding such as random, noiselets and chirp encoding have demonstrated image quality improvement in accelerated compressive sensing (CS) MRI applications. However, implementation of the non-Fourier encoding schemes in 2D spin echo sequence limits its use in practice, due to the fact that spin echo is inherently a slow sequence. In this work, we present a novel implementation of chirp encoding in fast 3D gradient echo (GRE) and MPRAGE sequences. The chirp encoding scheme is compared with conventional Fourier encoding for compressive sensing and susceptibility weighted imaging applications. The evaluation demonstrates that chirp encoding is able to preserve spatial resolution better than the Fourier encoding.

**Methods**

**
Pulse
sequence design
**

A
3D GRE pulse sequence is designed as shown in Fig.1, that encodes the primary phase encoding (PE) direction with the chirp and other two directions with Fourier. For* N *PE, a total of* N *different spatially selective RF excitation pulses is
required to acquire the chirp-encoded k-space data. Due to the hardware constraint
on the scanner, which limits the number of different RF pulses to 128, the RF
pulses are prepared in real time for each PE. The gradient required during the
excitation RF pulse is given by:

$$G_y= \frac{1}{\gamma \hspace{3pt} FOV_{pe} \hspace{3pt} \Delta t_{p} }$$

where *γ* is gyromagnetic ratio,* FOV _{pe} *is field of view in PE direction,

**CS Simulation**

Simulations were performed on a phantom image to determine the effect of chirp encoding in CS reconstruction. The k-space of a phantom image was under-sampled by a factor of 4 and 6, and the reconstructed images using multichannel compressive sensing (MCS)^{4} was evaluated. To simulate chirp-encoded data, the PE direction was encoded with chirp basis and readout was encoded with Fourier basis. The chirp basis is constructed by modulating each row of the Fourier basis with a modulation function defined as:

$$C(n) = e^{j (\Delta c \hspace{1 pt} n^2 + \Delta c)}$$

where *n* is phase encode number and Δc is chirp encoding factor.

**Multichannel compressive sensing reconstruction**

The acquired data were under-sampled using complete random under-sampling for chirp and variable density under-sampling for Fourier. The images were reconstructed with MCS^{4} reconstruction by solving the following optimization:

$$Min_{\widehat{x}} \hspace{3mm} \lambda_1\|\Psi \widehat{x}\|_{l_1} + \lambda_2\| TV \widehat{x}\|_{l_1} + \|Y-E \widehat{x}\|_{l_2}^2$$

where, ψ * *is the wavelet transform operator, *TV *is the total variation operator and λ_{1}, λ_{2} are the regularization parameters.

**Imaging Parameters**

Two sequences, 3D GRE and MPRAGE were evaluated for chirp and Fourier encoding. The imaging parameters for flow compensated GRE were: TE/TR = 20/30 *ms*, flip angle = 10^{0}, FOV = 256x256x192 *mm ^{3}* and resolution 1

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