Satoshi ITO^{1} and Yasumichi WAKATSUKI^{1}

^{1}Utsunomiya University, Utsunomiya, Japan

Spatial resolution of images in phase scrambling Fourier transform imaging can be improved by post-processing signal band extrapolation. However, the improvements is small in the central area of image space. In our research, super-resolution using deep convolutional neural network are applied to temporally resolution improved images through iterative reconstruction. Simulation and experimental results showed that spatial resolution was fairly improved in the central area as well as in the edge of images. Proposed method is applicable to phase varied images by using adequately estimated phase distribution map since signal under-sampling is not utilized in proposed method.

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

where $$$\rho(x,y)$$$ represents the spin density distribution in the subject, $$$\phi(x,y)$$$ is phase distribution, $$$c$$$ is the coefficient of quadratic phase shifting [1]. Eq.(1) can be rewritten as Eq.(2), where $$$a_1$$$ and $$$a_2$$$ are modulation factor which increase in proportion to $$$x$$$ and $$$y$$$, respectively. Consider small segmented image $$$\rho_B$$$ as shown in Fig.1. Amplitude modulation to $$$\rho_B$$$ makes its Fourier spectrum $$$S_B$$$ shift corresponding to the modulation factor $$$a_2$$$. Spectrum $$$S_B$$$ has an asymmetric frequency band, and the signal can be extrapolated so as to be symmetric with respect to the signal peak using real-value constraint of the image like Half-Fourier imaging. We have proposed a resolution improvement method that does not require estimation of phase distribution by adopting deep CNN, however [4], the resolution improvements is smaller compared to that of iterative reconstruction using properly estimated phase map. In this work, we propose an improved method in which spatial resolution is boosted in two stages. Figure 2 show the schematic of our work. Spatial resolution is improved by iterative reconstruction in the first stage. Iterative reconstruction can extend the PSFT signal as shown in Fig.1, however the resolution improvement is proportional to the distance from the center and therefore resolution improvements is small in the central region. The spatial resolution is boosted by following deep CNN super-resolution stage. Iterative reconstruction require precise phase distribution to correct the phase shift on the image. We estimate phase distribution using acquired k-space signal. Since subsampling is not executed and signal standard symmetrical sampling with respect to the origin of k-space is performed unlike Half Fourier imaging, phase map can be estimated with high accuracy using the acquired signal. We adopted deep residual learning for CNN training [5] which is known as high excellent denoising performances. The depth 17 of CNN was set 17 and corresponding receptive field size was 35x35. Three types of layers were used, (1) Conv+ReLU: for the first layer, 64 filters of size 3 x 3, 2) Conv+BN+ReLU: for layers 2 ~ 16, 64 filters of size 3 x 3 x 64, 3) Conv: for the last layer, 3x3x64 filter were used to reconstruct the output.

1. Maudsley AA., Dynamic Range Improvement in NMR Imaging Using Phase Scrambling. J Magn Reson 1988; 76, 287-305

2. Ito S, Liu. N, Yamada Y, Improving Super-resolution by adopting Phase-scrambling Fourier Imaging. ISMRM2007, 1907, Berlin, Germany

3. 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

4. Ito S, Super-resolution based on the signal extrapolation in Phase scrambling Fourier Transform Imaging using Deep Convolutional Neural Network, ISMRM2019, 4576, Montreal, Canada

5. 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 Signal band expansion of PSFT
signal; (a) When a quadratic phase shift is given to the object, higher spatial frequency
modulation is given to 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 an asymmetric
frequency band, (c) signal band $$$S_B$$$can be restored by using the real-value
constraint of segmented image $$$\rho_B$$$. Improvement of spatial resolution
is proportional to the spatio-temporal frequency.

Fig.2 Schematic of dual stage PSFT
super-resolution; (a) zero-data is extrapolated in signal space, and (b) PSFT signal is extrapolated in the zero-padded space by several time iterative reconstruction, (c) temporal reconstruct
image. Image (c) is used as input image of deep CNN to boost the spatial
resolution.

Fig.3 Results of simulation
experiments ($$$m/N=1.0$$$); Figs (a) and (b) are real-value image models and
(c) is MR image model with phase variation., (a,b,c-1) are fully scanned images
and (a,b,c-2 to 6) are close-up of fully scanned images, standard Fourier
zero-filled images, PSFT iterative images, PSFT CNN output image, PSFT
iterative and CNN images (proposed). As shown in red arrows, spatial resolution
in the central image area is boosted by adopting CNN super-resolution.

Fig.4 Improvement of spatial
resolution for $$$m/N=0.6, 0.8, 1.0$$$. Spatial resolution is fairly improved
in the central area of image space (left end of graph).

Fig.5 Application to experimentally
obtained PSFT signal imaging an orange. Quadratic field gradient coil was used to produce quadratic phase shifts; (a) zero-data extrapolated PSFT signal, (b)
fully scanned image as a reference image, (c) phase map of (c), simple inverse
Fourier transform of signal (a), and output image of proposed method with
$$$\\m/N=1.0$$$.