Making use of a partial order in solving inverse problems: II.

Korolev, Yury

doi:10.1088/0266-5611/30/8/085003

Cited by 6 publications

(17 citation statements)

References 12 publications

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“…the forward operator is exact), the set U h,δ is non-convex and the residual method results in a non-convex optimisation problem even for convex regularisation functionals. An alternative approach to modelling uncertainty in A and f using partially ordered spaces was proposed in [15,16]. Assume that U and F are Banach lattices, i.e.…”

Section: Introductionmentioning

confidence: 99%

“…Then, uncertainties in A and f can be characterised using intervals in appropriate partial orders, i.e. (1.4) where L r (U, F) ⊂ L(U, F) is the space of regular operators U → F. Assuming positivity of the exact solutionū and using the inequalities in (1.4), we can show that the exact solutionū is contained in the following feasible set [15]: Using the relationship between partial orders and norms in Banach lattices, one can prove the inclusion of the partial-order-based feasible set U (1.5) in the norm-based feasible set U h,δ (1.3) for appropriately chosen A h , u δ , h, δ. We briefly review the partial-order-based approach in Section 2.…”

Section: Introductionmentioning

confidence: 99%

“…We briefly review the partial-order-based approach in Section 2. While convergence of the minimisers of (1.6) to the exact solution can be guaranteed [15], it is not clear, whether the solution to (1.6) actually corresponds to a particular pair (A, f ) within the bounds (1.4). It seems more natural to look for approximate solutions in the following set:…”

Section: Introductionmentioning

confidence: 99%

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Image Reconstruction with Imperfect Forward Models and Applications in Deblurring

Korolev¹,

Lellmann²

2018

SIAM J. Imaging Sci.

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We present and analyse an approach to image reconstruction problems with imperfect forward models based on partially ordered spaces -Banach lattices. In this approach, errors in the data and in the forward models are described using order intervals. The method can be characterised as the lattice analogue of the residual method, where the feasible set is defined by linear inequality constraints. The study of this feasible set is the main contribution of this paper. Convexity of this feasible set is examined in several settings and modifications for introducing additional information about the forward operator are considered. Numerical examples demonstrate the performance of the method in deblurring with errors in the blurring kernel.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Image Reconstruction with Imperfect Forward Models and Applications in Deblurring

Korolev¹,

Lellmann²

2018

SIAM J. Imaging Sci.

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“…The reason for non-convexity is the fact that we used the operator norm to quantify the error in the operator. An alternative approach was proposed in [26]. Instead of the operator norm, it uses intervals in an appropriate partial order to quantify the error in the operator.…”

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confidence: 99%

Convergence rates and structure of solutions of inverse problems with imperfect forward models

2019

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The goal of this paper is to further develop an approach to inverse problems with imperfect forward operators that is based on partially ordered spaces. Studying the dual problem yields useful insights into the convergence of the regularised solutions and allow us to obtain convergence rates in terms of Bregman distances -as usual in inverse problems, under an additional assumption on the exact solution called the source condition. These results are obtained for general absolutely one-homogeneous functionals. In the special case of TV-based regularisation we also study the structure of regularised solutions and prove convergence of their level sets to those of an exact solution. Finally, using the developed theory, we adapt the concept of debiasing to inverse problems with imperfect operators and propose an approach to pointwise error estimation in TV-based regularisation.Keywords: inverse problems, imperfect forward models, total variation, extended support, Bregman distances, convergence rates, error estimation, debiasing where A : L 1 (Ω) → L ∞ (Ω) is a linear operator and Ω ⊂ R m is a bounded domain. We assume that there exists a non-negative solution of (1.1).For an appropriate functional J (·) : L 1 → R + ∪{∞} we consider non-negative J -minimising solutions, which solve the following problem:We assume that the feasible set in (1.2) has at least one point with a finite value of J and denote a (possibly non-unique) solution of (1.2) byū J . Throughout this paper it is assumed that the regularisation functional J (·) is convex, proper and absolutely one-homogeneous.In practice the data f are not known precisely and only their perturbed versionf is available. In this case, we cannot simply replace the constraint Au = f in (1.2) with Au =f , since the solutions of the original problem (1.1) would no longer be feasible in this case. Therefore, we need to relax the equality in (1.2) to guarantee the feasibility of solutions of the original problem (1.1). This is the idea of the residual method [20,23]. If the error in the data is bounded by some known constant δ, the residual method accounts to solving the following constrained problem: min( 1.3)The fidelity function becomes in this case the characteristic function of the convex set {u : Au− f δ}. In the linear case, the residual method is equivalent to Tikhonov regularisation min u∈L 1with the regularisation parameter α = α(f , δ) chosen according to Morozov's discrepancy principle [23]. In many practical situations not only the data contain errors, but also the forward operator, that generated the data, are not perfectly known. In order to guarantee the feasibility of solutions of the original problem (1.1) in the constrained problem (1.3), one needs to account for the errors in the operator in the feasible set. If the errors in the operator are bounded by a known constant h (in the operator norm), the feasible set can be amended as follows in order to guarantee feasibility of the solutions of the original problem (1.1):whereÃ is the noisy operator. This optimi...

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“…We now briefly outline the theoretical framework [32] that is the basis for our approach. Consider a linear operator equation Fig.…”

Section: Mathematical Basismentioning

confidence: 99%

Diffusion Tensor Imaging with Deterministic Error Bounds

2016

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Errors in the data and the forward operator of an inverse problem can be handily modelled using partial order in Banach lattices. We present some existing results of the theory of regularisation in this novel framework, where errors are represented as bounds by means of the appropriate partial order. We apply the theory to diffusion tensor imaging, where correct noise modelling is challenging: it involves the Rician distribution and the non-linear Stejskal-Tanner equation. Linearisation of the latter in the statistical framework would complicate the noise model even further. We avoid this using the error bounds approach, which preserves simple error structure under monotone transformations.

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Making use of a partial order in solving inverse problems: II.

Cited by 6 publications

References 12 publications

Image Reconstruction with Imperfect Forward Models and Applications in Deblurring

Image Reconstruction with Imperfect Forward Models and Applications in Deblurring

Convergence rates and structure of solutions of inverse problems with imperfect forward models

Diffusion Tensor Imaging with Deterministic Error Bounds

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