Numerical methods for the approximate solution of ill-posed problems on compact sets

Тихонов, А. Н.; Goncharsky, A. V.; Stepanov, V. V.; Yagola, A. G.

doi:10.1007/978-94-015-8480-7_3

Cited by 52 publications

(57 citation statements)

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“…(14) is "a free parameter" finally determined upon some criterion associated with the results of the functional Φ α A minimization for different α > 0. The problem of the proper choice of α may draw us into still more abstract and profound discussion on how to decide between many possible criteria (see, e.g., [16]). On the contrary, we did not dwell long in this issue and turned our attention to special numerical experiments.…”

Section: Inverse Problemmentioning

confidence: 99%

“…Once a solution of this minimization problem is found with an acceptably small value of the functional, one can derive the sought function a ay. As known [16], a solution of the so-formulated problem [a kind of normal solution of Eq. (13)] may be very unstable, if any.…”

Section: Inverse Problemmentioning

confidence: 99%

“…This inverse problem is ill-posed and can be only solved by employing a relevant regularization algorithm. The fundamental achievements in this area are connected with Lavrent'yev and Tikhonov [16]. As known, the idea of Tikhonov regularization is to reduce the relevant regularizing functional to a minimum.…”

Section: Introductionmentioning

confidence: 99%

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Profile reconstruction of periodic interface

et al. 2014

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The reconstruction problem for periodic (arbitrary profiled within a period) boundary between two homogeneous media is considered. Our approach to the solution of the inverse problem is based on the Tikhonov regularization technique, which requires successive selection of the boundaries on the basis of multiple solutions of the direct problem of wave diffraction by the candidate boundaries. The analytical numerical C method has been chosen as a simple but rather efficient tool for the direct problem solving. The scheme for numerical tests of algorithms and criteria for reconstruction accuracy have been suggested and verified. Results of numerical experiments that prove the validity of the approach are presented.

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Section: Inverse Problemmentioning

confidence: 99%

Section: Inverse Problemmentioning

confidence: 99%

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Profile reconstruction of periodic interface

et al. 2014

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“…Traditional methods such as inverse filtering [7], Tikhonov filtering [8], Weiner filtering [9], and Richardson-Lucy algorithm [10], [11] were proposed several decades ago for the inverse problem. A non-linear iterative method, Maximum Likelihood Expectation Maximization (MLEM), which is similar to the Richardson-Lucy algorithm, has been designed for solving both the blind and the non-blind Poisson image deconvolution problems [12], [13], [14], [15].…”

Section: Related Workmentioning

confidence: 99%

Three-Dimensional Deconvolution of Wide Field Microscopy with Sparse Priors: Application to Zebrafish Imagery

Dong

Shao

Frangi

et al. 2014

2014 22nd International Conference on Pattern Recognition

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Zebrafish, as a popular experimental model organism, has been frequently used in biomedical research. For observing, analysing and recording labelled transparent features in zebrafish images, it is often efficient and convenient to adopt the fluorescence microscopy. However, the acquired z-stack images are always blurred, which makes deblurring/deconvolution critical for further image analysis. In this paper, we propose a Bayesian Maximum a-Posteriori (MAP) method with the sparse image priors to solve three-dimensional (3D) deconvolution problem for Wide Field (WF) fluorescence microscopy images from zebrafish embryos. The novel sparse image priors include a global Hyper-Laplacian model and a local smooth region mask. These two kinds of prior are deployed for preserving sharp edges and suppressing ringing artifacts, respectively. Both synthetic and real WF fluorescent zebrafish embryo data are used for evaluation. Experimental results demonstrate the potential applicability of the proposed method for 3D fluorescence microscopy images, compared with state-of-the-art 3D deconvolution algorithms.

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“…In cooperation with appropriate fluorescent probes, FMT allows for three-dimensional (3-D) imaging of molecular functions and the visualization of biological and physiological processes, which make it powerful in clinical and preclinical research, e.g., cancer and drug development. In the early stage of FMT development, l 2 -norm regularization method, such as Tikhonov regularization, 7,8 was widely used to deal with the ill-posedness of FMT. However, the inherently ill-posed nature of the inverse problem along with complex scattering and absorbing in the process of near-infrared photons propagation in biological tissues makes FMT challenging.…”

Section: Introductionmentioning

confidence: 99%

Laplacian manifold regularization method for fluorescence molecular tomography

Wang

et al. 2017

J. Biomed. Opt

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Sparse regularization methods have been widely used in fluorescence molecular tomography (FMT) for stable three-dimensional reconstruction. Generally, ? 1 -regularization-based methods allow for utilizing the sparsity nature of the target distribution. However, in addition to sparsity, the spatial structure information should be exploited as well. A joint ? 1 and Laplacian manifold regularization model is proposed to improve the reconstruction performance, and two algorithms (with and without Barzilai–Borwein strategy) are presented to solve the regularization model. Numerical studies and in vivo experiment demonstrate that the proposed Gradient projection-resolved Laplacian manifold regularization method for the joint model performed better than the comparative algorithm for ? 1 minimization method in both spatial aggregation and location accuracy.

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Numerical methods for the approximate solution of ill-posed problems on compact sets

Cited by 52 publications

References 0 publications

Profile reconstruction of periodic interface

Profile reconstruction of periodic interface

Three-Dimensional Deconvolution of Wide Field Microscopy with Sparse Priors: Application to Zebrafish Imagery

Laplacian manifold regularization method for fluorescence molecular tomography

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