On the Construction of Non Linear Adjoint OperatorsApplication to L1-Penalty Dynamic Image Reconstruction

Résumé

The purpose of this work is to develop a methodology for the ad- joint 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 recon- struction. 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 general- ized the concept of adjoint operator under the basis of Taylor’s formula, using Gateaux derivatives in order to construct a lin- ˆ earised adjoint operator associated to the non linear operator. The proposed approach has been validated in a Magnetic Reso- nance Imaging (MRI) reconstruction framework with Cartesian subsampled k-space data using Compressed Sensing based tech- niques and a groupwise registration algorithm for motion com- pensation. The proposed algorithm has shown to be able to ef- fectively deal with the presence of both physiological motion and subsampling artefacts, increasing accuracy and robustness of the reconstruction as compared with its linear counterpart.