Satoshi ITO^{1}

The signal obtained in phase scrambling Fourier transform imaging can be extrapolated beyond sampling length after data acquisition like Half-phase encoding method. To realize the method for phase varied images, precise phase distribution map is required. In this paper, a new post-processing super resolution in PSFT imaging is proposed in which deep convolution neural network (CNN) is used and phase map is not required. Simulation and experimental results showed that spatial resolution was fairly improved with signal extrapolation and the improvement of spatial resolution is proportional to the strength of phase scrambling coefficient.

Phase-scrambling Fourier transform imaging (PSFT) is described as Eq.(1),

$$v(k_x,k_y)= \int \hspace{-2.0mm} \int^{\infty}_{-\infty} \left\{ \rho(x,y) e^{-j c (x^2+y^2)} \right\} e^{-j(k_x x+k_y y)}dxdy ...(1),$$ $$ = \int \hspace{-2.0mm} \int^{\infty}_{-\infty} \left\{ \rho(x,y) e^{-j \{a_1(x) x+a_2(y) y\} } \right\} e^{-j(k_x x+k_y y)}dxdy ...(2),$$ $$ a_1(x) = c x, a_2(x)=c y ...(3), $$

where $$$\rho(x,y)$$$ represents the spin density distribution in the subject, $$$c$$$ is the coefficient of quadratic phase shifting [1]. Eq.(1) can be rewritten as Eq.(2), in which amplitude of $$$\rho(x,y)$$$ is modulated by $$$ \exp{-j \{a_1(x) x+a_2(y) y\} }$$$. Consider small segmented image $$$\rho_B$$$ as shown in Fig.1, then spatiotemporal frequency $$$a_1(x), a_2(y)$$$ can be approximated as a constant in a small segment. Amplitude modulation to $$$\rho_B$$$ makes its Fourier spectrum $$$S_B$$$ shift in accordance with the modulation frequency. Since spectrum $$$S_B$$$ has asymmetric frequency band for k+ and k- directions, and one of which has wider frequency band from the peak of $$$S_B$$$ compared to standard spectrum of $$$S_A$$$, obtained image corresponding to $$$\rho_B$$$ has higher spatial resolution than image $$$\rho_A$$$. In this work, we propose a new faster imaging method in which super-resolution is executed without using phase distribution map. Figure 2 show the scheme of this work. Interpolated images are calculated using zero-data extrapolated PSFT signals, and obtained images are used as input images to convolutional neural network (CNN). We adopted Deep residual learning CNN [4] which is known as high excellent denoising performances.

- Maudsley AA., Dynamic Range Improvement in NMR Imaging Using Phase Scrambling. J Magn Reson 1988; 76, 287-305.
- Ito S, Liu. N, Yamada Y, Improving Super-resolution by adopting Phase-scrambling Fourier Imaging. ISMRM2007, 1907, Berlin, Germany
- Ito S, Liu. N, Yamada, Improvement of Spatial Resolution in Magnetic Resonance Imaging Using Quadratic Phase Modulation. IEEE International Conference on Image Processing 2009; 2497-2500, Cairo, Egypt
- Zhang K，Zuo W，Chen Y et al: Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising. IEEE Tran Image Proc 2017; 26, 3142-3155

Fig.1
Amplitude modulation in PSFT signal; (a) quadratic phase is given on the
object, high spatial frequency modulation is given on the segmented image
$$$\rho_B$$$ than segmented image $$$\rho_A$$$, (b) Frequency spectrum
$$$S_B$$$ corresponding to segmented images $$$\rho_B$$$ is shifted by
frequency modulation and has asymmetric frequency band for k+ and k- direction.
The k- band is wider than k+ band and standard $$$S_A$$$ band, (c) k +band can
be restored (extrapolated) by giving the real-value constraint of segmented
image $$$\rho_B$$$.

Fig.2 Scheme of image super-resolution; zero-data
extrapolated signal (a) is used to reconstruct interpolated image (b). image
(b) is used as input image of Deep CNN.

Fig.3 Results of simulation experiments. (a), (b), (c)
are zero-data extrapolated PSFT signal for $$$\alpha=0.2, 0.6, 1.0$$$
respectively, (d),(e),(f) are obtained images in proposed method, (g) close-up
fully scanned image, (h) simple inverse Fourier transform image of signal (c),
(i) unsharped image of (h), (j),(k),(l) are close-up images of (d),(e),(f), respectively. (m), (n), (o) are 1-dimentional
plot of output PSFT signal corresponding to images (j), (k), (l).

Fig.4 Comparison of PSNR and phase scrambling coefficient
$$$\alpha$$$.

Fig.5 Application to experimentally obtained PSFT
signal. (a) zero-data extrapolated PSFT signal, quadratic field gradient coil
was used to produce quadratic phase shifts, (b) simple inverse Fourier
transform of signal (a), output image of proposed method using
$$$\alpha=1.0$$$.
(d), fully scanned image as a reference
image.