On the Construction of Non Linear Adjoint Operators: Application to L1-Penalty Dynamic Image Reconstruction

The purpose of this work is to develop a methodology for the adjoint operators application in non linear optimization problems. The use of adjoint operators is very popular for numerical control theory; one of its main applications is devised for image reconstruction. Most of these reconstruction techniques are limited to linear L1-constraints whose adjoints are well-defined. We aim to extend these image reconstruction techniques allowing the terms involved to be non linear. For these purpose, we have generalized the concept of adjoint operator under the basis of Taylor’s formula, using Gateaux derivatives in order to construct a linearised adjoint operator associated to the non linear operator. The proposed approach has been validated in a Magnetic Resonance Imaging (MRI) reconstruction framework with Cartesian subsampled k-space data using Compressed Sensing based techniques and a groupwise registration algorithm for motion compensation.
The proposed algorithm has shown to be able to effectively deal with the presence of both physiological motion and subsampling artefacts, increasing accuracy and robustness of the reconstruction as compared with its linear counterpart.