We have proposed a new fast image reconstruction method in which a regularly undersampled signal is used instead of random sampling¹. Proposed method has the advantage that obtained image quality do not depend on the randomness of signal under-sampling. In the clinical imaging, a rapid spatial phase variations are sometimes imposed on the reconstructed image. In this paper, further examination about the experimental and reconstruction parameters considering spatial phase variation is performed to improve the practical use of proposed method.

In the proposed method, phase-scrambling Fourier transform imaging (PSFT)² in which a quadratic phase scrambling is added to the conventional FT imaging is adopted. $$v(k_x,k_y)= \int \hspace{-2.0mm} \int^{\infty}_{-\infty} \left\{ \rho(x,y) e^{-j\gamma b \tau(x^2+y^2)} \right\} e^{-j(k_x x+k_y y)}dxdy ...(1),$$ where $$$\rho(x,y)$$$ represents the spin density distribution in the subject, $$$\gamma$$$ is the gyromagnetic ratio, and $$$b$$$ and $$$\tau$$$ are the coefficient and impressing time, respectively, of the quadratic field gradient. The coefficient of phase scrambling $$$\gamma b \tau$$$ is normalized as $$$\gamma b \tau=h \gamma b \tau \prime$$$ ( $$$\gamma b \tau \prime=\pi/(N \Delta x^2) )$$$ (N: size of image, $$$\Delta x$$$ : pixel size).

Let $$$F_u$$$, $$$\Phi$$$, $$$\Psi$$$ and $$$Q$$$ be a under-sampling operator in Fourier space, measurement matrix, sparsifying function and quadratic phase scrambling function in Eq.(1), then sparsified image is described as，$$${\breve \rho}= \Psi \rho $$$, $$$\rho $$$ can be well recovered through nonlinear optimizations,

$$$s=F_u Q \rho=F_u Q \Psi^{-1}{\breve \rho} $$$, $$$s= \Phi {\breve \rho}$$$, where $$$\Phi=F_u Q \Psi^{-1}.$$$

Phase scrambling function Q play a in important role to diffuse and spread the artifacts caused by under-sampling, therefore, it is not necessary to randomly pick-up signal as is used in conventional compressed sensing(CS). In the previous work, we supposed the case when phase on the image was small and it could be estimated well by either an acquired image or a pre-scanned images and real-value image constraint was used in the reconstruction procedure³. In this paper, complex sparsifying function is used to meet the rapid spatial phase changes. We utilized the multi-frame FREBAS transform³ as sparsifying function that can remove the aliasing artifacts effectively using a large number basis functions used for multi-frame decomposition

MR normal volunteer images were collected using a Toshiba 1.5T MRI scanner. Flow-sensitive black blood$$$^{4}$$$ images were acquired in order to obtain images that have locally strong phase distortions (TE/TR = 40/50 ms, 256x256 matrix, slice thickness: 1.5 mm ). PSFT signals were calculated using the acquired images data according to the Eq.(1) in the simulation experiments.

Figures 1(a) and (d) show the phase and magnitude of fully scanned image. Initial image is very important to the fast convergence of reconstruction. Fig.(b) shows the simple zero-filled reconstructed image and (c) shows the image using linearly interpolated signal in Fresnel signal domain¹. We started the reconstruction using the image (c). Two different intervals of signal selection were mixed in our sampling strategy, since when only a single interval is used, aliasing artifacts sometimes remain as pointed by red arrows in Fig.1(e). The combination of sampling every third point and every eleventh point results in scanning 42% of the signal ( (85 + 23) /256= 0.428). Aliasing artifacts are fairly reduced as shown in Fig.(f). Figure 2(a) shows the comparison of a PSNR evaluation with PSFT-CS with random sampling as a function of h using 20 images. Dashed line are the PSNRs for the case when image are supposed to be a real-value function. Figure 3(b) shows the PSNR as a function of reduction factor of signal. The proposed method has almost the same PSNR as does PSFT-CS. PSNR of PSFT-CS varies with the seed of the random number used to select the phase-encoding direction. Figure 3(a)-(c) show the magnitude images for h = 0.4, 0.6, and 1.0, respectively and (d) show the image obtained by PSFT-CS. The best images obtained by the proposed method were with almost h = 1.0, as shown in Fig. 2 and Fig.3(c). Fig.4 shows the results using actual PSFT signal (42%, h=1.0) acquired 0.02T prototype MRI. Comparative image to PSFT-CS was obtained in proposed method. Without a real-value constraint on the spin density function, quadratic phase scrambling is the way to spread the error; therefore, for phase-varied images, PSNR attains a maximum value at a higher value of h. In a practical situation, a higher h value is preferable for an image that has rather strong phase variation.

A new fast image reconstruction method for images with rapid phase changes using a regularly undersampled signal is proposed and demonstrated. By combining two different interval of signal selection, fairly good complex images could be reconstructed.