Indirect image registration is a promising technique to improve image reconstruction quality by providing a shape prior for the reconstruction task. In this paper, we propose a novel hybrid method that seeks to reconstruct high quality images from few measurements whilst requiring low computational cost. With this purpose, our framework intertwines indirect registration and reconstruction tasks is a single functional. It is based on two major novelties. Firstly, we introduce a model based on deep nets to solve the indirect registration problem, in which the inversion and registration mappings are recurrently connected through a fixed-point interaction based sparse optimisation. Secondly, we introduce specific inversion blocks, that use the explicit physical forward operator, to map the acquired measurements to the image reconstruction. We also introduce registration blocks based deep nets to predict the registration parameters and warp transformation accurately and efficiently. We demonstrate, through extensive numerical and visual experiments, that our framework outperforms significantly classic reconstruction schemes and other bi-task method; this in terms of both image quality and computational time. Finally, we show generalisation capabilities of our approach by demonstrating their performance on fast Magnetic Resonance Imaging (MRI), sparse view computed tomography (CT) and low dose CT with measurements much below the Nyquist limit.
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