On Learned Operator Correction in Inverse Problems

Lunz, Sebastian; Hauptmann, Andreas; Tarvainen, Tanja; Schönlieb, Carola‐Bibiane; Arridge, Simon

doi:10.1137/20m1338460

Cited by 40 publications

(70 citation statements)

References 37 publications

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“…This is possibly because CNNs can learn and compensate non-Gaussian modelling errors more efficiently than BAE, which assumes modelling errors as Gaussian. The authors refer to [50], [51] for more discussions on modelling error corrections using CNNs.…”

Section: Discussionmentioning

confidence: 99%

A Model-Based Iterative Learning Approach for Diffuse Optical Tomography

Mozumder

Hauptmann

Nissilä

et al. 2022

IEEE Trans. Med. Imaging

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Diffuse optical tomography (DOT) utilises near-infrared light for imaging spatially distributed optical parameters, typically the absorption and scattering coefficients. The image reconstruction problem of DOT is an ill-posed inverse problem, due to the non-linear light propagation in tissues and limited boundary measurements. The ill-posedness means that the image reconstruction is sensitive to measurement and modelling errors. The Bayesian approach for the inverse problem of DOT offers the possibility of incorporating prior information about the unknowns, rendering the problem less ill-posed. It also allows marginalisation of modelling errors utilising the socalled Bayesian approximation error method. A more recent trend in image reconstruction techniques is the use of deep learning , which has shown promising results in various applications from image processing to tomographic reconstructions. In this work, we study the non-linear DOT inverse problem of estimating the (absolute) absorption and scattering coefficients utilising a 'model-based' learning approach, essentially intertwining learned components with the model equations of DOT. The proposed approach was validated with 2D simulations and 3D experimental data.We demonstrated improved absorption and scattering estimates for targets with a mix of smooth and sharp image features, implying that the proposed approach could learn image features that are difficult to model using standard Gaussian priors. Furthermore, it was shown that the approach can be utilised in compensating for modelling errors due to coarse discretisation enabling computationally efficient solutions. Overall, the approach provided improved computation times compared to a standard Gauss-Newton iteration.

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Section: Discussionmentioning

confidence: 99%

A Model-Based Iterative Learning Approach for Diffuse Optical Tomography

Mozumder

Hauptmann

Nissilä

et al. 2022

IEEE Trans. Med. Imaging

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“…While the scope of digital twins' applications spans beyond SHM alone, its basic aim is to provide information on the current or future state of an asset by combining real-time data, and a physical/data-driven model offers many potential avenues for engagement with the inverse problems community. Nonetheless, in specifically considering a classical SHM application, such as damage localization [230], developments stemming from the inverse community, including, for example, state estimation [231][232][233], uncertainty/model error approximation/compensation [194,234,235], regularization [236] and model reduction [237], have excellent potential for enriching or enhancing digital twin frameworks. As a whole, the future outlook for the integration and advancement of inverse methodologies in SHM is very bright.…”

Section: (E) Digital Twins and Outlookmentioning

confidence: 99%

“…Especially for the cases involving intractable forward problems, model reduction techniques have been promising [13,[299][300][301], but these are either difficult to design by hand or are restricted by overly simplistic assumptions. Here, datadriven approaches are a powerful alternative to compensate for modelling errors [194,234,302] or reducing computational cost of iterative optimization schemes by model approximations [32,303]. Finally, we note that recent developments in geometric learning extend deep networks on Euclidean meshes to general meshes, such as finite elements, by a embedding them into graph structures essentially using the underlying geometry [304,305].…”

Section: (A) Machine-learned Inversionmentioning

confidence: 99%

Structural engineering from an inverse problems perspective

et al. 2022

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The field of structural engineering is vast, spanning areas from the design of new infrastructure to the assessment of existing infrastructure. From the onset, traditional entry-level university courses teach students to analyse structural responses given data including external forces, geometry, member sizes, restraint, etc.—characterizing a forward problem (structural causalities → structural response). Shortly thereafter, junior engineers are introduced to structural design where they aim to, for example, select an appropriate structural form for members based on design criteria, which is the inverse of what they previously learned. Similar inverse realizations also hold true in structural health monitoring and a number of structural engineering sub-fields (response → structural causalities). In this light, we aim to demonstrate that many structural engineering sub-fields may be fundamentally or partially viewed as inverse problems and thus benefit via the rich and established methodologies from the inverse problems community. To this end, we conclude that the future of inverse problems in structural engineering is inexorably linked to engineering education and machine learning developments.

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“…This can be a strength when the original updates δx k+1 converge toward the true solution. Alternatively, if the forward model is not accurate, the GCN can compensate and correct for the wrong components and extract useful information for the updates, acting as a learned model correction [13], [14]. We will see this correcting nature in the experiments (e.g.…”

Section: B Network Structurementioning

confidence: 99%

“…where γ ∈ R + is the smoothing parameter that can be varied [18], and E = diag (Lσ) 2 + γ is a diagonal matrix. We will compare the GCNM recosntructions to TV reconstructions using (13).…”

Section: B the Inverse Problem For Eitmentioning

confidence: 99%

Graph Convolutional Networks for Model-Based Learning in Nonlinear Inverse Problems

Herzberg

Rowe

Hauptmann

et al. 2021

IEEE Trans. Comput. Imaging

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The majority of model-based learned image reconstruction methods in medical imaging have been limited to uniform domains, such as pixelated images. If the underlying model is solved on nonuniform meshes, arising from a finite element method typical for nonlinear inverse problems, interpolation and embeddings are needed. To overcome this, we present a flexible framework to extend model-based learning directly to nonuniform meshes, by interpreting the mesh as a graph and formulating our network architectures using graph convolutional neural networks. This gives rise to the proposed iterative Graph Convolutional Newton-type Method (GCNM), which includes the forward model in the solution of the inverse problem, while all updates are directly computed by the network on the problem specific mesh. We present results for Electrical Impedance Tomography, a severely ill-posed nonlinear inverse problem that is frequently solved via optimization-based methods, where the forward problem is solved by finite element methods. Results for absolute EIT imaging are compared to standard iterative methods as well as a graph residual network. We show that the GCNM has strong generalizability to different domain shapes and meshes, out of distribution data as well as experimental data, from purely simulated training data and without transfer training.

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On Learned Operator Correction in Inverse Problems

Cited by 40 publications

References 37 publications

A Model-Based Iterative Learning Approach for Diffuse Optical Tomography

A Model-Based Iterative Learning Approach for Diffuse Optical Tomography

Structural engineering from an inverse problems perspective

Graph Convolutional Networks for Model-Based Learning in Nonlinear Inverse Problems

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