De-Aliasing for Under-sampling in Phase Scrambling Fourier Transform Imaging using Alias-free Reconstruction and Deep Convolutional Neural Network
Satoshi ITO1 and Tsukasa SAITO1

1Utsunomiya University, Utsunomiya, Japan


Alias-free image reconstruction is feasible in phase scrambling Fourier transform imaging. When small down-scaling factor is used in that method, the size of reconstructed images become small and aliased image are separated in the scaled space. In this work, a new fast imaging method in which aliasing artifacts due to under-sampling of signal is removed 2-steps; one is down-scaled space introduced by alias-free reconstruction and the second is the denoising using deep convolution network. It was shown that proposed method provide higher PSNR images compared to random sampling compressed sensing and has an advantage in low sampling rate image acquisition.


When the size of imaging object is larger than the field-of-view in magnetic resonance imaging, aliasing artifacts will appear on the reconstructed image. We have proposed a new image reconstruction technique, in which images at optional scaling can be obtained and hence alias-free images can be reproduced from the data that occur aliasing artifact in Fourier transform image reconstruction technique [1,2]. Since alias-free image is realized by expanding the pseudo FOV, the spatial resolution of that image must be reduced. In this work, we propose a new faster imaging method in which equi-spaced under-sampling is adopted and aliasing artifacts are removed by aliasing control in the scaled space introduced by alias-free reconstruction, and following deep convolutional neural network.


Phase-scrambling Fourier transform imaging (PSFT) [3] is adopted in proposed method,

$$v(k_x,k_y)= \int \hspace{-2.0mm} \int^{\infty}_{-\infty} \left\{ \rho(x,y) e^{-j q (x^2+y^2)} \right\} e^{-j(k_x x+k_y y)}dxdy ...(1),$$ $$ \rho_\alpha(x',y')= \alpha^2 \rho(\alpha x, \alpha y) e^{-j c \left(\frac{\alpha-1}{\alpha} \right) \left[ (\alpha x)^2 + (\alpha y)^2 \right]} ...(2), $$

where $$$\rho(x,y)$$$ represents the spin density distribution in the subject, $$$c$$$ is the coefficient of quadratic phase shifting. The coefficient $$$c$$$ is normalized as $$$c=\alpha (\pi/N)$$$, where phase changes with neighboring pixel become $$$\pi$$$ at the end of image space when $$$\alpha =1.0 $$$ (N: matrix size of image). Figure 1 shows the application of alias-free reconstruction (AFR) [1,2]. Most of aliasing artifacts are removed, however, the spatial resolution is reduced due to the shrinkage of image to fit in the size of signal matrix as shown in Fig.1(b).

Regarding to $$$\alpha$$$, optional value can be used in reconstruction process irrespective of actual parameter used in the data acquisition and in that time scaling of images will be realized.When AFR is executed in a high down-scaling factor using a zero-filled under-sampled signal (Fig.1(c)), main image components and aliasing components will be separated in the scaled space as shown in Fig.1(e). After removal of main aliasing components shown as red squared line and followed by the inverse of alias-free reconstruction, almost aliasing artifacts are removed as shown in Fig.1(f). Since some aliasing artifacts are still remained on the image, deep convolutional neural network (CNN) is adopted to remove the remained artifacts. Deep CNN [4] we utilized was known as high de-aliasing performance without sacrificing spatial resolution.

Results and Discussions

In simulation experiments, PSFT signal is calculated using the MR volunteer image data according to the Eq. (1). Calculated signals were under-sampled at an equal interval to be 2x, 3x and 4x acceleration factor. Imaging parameters are set as $$$\alpha_{true}=1.0$$$ for data acquisition, and $$$\alpha$$$ for reconstruction is listed in Fig.3 The structure of deep CNN [4] is as follows; depth: 17, receptive field size: 35, 17 layer, filter size: 3x3x64. 50 images were used for learning of deep CNN network. Figures (a), (e), (i) are down-scaled images using alias-free reconstruction for 2x, 3x, 4x factor. Most of the aliasing artifacts were removed by replacing these separated aliased images surrounded by red dashed line with zero data. Obtained reduce aliased images shown in Figs (b), (f), (j) were used as the input images of deep CNN network. Obtained images by CNN are shown in Fig.3 (c),(g),(k). Figs. (b),(f),(j) show that remained artifacts are clearly removed by deep CNN network without conspicuous degradation of spatial resolution. Figure 4 shows the PNSR characteristics with reference to signal reduction factor using 20 phase varied images. Proposed method are compared with CS iterative reconstruction using PSFT signal using random sampling (PSFT-CS) [5] and CS iterative reconstruction (FT-CS). Figure 4 indicates that proposed method shows higher PSNR especially for lower sampling rate, 25% and 33%. The reason is that 1) aliasing artifacts are separated and they can be removed effectively by AFR, 2) spatial resolution is not severely sacrificed by under-sampling since equi-space sampling is executed, 2) remained aliasing artifacts are fairly removed by deep CNN. Figure 4 shows the results of application to experimentally obtained PSFT signal. Fully scanned signal was acquired using 0.2T MRI and then the obtained signal was under-sampled. Imaging parameters are the same as Fig.3. Figure (b) shows the fully scanned image, and (c), (d), (e) show the image with acceleration factor 2x, 3x and 4x, respectively. Even though aliasing artifacts were slightly remained on the images, high resolution images are obtained even acceleration factor 4x case (e).


A new fast imaging method equi-spaced under-sampled signal in PSFT is proposed. Aliasing artifacts are removed using alias-free reconstruction and deep CNN


This study was supported in part by JSPS KAKENHI(16K06379). We would like to thank Canon Medical Systems.


  1. Ito S, Nakamura S, Yamada Y et al. Anti-alias Imaging by Fresnel Scalable Image Reconstruction. ISMRM2006, 693, Seattle, USA
  2. Ito S, Yamada Y, Alias-free image reconstruction using Fresnel transform in the phase-scrambling Fourier imaging technique. Magn Reson Med 2008; 60, 422-430
  3. Maudsley AA, Dynamic Range Improvement in NMR Imaging Using Phase Scrambling. J Magn Reson 1988; 76, 287-305.
  4. 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
  5. Puy G, Wiaux Y, Gruetter R et al. Spread Spectrum for Accelerated Acquisition in Magnetic Resonance Imaging. IEEE Trans Med Imag 2012; 31, 586-598


Fig.1 Aliasing artifacts in PSFT imaging; (a) under-sampled PSFT signal, (b) alias-free image. Spatial resolution is reduced due to reduced signal data size, (c) zero-filled under-sampled PSFT signal, (d) reconstructed image by simple inverse FFT of (c), scaled image by alias-free image reconstruction with high down-scaling factor, (f) reduced aliased image by removing the red rectangular region shown in (e). Most of aliasing artifacts are removed.

Fig.2 Illustrative of proposed de-aliasing method; reduced aliased images are reconstructed by alias-free image reconstruction and reduced aliased images de-aliased using deep CNN. Obtained image is updated by replacing the acquired signal in k-space.

Fig.3 Results of simulation experiments; (a),(e),(i) scaled space images by alias-free reconstruction for 2x, 3x and 4x acceleration factor, (b),(f),(j) reduced aliased images by removing the red rectangular region shown in (a),(e),(i), respectively, (c),(g),(k) obtained images by proposed method, (d),(h),(l) residual of obtained images.

Fig.4 Comparison of PSNR with PSFT compressed sensing and FT compressed sensing with reference to signal reduction factor. Averaged PSNR of 20 phase varied images.

Fig.5 Application of proposed method to experimentally obtained PSFT signal; (a) obtained PSFT signal imaging orange, (b) fully scanned (256x256) images, (c),(d),(e) obtained images for 2x, 3x and 4x acceleration factor. Although slight residuals of artifacts were observed, images with good resolution were obtained.

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)