Anatole Moreau^{1,2}, Florent Gbelidji^{1,3}, Boris Mailhe^{1}, Simon Arberet^{1}, Xiao Chen^{1}, Marcel Dominik Nickel^{4}, Berthold Kiefer^{4}, and Mariappan Nadar^{1}

In this work we introduce RadixNet, a fast, scalable, transform network architecture based on the Cooley-Tukey FFT, and use it in a fully-learnt iterative reconstruction with a residual dense U-Net image regularization. Results show that fast transform networks can be trained at 256x256 dimensions and outperform the FFT.

A reconstruction network was trained to reconstruct 2D MR images from under-sampled k-space data. The network structure was chosen to mimic 10 iterations of a proximal gradient algorithm. At each iteration, the network performs a gradient descent step followed by a regularization step, as described by the equation $$$X_n \leftarrow R_n (X_{n-1} +\alpha_n F_n^h(Y-\Pi F_n X_{n-1})) $$$ with $$$X_n$$$ the image reconstructed at nth iteration, $$$Y$$$ the under-sampled k-space measurements, $$$\Pi$$$ the fixed under-sampling operator, $$$F_n$$$ and $$$F_n^H$$$ the forward and adjoint measurement operators, $$$\alpha_n$$$ the trainable gradient step size and $$$R_n$$$ the regularizer. Regularizer networks were implemented using residual dense U-Nets with 2 scales, 8 channels in the hidden layers and depth 1 at each scale.

The operators $$$F_n$$$ and $$$F_n^h$$$ were also trained. In order to enable training while maintaining the low complexity of the FFT, a new network architecture named RadixNet is introduced (see Fig.1.) RadixNet is a recursive complex-valued transform network composed of 4 blocks reproducing the structure of the Cooley-Tukey FFT algorithm: a convolution to split the input into even and odd channels, recursive calls to another RadixNet on each decimated channel, a diagonal operator to apply twiddle factors and a final convolution to perform the butterfly. Each block can be made deeper and nonlinear. The overall complexity remains in $$$O(N \log N)$$$ as long as the split layer does not expand the total data size. RadixNet can be extended to multidimensional operators by splitting the input along all dimensions simultaneously (e.g. (2, 2) stride and 4 output channels in 2D).

Two networks were trained to reconstruct 2D axial T1w brain images from the Human Connectome Project [4]. 900 subjects were used with an 850 training / 50 validation split and 64 slices per subject for a total of 57600 images of fixed dimensions 256x256. A fixed variable density sampling pattern with 5x acceleration and front-to-back readout direction was used to generate the input k-space. One network was trained with 2D forward and adjoint FFTs as operators, the other with shallow linear 2D-RadixNets initialized with the Cooley-Tukey coefficients. The weights of the U-Net were initialized to i.i.d. Gaussian values with 0 mean and standard deviation 1e-3. Training used an $$$L_2$$$ loss and the Adam algorithm with mini-batch size 16 and learning rate 1e-4.

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