Change Point Estimation of Bilevel Functions

2005

We present two novel methods for the blind extraction of multiple binary signals from a single observation generated from linear convolution and superposition. Both methods are based on two features: (a) the constellation structure of the successor set for each output value, and (b) the concept of system deflation, ie. the recursive shortening of the channel until it is reduced into an instantaneous mixture. The two methods differ in complexity as the latter requires significantly smaller amounts of data. Both methods are fast for small problem sizes but their complexities increase exponentially as the number of inputs and/or channel length increases. We present the complete analytical solution to the noiseless case and we thoroughly treat the related mathematical tools and concepts. The noisy case is briefly discussed as the complete treatment would exceed the scope of the present paper.

Bernoulli, 2016

Quantile estimation in deconvolution problems is studied comprehensively. In particular, the more realistic setup of unknown error distributions is covered. Our plug-in method is based on a deconvolution density estimator and is minimax optimal under minimal and natural conditions. This closes an important gap in the literature. Optimal adaptive estimation is obtained by a data-driven bandwidth choice. As a side result, we obtain optimal rates for the plug-in estimation of distribution functions with unknown error distributions. The method is applied to a real data example.

1998

In this paper a new blind deconvolution algorithm as modification of the Bellini's 'Bussgang' is presented. Firstly, a novel version based on stochastic Gradient Steepest Descent error minimization technique is proposed. Then the Bayesian estimator used by Bellini is approximated with a flexible 'sigmoid' parameterized with adjustable amplitude and slope, and a gradient-based technique is proposed to adapt such parameters in order to avoid their unsuitable choices. Experimental results are shown to assess the usefulness of the new equalization method.

IEEE Transactions on Signal Processing, 2005

We propose a relative optimization framework for quasi maximum likelihood (QML) blind deconvolution and the relative Newton method as its particular instance. Special Hessian structure allows fast Newton system construction and solution, resulting in a fast-convergent algorithm with iteration complexity comparable to that of gradient methods. We also propose the use of rational IIR restoration kernels, which constitute a richer family of filters than the traditionally used FIR kernels. We discuss different choices of non-linear functions suitable for deconvolution of super-and sub-Gaussian sources, and formulate the conditions, under which the QML estimation is stable. Simulation results demonstrate the efficiency of the proposed methods.

Applicable Analysis, 2021

Blind image deconvolution recovers a deblurred image and the blur kernel from a blurred image. From a mathematical point of view, this is a strongly ill-posed problem and several works have been proposed to address it. One successful approach proposed by Chan and Wong, consists in using the total variation (TV) as a regularization for both the image and the kernel. These authors also introduced an Alternating Minimization (AM) algorithm in order to compute a physical solution. Unfortunately, Chanâs approach suffers in particular from the ringing and staircasing effects produced by the TV regularization. To address these problems, we propose a new model based on Bilateral Total Variation (BTV) regularization of the sharp image keeping the same regularization for the kernel. We prove the existence of a minimizer of a proposed variational problem in a suitable space using a relaxation process. We also propose an AM algorithm based on our model. The efficiency and robustness of our model are illustrated and compared with the TV method through numerical simulations.

2005

The present paper deals with deconvolution for linear systems within a discrete time framework. The originality of the work stands in the integration of both a filtering step and a level constraint (positivity constraint) within the estimation algorithm. After setting the structure of the filter, an iterative scheme based on Lagrange multipliers optimization technique is developed. The regularized optimization criterion is based on the fit to filtered data principle, the positive signal to be restored is considered as the square of an exogenous unconstrained signal. An example shows the enhancement brought by this approach. Copyright © 2005 IFAC

Journal of Computational and Applied Mathematics, 2006

The need for image restoration arises in many applications of various scientific disciplines, such as medicine and astronomy and, in general, whenever an unknown image must be recovered from blurred and noisy data [M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging, Institute of Physics Publishing, Philadelphia, PA, USA, 1998]. The algorithm studied in this work restores the image without the knowledge of the blur, using little a priori information and a blind inverse filter iteration. It represents a variation of the methods proposed in Kundur and Hatzinakos [A novel blind deconvolution scheme for image restoration using recursive filtering, IEEE Trans. Signal Process. 46 375-390] and Ng et al. [Regularization of RIF blind image deconvolution, IEEE Trans. Image Process. 9 ]. The problem of interest here is an inverse one, that cannot be solved by simple filtering since it is ill-posed. The imaging system is assumed to be linear and space-invariant: this allows a simplified relationship between unknown and observed images, described by a point spread function modeling the distortion. The blurring, though, makes the restoration ill-conditioned: regularization is therefore also needed, obtained by adding constraints to the formulation. The restoration is modeled as a constrained minimization: particular attention is given here to the analysis of the objective function and on establishing whether or not it is a convex function, whose minima can be located by classic optimization techniques and descent methods. Numerical examples are applied to simulated data and to real data derived from various applications. Comparison with the behavior of methods [D. Kundur, D. Hatzinakos, A novel blind deconvolution scheme for image restoration using recursive filtering, IEEE Trans. Signal Process. 46 375-390] and [M. Ng, R.J. Plemmons, S. Qiao, Regularization of RIF Blind Image Deconvolution, IEEE Trans. Image Process. 9(6) (2000) 1130-1134] show the effectiveness of our variant.

2005 IEEE Instrumentationand Measurement Technology Conference Proceedings, 2005

Inverse filtering requires the knowledge of the transfer function of the measurement system, which can be estimated on the base of measurements (system identification). The quality of the system identification influences the quality of the signal reconstruction. We investigate the influence of the identification on the signal reconstruction in the case of ill-posed problems. It is shown that overfiltering the noise in the identification stage introduces bias in the pass-and transition region of the transfer function, which causes trouble in the signal reconstruction stage. Underfiltering the noise in the identification stage also causes bias, but its effect is mostly in the stopband, which can be suppressed in the signal reconstruction stage.

Applied Optics, 2003

A new method for deconvolution of one-dimensional and multidimensional data is suggested. The proposed algorithm is local in the sense that the deconvolved data at a given point depend only on the value of the experimental data and their derivatives at the same point. In a regularized version of the algorithm the deconvolution is constructed iteratively with the help of an approximate deconvolution operator that requires only the low-order derivatives of the data and low-order integral moments of the point-spread function. This algorithm is expected to be particularly useful in applications in which only partial knowledge of the point-spread function is available. We tested and compared the proposed method with some of the popular deconvolution algorithms using simulated data with various levels of noise.

2021

We address the problem of optimal scale-dependent parameter learning in total variation image denoising. Such problems are formulated as bilevel optimization instances with total variation denoising problems as lower-level constraints. For the bilevel problem, we are able to derive M-stationarity conditions, after characterizing the corresponding Mordukhovich generalized normal cone and verifying suitable constraint qualification conditions. We also derive B-stationarity conditions, after investigating the Lipschitz continuity and directional differentiability of the lower-level solution operator. A characterization of the Bouligand subdifferential of the solution mapping, by means of a properly defined linear system, is provided as well. Based on this characterization, we propose a two-phase non-smooth trust-region algorithm for the numerical solution of the bilevel problem and test it computationally for two particular experimental settings.