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Translation invariant diagonal frame decomposition of inverse problems and their regularization
(2022)
Solving inverse problems is central to a variety of important applications, such as biomedical image reconstruction and non-destructive testing. These problems are characterized by the sensitivity of direct solution methods with respect to data perturbations. To stabilize the reconstruction process, regularization methods have to be employed. Well-known regularization methods are based on frame expansions, such as the wavelet-vaguelette (WVD) decomposition, which are well adapted to the underlying signal class and the forward model and allow efficient implementation. However, it is well known that the lack of translational invariance of wavelets and related systems leads to specific artifacts in the reconstruction. To overcome this problem, in this paper we introduce and analyze the concept of translation invariant diagonal frame decomposition (TI-DFD) of linear operators. We prove that a TI-DFD combined with a regularizing filter leads to a convergent regularization method with optimal convergence rates. As illustrative example, we construct a wavelet-based TI-DFD for one-dimensional integration, where we also investigate our approach numerically. The results indicate that filtered TI-DFDs eliminate the typical wavelet artifacts when using standard wavelets and provide a fast, accurate, and stable solution scheme for inverse problems.
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.
We present two methods that combine image reconstruction and edge detection in computed tomography (CT) scans. Our first method is as an extension of the prominent filtered backprojection algorithm. In our second method we employ ℓ1-regularization for stable calculation of the gradient. As opposed to the first method, we show that this approach is able to compensate for undersampled CT data.
In this paper, we consider the problem of feature reconstruction from incomplete X-ray CT data. Such incomplete data problems occur when the number of measured X-rays is restricted either due to limit radiation exposure or due to practical constraints, making the detection of certain rays challenging. Since image reconstruction from incomplete data is a severely ill-posed (unstable) problem, the reconstructed images may suffer from characteristic artefacts or missing features, thus significantly complicating subsequent image processing tasks (e.g., edge detection or segmentation).
In this paper, we introduce a framework for the robust reconstruction of convolutional image features directly from CT data without the need of computing a reconstructed image first. Within our framework, we use non-linear variational regularization methods that can be adapted to a variety of feature reconstruction tasks and to several limited data situations. The proposed variational regularization method minimizes an energy functional being the sum of a feature dependent datafitting term and an additional penalty accounting for specific properties of the features. In our numerical experiments, we consider instances of edge reconstructions from angular under-sampled data and show that our approach is able to reliably reconstruct feature maps in this case.
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.
We present two methods that combine image reconstruction and edge detection in computed tomography (CT) scans. Our first method is as an extension of the prominent filtered backprojection algorithm. In our second method we employ l1-regularization for stable calculation of the gradient. As opposed to the first method, we show that this approach is able to compensate for undersampled CT data.