  This paper focuses on the development of a space-variant regularization model
for solving an under-determined linear inverse problem. The case study is a
medical image reconstruction from few-view tomographic noisy data. The primary
objective of the proposed optimization model is to achieve a good balance
between denoising and the preservation of fine details and edges, overcoming
the performance of the popular and largely used Total Variation (TV)
regularization through the application of appropriate pixel-dependent weights.
The proposed strategy leverages the role of gradient approximations for the
computation of the space-variant TV weights. For this reason, a convolutional
neural network is designed, to approximate both the ground truth image and its
gradient using an elastic loss function in its training. Additionally, the
paper provides a theoretical analysis of the proposed model, showing the
uniqueness of its solution, and illustrates a Chambolle-Pock algorithm tailored
to address the specific problem at hand. This comprehensive framework
integrates innovative regularization techniques with advanced neural network
capabilities, demonstrating promising results in achieving high-quality
reconstructions from low-sampled tomographic data.
