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Inverse problems are at the heart of many practical problems such as image reconstruction or nondestructive testing. A characteristic feature is their instability with respect to data perturbations. To stabilize the inversion process, regularization methods must be developed and applied. In this paper, we introduce the concept of filtered diagonal frame decomposition, which extends the classical filtered SVD to the case of frames. The use of frames as generalized singular systems allows a better match to a given class of potential solutions and is also beneficial for problems where the SVD is not analytically available. We show that filtered diagonal frame decompositions yield convergent regularization methods, derive convergence rates under source conditions and prove order optimality. Our analysis applies to bounded and unbounded forward operators. As a practical application of our tools, we study filtered diagonal frame decompositions for inverting the Radon transform as an unbounded operator on L2(R2).
Most of the common model-based reconstruction schemes in magnetic particle imaging (MPI) use idealized assumptions, e.g., of an ideal field-free-line (FFL) topology. However, the magnetic fields that are generated in real MPI scanners have distortions and, therefore, model-based approaches often lead to inaccurate reconstructions and may contain artifacts. In order to improve the reconstruction quality in MPI, more realistic MPI models need to be derived. In the present work, we address this problem and present a hybrid model for MPI that allows us to incorporates real measurements of the applied magnetic fields. We will explain that the measurements, that are needed to setup a model for the magnetic fields, can be obtained in a novel calibration procedure that is independent of the resolution and which is much less time-consuming than the one employed in measurement-based MPI reconstructions.We will also present a discretization strategy for this model, that can be used in context of algebraic reconstructions. The presented approach was validated on simulated data in [1], however, its evaluation on real data is a topic for future research.
In a variety of tomographic applications, data cannot be fully acquired, leading to severely underdetermined image reconstruction. Conventional methods result in reconstructions with significant artifacts. In order to remove these artifacts, regularization methods have to be applied that incorporate additional information. An important example is TV reconstruction which is well known to efficiently compensate for missing data and well reduces reconstruction artifacts. At the same time, however, tomographic data is also contaminated by noise, which poses an additional challenge. The use of a single regularizer within a variational regularization framework must therefore account for both the missing data and the noise. However, a single regularizer may not be ideal for both tasks. For example, the TV regularizer is a poor choice for noise reduction over different scales, in which case ℓ1 curvelet regularization methods work well. To address this issue, in this paper we introduce a novel variational regularization framework that combines the advantages of two different regularizers. The basic idea of our framework is to perform reconstruction in two stages, where the first stage mainly aims at accurate reconstruction in the presence of noise, and the second stage aims at artifact reduction. Both reconstruction stages are connected by a data proximity condition. The proposed method is implemented and tested for limited-view CT using a combined curvelet-TV approach. We define and implement a curvelet transform adapted to the limited view problem and demonstrate the advantages of our approach in a series of numerical experiments in this context.
In various fields of applications, inverse problems are characterized by their sensitivity to data perturbations which can cause severe reconstruction errors. Hence, regularization procedures are employed in order to ensure stability and reconstruction quality. To overcome limitations of classical approaches such as the filtered singular value decomposition (SVD), frame based
diagonalization methods have been studied in the recent years, e.g., wavelet-vagulette (WVD) decomposition. While these methods can be well adapted to the problem at hand, it is well-known, that the lack of translation invariance in multiscale systems can cause specific artifacts in the recovered object. Thus, to overcome these drawbacks we use the translation invariant diagonal frame decomposition (TI-DFD) of linear operators. For illustration, we construct a TI-WVD for one-dimensional integration operator, and confirm our theoretical findings by numerical simulations.
Inverse problems are inherently ill-posed and therefore require regularization techniques to achieve a stable solution. While traditional variational methods have wellestablished theoretical foundations, recent advances in machine learning based approaches have shown remarkable practical performance. However, the theoretical foundations of learning-based methods in the context of regularization are still underexplored. In this paper, we propose a general framework that addresses the current gap between learning-based methods and regularization strategies. In particular, our approach emphasizes the crucial role of data consistency in the solution of inverse problems and introduces the concept of data-proximal null-space networks as a key component for their solution. We provide a complete convergence analysis by extending the concept of regularizing null-space networks with data proximity in the visual part. We present numerical results for limited-view computed tomography to illustrate the validity of our framework.
In this article, we address the challenge of solving the ill-posed reconstruction problem in computed tomography using a translation invariant diagonal frame decomposition (TIDFD). First, we review the concept of a TI-DFD for general linear operators and the corresponding filter-based regularization concept. We then introduce the TI-DFD for the Radon transform on L 2 (R 2) and provide an exemplary construction using the TI wavelet transform. Presented numerical results clearly demonstrate the benefits of our approach over non-translation invariant counterparts.