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We consider reconstruction of a wave field distribution in an input/object plane from data in an output/ diffraction (sensor) plane. We provide digital modeling both for the forward and backward wave field propagation. A novel algebraic matrix form of the discrete diffraction transform (DDT) originated in Katkovnik et al. [Appl. Opt. 47, 3481 (2008)] is proposed for the forward modeling that is aliasing free and precise for pixelwise invariant object and sensor plane distributions. This 'matrix DDT' is a base for formalization of the object wave field reconstruction (backward propagation) as an inverse problem. The transfer matrices of the matrix DDT are used for calculations as well as for the analysis of conditions when the perfect reconstruction of the object wave field distribution is possible. We show by simulation that the developed inverse propagation algorithm demonstrates an improved accuracy as compared with the standard convolutional and discrete Fresnel transform algorithms. OCIS codes: 070.2025, 100.3010, 100.3190.

The paper attacks absolute phase estimation with a two-step approach: the first step applies an adaptive local denoising scheme to the modulo-2[pi] noisy phase; the second step applies a robust phase unwrapping algorithm to the denoised modulo-2[pi] phase obtained in the first step. The adaptive local modulo-2[pi] phase denoising is a new algorithm based on local polynomial approximations. The zero-order and the first-order approximations of the phase are calculated in sliding windows of varying size. The zero-order approximation is used for pointwise adaptive window size selection, whereas the first-order approximation is used to filter the phase in the obtained windows. For phase unwrapping, we apply the recently introduced robust (in the sense of discontinuity preserving) PUMA unwrapping algorithm [IEEE Trans. Image Process. 16, 698 (2007)] to the denoised wrapped phase. Simulations give evidence that the proposed algorithm yields state-of-the-art performance, enabling strong noise attenuation while preserving image details. OCIS codes: 100.5088, 110.5086, 280.0280.

The article discusses a newly developed simple and usable noise model, which consists of a Poissonian part for the modeling of photon sensing and a Gaussian part for the remaining disturbances in the output. The model presents the pointwise standard-deviation of noise as a function of the pixel-raw-data output and is widely used in digital imaging sensors.

A discrete diffraction transform (DDT) is a novel discrete wavefield propagation model that is aliasing free for a pixelwise invariant object distribution. For this class of distribution, the model is precise and has no typical discretization effects because it corresponds to accurate calculation of the diffraction integral. A spatial light modulator (SLM) is a good example of a system where a pixelwise invariant distribution appears. Frequency domain regularized inverse algorithms are developed for reconstruction of the object wavefield distribution from the distribution given in the sensor plane. The efficiency of developed frequency domain algorithms is demonstrated by simulation. OCIS codes: 070.2025, 100.3010, 100.3190.

Local polynomial approximation (LPA) technique was used for estimating the absolute phase reconstruction from noisy wrapped phase data. Findings revealed a reasonable compromise between the noise smoothing and preservation of details in phase image due to window size adaptation.

A new method is described for spatially adaptive estimation and the local likelihood is used for nonparametric modeling of estimated signals. The new algorithm has performed well for image denoising and it is compared with the intersection of confidence interval (ICI) technique that has used a selection of an adaptive neighborhood for estimation.

Direction-of-arrival (DOA) estimation in a phased-array system is considered for a non-Gaussian noise environment. In this paper, a robust M-estimation scheme based on Huber's loss function with an adaptive threshold is proposed. The method is evaluated using experimental data from the University of California, Los Angeles, 60-GHz antennaarray system. It is shown that the proposed method enables a decreased sensitivity of the DOA estimates with respect to noise components having heavy-tailed distributions. Index Terms--Adaptive threshold, non-Gaussian noise, robust direction-of-arrival (DOA) estimation, 60-GHz antenna array.

A new image denoising strategy based on enhanced sparse representation in transform domain achieved an excellent denoising performance in both peak signal-to-noise ratio and subjective visual quality.

The shape-adaptive discrete cosine transform (SA-DCT) transform is found to be extremely useful for the denoising and deblocking of grayscale and color images, hence providing clean reconstructed edges without any unpleasant ringing artifacts due to the adaptive support provided by the structural constraints.

In this paper, new robust Al-estimation techniques are developed for combating impulsive noise and multiple-access interference in communication systems. Power functions of the proposed robust estimators are obtained by minimizing a non-quadratic residual function derived from the Huber's minimax robust estimation theory. Maximum peaks of these power functions are used for estimation of communication signals as well as direction of arrival. A strong advantage of the proposed robust M-estimation algorithms is a decreased sensitivity of the estimates with respect to an actual unknown distribution of random noises and interferences. Simulation results demonstrate that the proposed robust algorithms, the robust median, and robust Huber estimators, offer significant performance gain over the conventional and minimum-variance distortionless-response estimators, with the best results given by the robust Huber estimator. Index Terms--Antenna array, direction-of-arrival (DOA) estimation, impulsive noise, M-estimation, wireless communications.

A novel nonparametric regression method for deblurring noisy images is proposed. The regularized inverse and Wiener inverse filters serve as deblurring operator used jointly with the local polynomial approximation-design directional kernel filters.

Modern standard high-resolution array signal processing techniques are based on simultaneous sampling of the whole multiple-sensor array. A switch antenna array system is a promising substitute for the multiple-channel array due to its lower cost and a simpler front-end circuitry. A system composed of a single transmitter, M receiving antennas, and a single receiving channel is explained.

A windowed local polynomial approximation (LPA) of time-varying directions of arrival is used to develop an LPA-beamformer for localization of rapidly moving sources. Closed-form expressions are obtained for the bias and variance of the LPA-beamformer in the moving source case and the problem of the optimal window choice is discussed. Index Terms--Array processing, beamforming, moving sources.

The L-Wigner distributions are defined in order to improve the concentration of signal's time-frequency representation. For a finite distribution order and nonlinear instantaneous frequency (IF) it gives biased IF estimates. In the case of noisy signals the optimal window length and distribution order depend on the noise variance and unknown IF. In this correspondence an adaptive IF estimator with the time-varying and data-driven window length and distribution order is developed. Based on the analysis that has been done here, lower order time-frequency distributions are introduced. Index Terms:Estimation, instantaneous frequency, time-frequency analysis, Wigner distribution, window optimization.

A unique strategy for solving a problem of bandwidth selection for filtering a signal with additive noise is shown. The linear local polynomial approximation is employed to demonstrate the efficiency of the technique, while a development to nonlinear observations and robust M-estimates is possible. The algorithm is simple to implement and needs the computation of the estimates and their standard deviations for a set of bandwidth values.

Minimax Huber's robust statistics is an alternative to the maximum likelihood approach in the estimation of the distribution of random errors. A new periodogram is developed based on the nonlinear estimates of the amplitude C. The M-periodogram is robust in the estimation of the distribution of errors with unknown but heavy-tailed distributions. A key point of the robust M-periodogram is the choice of the nonquadratic residual function F.

A new form of the Fourier transform and the local polynomial periodogram is developed as an estimator of the instantaneous frequency (IF) of a harmonic complex-valued signal, using the local polynomial approximation of the time-varying phase. An examination of the variance and bias of the estimate for the short- and long-time asymptotic behavior of the IF estimates shows that the orders for the optimal asymptotic mean squared error of the estimates of omega(super k-1)(t) are 0(N-(super 2k+1)) and 0(N-2(m-k+1)/2m+3), respectively, for a polynomial omega(t) of the degree m-1 and arbitrary smooth omega(t) with a bounded mth derivative.

An adaptive estimator for calculating instantaneous frequency (IF) of a harmonic complex-valued signal with additive noise was created using the Wigner distribution. This estimation tool uses an algorithm with a time-varying and data-driven window length that was observed to generate IF calculations based on the formula for the variance of the estimate. Simulation testing also showed that this adaptive estimator has good accuracy and is capable of enhancing time-frequency representation of signal with a nonlinear IF.

The local polynomial approximation of time-varying phase is used in order to estimate the instantaneous frequency and its derivatives for a complex-valued harmonic signal given by discrete-time observations with a noise. The considered estimators are high-order nonparametric generalizations of the short-time Fourier transform and the Wigner-Ville distribution. The asymptotic variance and bias of the estimates are obtained. Index Terms - Fourier analysis, nonparametric estimation, spectral analysis, time-frequency distribution, time-varying frequency.

A study was conducted to analyze the problem of estimating the instantaneous frequency (IF). The asymptotic minimax lower bound was determined for the mean squared error of estimation while the local polynomial periodogram was represented as a nonparametric estimator of the IF. Results indicated that the optimal choice of the window size in the local polynomial periodogram estimator of the IF and its derivatives correlated with mean squared errors that were different from the minimax lower bounds.