Implementing iterative algorithms with Unet

[fadinammour]

• adapt Utsav's code from https://github.com/utsav-akhaury/ADMM/tree/main/ADMM_PnP_Radio
• use the Tikhonov filtered image as initialization for the solution (first guess)
• add shape constraint to loss function

[fadinammour]

Both ADMMnet and FBSnet do not converge. This is probably due to the use of the Unet in a iterative scheme without any contractiveness guarantee on the network (see https://arxiv.org/pdf/1605.01710.pdf).

One way to enforce contractiveness is to perform a spectral normalization to each layer of the network like so https://www.tensorflow.org/addons/api_docs/python/tfa/layers/SpectralNormalization.

Meanwhile, in the case of the FBSnet, an early stopping with only 5 iterations gives a satisfying reconstruction (with a diverging algorithm), see https://github.com/CosmoStat/ShapeDeconv/blob/master/notebooks/FBnet/radio/FBS_radio_test_dataset.ipynb.

The next proposed steps are:

• normalizing the Unet's layers.
• testing the early stopping reconstruction.

[fadinammour]

TensorFlow addons cannot be used with the version 1 of TensorFlow (see compatibility chart in https://github.com/tensorflow/addons). Another solution should be used for spectral normalization.