Convergent Data-driven Regularizations for CT Reconstruction

Kabri, Samira; Auras, Alexander; Riccio, Danilo; Bauermeister, Hartmut; Benning, Martin; Moeller, Michael; Burger, Martin

doi:10.48550/arxiv.2212.07786

Cited by 2 publications

(3 citation statements)

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“…Here, potential generalizations also could incorporate alternative loss functions in the integrand of the reconstruction training loss, which might result in approximations of alternative estimators induced by Bayes costs [12]. In this context, as a starting point, one may limit oneself to linear estimators to further investigate the relation to learned MAP estimators, respectively Tikhonov regularization, as in [2,11]. In order to obtain convergence guarantees as well as data dependence, when desirable, one can explore a noise-controlled convex combination of both training losses.…”

Section: Discussion and Outlookmentioning

confidence: 99%

Bayesian view on the training of invertible residual networks for solving linear inverse problems ^*

Arndt,

Dittmer,

Heilenkötter

et al. 2024

Inverse Problems

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Learning-based methods for inverse problems, adapting to the data's inherent structure, have become ubiquitous in the last decade. Besides empirical investigations of their often remarkable performance, an increasing number of works address the issue of theoretical guarantees. Recently, [3] exploited invertible residual networks (iResNets) to learn provably convergent regularizations given reasonable assumptions. They enforced these guarantees by approximating the linear forward operator with an iResNet. Supervised training on relevant samples introduces data dependency into the approach. An open question in this context is to which extent the data's inherent structure influences the training outcome, i.e., the learned reconstruction scheme. Here, we address this delicate interplay of training design and data dependency from a Bayesian perspective and shed light on opportunities and limitations. We resolve these limitations by analyzing reconstruction-based training of the inverses of iResNets, where we show that this optimization strategy introduces a level of data-dependency that cannot be achieved by approximation training. We further provide and discuss a series of numerical experiments underpinning and extending the theoretical findings.

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Section: Discussion and Outlookmentioning

confidence: 99%

Bayesian view on the training of invertible residual networks for solving linear inverse problems ^*

Arndt,

Dittmer,

Heilenkötter

et al. 2024

Inverse Problems

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show abstract

“…Our iResNet method using the diagonal architecture can be seen as a generalization of the learned linear spectral regularization considered in [22]. Using a different loss, which measures the reconstruction error, the authors of [22] obtain learned filter functions corresponding to Tikhonov regularization with data-and noise-dependent regularization parameters for each singular vector. An extension of the iResNet approach to this kind of training loss is thus desirable for future comparison.…”

Section: Discussion and Outlookmentioning

confidence: 99%

“…An earlier work [12] proposed a data-driven approach to a learned linear spectral regularization; more recently, [7,22] considered convergence aspects. In [22], the authors learn a scalar for each singular function direction in a filter-based reconstruction scheme, i.e. a linear regularization scheme.…”

Section: Related Workmentioning

confidence: 99%

Invertible residual networks in the context of regularization theory for linear inverse problems

Arndt,

Denker,

Dittmer

et al. 2023

Inverse Problems

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Learned inverse problem solvers exhibit remarkable performance in applications like image reconstruction tasks. These data-driven reconstruction methods often follow a two-step procedure. First, one trains the often neural network-based reconstruction scheme via a dataset. Second, one applies the scheme to new measurements to obtain reconstructions. We follow these steps but parameterize the reconstruction scheme with invertible residual networks (iResNets). We demonstrate that the invertibility enables investigating the influence of the training and architecture choices on the resulting reconstruction scheme. For example, assuming local approximation properties of the network, we show that these schemes become convergent regularizations. In addition, the investigations reveal a formal link to the linear regularization theory of linear inverse problems and provide a nonlinear spectral regularization for particular architecture classes. On the numerical side, we investigate the local approximation property of selected trained architectures and present a series of experiments on the MNIST dataset that underpin and extend our theoretical findings.

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Convergent Data-driven Regularizations for CT Reconstruction

Cited by 2 publications

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Bayesian view on the training of invertible residual networks for solving linear inverse problems ^*

Bayesian view on the training of invertible residual networks for solving linear inverse problems ^*

Invertible residual networks in the context of regularization theory for linear inverse problems

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Convergent Data-driven Regularizations for CT Reconstruction

Cited by 2 publications

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Bayesian view on the training of invertible residual networks for solving linear inverse problems *

Bayesian view on the training of invertible residual networks for solving linear inverse problems *

Invertible residual networks in the context of regularization theory for linear inverse problems

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Bayesian view on the training of invertible residual networks for solving linear inverse problems ^*

Bayesian view on the training of invertible residual networks for solving linear inverse problems ^*