We study a unified form of Tikhonov regularization for ill-posed problems with a general data similarity term. We discuss sufficient conditions on this generalized Tikhonov functional that guarantee existence and stability of solutions. Furthermore, we show that some particular cases of similarity functionals and regularization techniques can be cast into a unified theoretical framework. In particular, we consider the cases of $ p $-power norm, Bregman distance and mutual information as examples of a data similarity term, and Tikhonov regularization of order one or using a Sobolev norm, total variation penalization and powers of semi-norms associated to closed operators as examples of a regularization term. Finally, we take up the case of the image registration problem.
Citation: El Mostafa Kalmoun, Fatimah Allami. On the existence and stability of minimizers for generalized Tikhonov functionals with general similarity data[J]. AIMS Mathematics, 2021, 6(3): 2764-2777. doi: 10.3934/math.2021169
We study a unified form of Tikhonov regularization for ill-posed problems with a general data similarity term. We discuss sufficient conditions on this generalized Tikhonov functional that guarantee existence and stability of solutions. Furthermore, we show that some particular cases of similarity functionals and regularization techniques can be cast into a unified theoretical framework. In particular, we consider the cases of $ p $-power norm, Bregman distance and mutual information as examples of a data similarity term, and Tikhonov regularization of order one or using a Sobolev norm, total variation penalization and powers of semi-norms associated to closed operators as examples of a regularization term. Finally, we take up the case of the image registration problem.
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El Mostafa Kalmoun, Fatimah Allami. On the existence and stability of minimizers for generalized Tikhonov functionals with general similarity data[J]. AIMS Mathematics, 2021, 6(3): 2764-2777. doi: 10.3934/math.2021169
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