Inverse Problems and Imaging最新文献

A general framework for the tomographical description of states, that includes, among other tomographical schemes, the classical Radon transform, quantum state tomography and group quantum tomography, in the setting of begin{document}$ C^* $end{document}-algebras is presented. Given a begin{document}$ C^* $end{document}-algebra, the main ingredients for a tomographical description of its states are identified: A generalized sampling theory and a positive transform. A generalization of the notion of dual tomographic pair provides the background for a sampling theory on begin{document}$ C^* $end{document}-algebras and, an extension of Bochner's theorem for functions of positive type, the positive transform.

The abstract theory is realized by using dynamical systems, that is, groups represented on begin{document}$ C^* $end{document}-algebra. Using a fiducial state and the corresponding GNS construction, explicit expressions for tomograms associated with states defined by density operators on the corresponding Hilbert spade are obtained. In particular a general quantum version of the classical definition of the Radon transform is presented. The theory is completed by proving that if the representation of the group is square integrable, the representation itself defines a dual tomographic map and explicit reconstruction formulas are obtained by making a judiciously use of the theory of frames. A few significant examples are discussed that illustrates the use and scope of the theory.

This paper concerns the problem of phase retrieval from Fourier measurements with random masks. Here we focus on researching two kinds of random masks. Firstly, we utilize the Fourier measurements with real masks to estimate a general signal begin{document}$ mathit{boldsymbol{x}}_0in mathbb{R}^d $end{document} in noiseless case when begin{document}$ d $end{document} is even. It is demonstrated that begin{document}$ O(log^2d) $end{document} real random masks are able to ensure accurate recovery of begin{document}$ mathit{boldsymbol{x}}_0 $end{document}. Then we find that such real masks are not adaptable to reconstruct complex signals of even dimension. Subsequently, we prove that begin{document}$ O(log^4d) $end{document} complex masks are enough to stably estimate a general signal begin{document}$ mathit{boldsymbol{x}}_0in mathbb{C}^d $end{document} under bounded noise interference, which extends E. Candès et al.'s work. Meanwhile, we establish tighter error estimations for real signals of even dimensions or complex signals of odd dimensions by using begin{document}$ O(log^2d) $end{document} real masks. Finally, we intend to tackle with the noisy phase problem about an begin{document}$ s $end{document}-sparse signal by a robust and efficient approach, namely, two-stage algorithm. Based on the stable guarantees for general signals, we show that the begin{document}$ s $end{document}-sparse signal begin{document}$ mathit{boldsymbol{x}}_0 $end{document} can be stably recovered from composite measurements under near-optimal sample complexity up to a begin{document}$ log $end{document} factor, namely, begin{document}$ O(slog(frac{ed}{s})log^4(slog(frac{ed}{s}))) $end{document}

In this paper, we propose a nonconvex regularization model for images damaged by Cauchy noise and blur. This model is based on the method of the total variational proposed by Federica, Dong and Zeng [SIAM J. Imaging Sci.(2015)], where a variational approach for restoring blurred images with Cauchy noise is used. Here we consider the nonconvex regularization, namely a weighted difference of begin{document}$ l_1 $end{document}-norm and begin{document}$ l_2 $end{document}-norm coupled with wavelet frame, the alternating direction method of multiplier is carried out for this minimization problem, we describe the details of the algorithm and prove its convergence. Numerical experiments are tested by adding different levels of noise and blur, results show that our method can denoise and deblur the image better.