Multi-Parameter Regularization Method for Synthetic Aperture Imaging Radiometers

Synthetic aperture imaging radiometers (SAIRs) are powerful passive microwave systems for high-resolution imaging by use of synthetic aperture technique. However, the ill-posed inverse problem for SAIRs makes it difficult to reconstruct the high-precision brightness temperature map. The traditional regularization methods add a unique penalty to all the frequency bands of the solution, which may cause the reconstructed result to be too smooth to retain certain features of the original brightness temperature map such as the edge information. In this paper, a multi-parameter regularization method is proposed to reconstruct SAIR brightness temperature distribution. Different from classical single-parameter regularization, the multi-parameter regularization adds multiple different penalties which can exhibit multi-scale characteristics of the original distribution. Multiple regularization parameters are selected by use of the simplified multi-dimensional generalized cross-validation method. The experimental results show that, compared with the conventional total variation, Tikhonov, and band-limited regularization methods, the multi-parameter regularization method can retain more detailed information and better improve the accuracy of the reconstructed brightness temperature distribution, and exhibit superior noise suppression, demonstrating the effectiveness and the robustness of the proposed method.

Two-Parameter Regularization Method for a Nonlinear Backward Heat Problem with a Conformable Derivative

In this paper, we consider the nonlinear inverse-time heat problem with a conformable derivative concerning the time variable. This problem is severely ill posed. A new method on the modified integral equation based on two regularization parameters is proposed to regularize this problem. Numerical results are presented to illustrate the efficiency of the proposed method.

Adaptive multi-parameter regularization approach to construct the distribution function of relaxation times

Determination of the distribution function of relaxation times (DFRT) is an approach that gives us more detailed insight into system processes, which are not observable by simple electrochemical impedance spectroscopy (EIS) measurements. DFRT maps EIS data into a function containing the timescale characteristics of the system under consideration. The extraction of such characteristics from noisy EIS measurements can be described by Fredholm integral equation of the first kind that is known to be ill-posed and can be treated only with regularization techniques. Moreover, since only a finite number of EIS data may actually be obtained, the above-mentioned equation appears as after application of a collocation method that needs to be combined with the regularization. In the present study, we discuss how a regularized collocation of DFRT problem can be implemented such that all appearing quantities allow symbolic computations as sums of table integrals. The proposed implementation of the regularized collocation is treated as a multi-parameter regularization. Another contribution of the present work is the adjustment of the previously proposed multiple parameter choice strategy to the context of DFRT problem. The resulting strategy is based on the aggregation of all computed regularized approximants, and can be in principle used in synergy with other methods for solving DFRT problem. We also report the results from the experiments that apply the synthetic data showing that the proposed technique successfully reproduced known exact DFRT. The data obtained by our techniques is also compared to data obtained by well-known DFRT software (DRTtools).

Estimation of a fold convolution in additive noise model with compactly supported noise density

Consider the model Y = X + Z , where Y is an observable random variable, X is an unobservable random variable with unknown density f , and Z is a random noise independent of X . The density g of Z is known exactly and assumed to be compactly supported. We are interested in estimating the m- fold convolution fm=f*...*f on the basis of independent and identically distributed (i.i.d.) observations Y1,..,Yn drawn from the distribution of Y . Based on the observations as well as the ridge-parameter regularization method, we propose an estimator for the function fm depending on two regularization parameters in which a parameter is given and a parameter must be chosen. The proposed estimator is shown to be consistent with respect to the mean integrated squared error under some conditions of the parameters. After that we derive a convergence rate of the estimator under some additional regular assumptions for the density f .