Vladimir Katkovnik

Pointwise Shape-Adaptive DCT for High-Quality Denoising and Deblocking of Grayscale and Color Images

The shape-adaptive discrete cosine transform (SA-DCT) transform can be computed on a support of arbitrary shape, but retains a computational complexity comparable to that of the usual separable block-DCT (B-DCT). Despite the near-optimal decorrelation and energy compaction properties, application of the SA-DCT has been rather limited, targeted nearly exclusively to video compression. In this paper, we present a novel approach to image filtering based on the SA-DCT. We use the SA-DCT in conjunction with the Anisotropic Local Polynomial Approximation-Intersection of Confidence Intervals technique, which defines the shape of the transforms support in a pointwise adaptive manner. The thresholded or attenuated SA-DCT coefficients are used to reconstruct a local estimate of the signal within the adaptive-shape support. Since supports corresponding to different points are in general overlapping, the local estimates are averaged together using adaptive weights that depend on the regions statistics. This approach can be used for various image-processing tasks. In this paper, we consider, in particular, image denoising and image deblocking and deringing from block-DCT compression. A special structural constraint in luminance-chrominance space is also proposed to enable an accurate filtering of color images. Simulation experiments show a state-of-the-art quality of the final estimate, both in terms of objective criteria and visual appearance. Thanks to the adaptive support, reconstructed edges are clean, and no unpleasant ringing artifacts are introduced by the fitted transform

From Local Kernel to Nonlocal Multiple-Model Image Denoising

We review the evolution of the nonparametric regression modeling in imaging from the local Nadaraya-Watson kernel estimate to the nonlocal means and further to transform-domain filtering based on nonlocal block-matching. The considered methods are classified mainly according to two main features: local/nonlocal and pointwise/multipoint. Here nonlocal is an alternative to local, and multipoint is an alternative to pointwise. These alternatives, though obvious simplifications, allow to impose a fruitful and transparent classification of the basic ideas in the advanced techniques. Within this framework, we introduce a novel single- and multiple-model transform domain nonlocal approach. The Block Matching and 3-D Filtering (BM3D) algorithm, which is currently one of the best performing denoising algorithms, is treated as a special case of the latter approach.

Instantaneous frequency estimation using the Wigner distribution with varying and data-driven window length

The estimation of the instantaneous frequency (IF) of a harmonic complex-valued signal with an additive noise using the Wigner distribution is considered. If the IF is a nonlinear function of time, the bias of the estimate depends on the window length. The optimal choice of the window length, based on the asymptotic formulae for the variance and bias, can be used in order to resolve the bias-variance tradeoff. However, the practical value of this solution is not significant because the optimal window length depends on the unknown smoothness of the IF. The goal of this paper is to develop an adaptive IF estimator with a time-varying and data-driven window length, which is able to provide quality close to what could be achieved if the smoothness of the IF were known in advance. The algorithm uses the asymptotic formula for the variance of the estimator only. Its value may be easily obtained in the case of white noise and relatively high sampling rate. Simulation shows good accuracy for the proposed adaptive algorithm.

A new method for varying adaptive bandwidth selection

A novel approach is developed to solve a problem of varying bandwidth selection for filtering a signal given with an additive noise. The approach is based on the intersection of confidence intervals (ICI) rule and gives the algorithm, which is simple to implement and adaptive to unknown smoothness of the signal.

Compressed Sensing Image Reconstruction Via Recursive Spatially Adaptive Filtering

We introduce a new approach to image reconstruction from highly incomplete data. The available data are assumed to be a small collection of spectral coefficients of an arbitrary linear transform. This reconstruction problem is the subject of intensive study in the recent field of compressed sensing (also known as compressive sampling). Our approach is based on a quite specific recursive filtering procedure. At every iteration the algorithm is excited by injection of random noise in the unobserved portion of the spectrum and a spatially adaptive image denoising filter, working in the image domain, is exploited to attenuate the noise and reveal new features and details out of the incomplete and degraded observations. This recursive algorithm can be interpreted as a special type of the Robbins-Monro stochastic approximation procedure with regularization enabled by a spatially adaptive filter. Overall, we replace the conventional parametric modeling used in CS by a nonparametric one. We illustrate the effectiveness of the proposed approach for two important inverse problems from computerized tomography: Radon inversion from sparse projections and limited-angle tomography. In particular we show that the algorithm allows to achieve exact reconstruction of synthetic phantom data even from a very small number projections. The accuracy of our reconstruction is in line with the best results in the compressed sensing field.

Robust M-periodogram

A maximum likelihood-type M-periodogram is developed for observations contaminated by impulse random errors having unknown heavy-tailed error distributions. The periodogram is defined as being a squared amplitude of a harmonic signal fitting the observations with, a nonquadratic residual loss function given by the Hubers (1981) minimax robust statistics.

A new form of the Fourier transform for time-varying frequency estimation

The local polynomial time-frequency transform (LPTFT) and the local polynomial periodogram (LPP) are proposed in order to estimate a rapidly time-varying frequency ω(t) of a harmonic signal m(t) = A exp(jω(t)t). The LPTFT gives a time-frequency energy distribution over the t −(ω(t),dω(t)/dt,…,dm−1ω(t)/dtm−1) space, where m is a degree of the LPTFT. The LPTFT enables one to estimate both the time-varying frequency and its derivatives. The technique is based on fitting a local polynomial approximation of the frequency which implements a high-order nonparametric regression. The a priori information about bounds for the frequency and its derivatives can be incorporated to improve the accuracy of the estimation. The estimator is shown to be strongly consistent and Gaussian for a polynomial frequency. The asymptotic covariance matrix and bias of the estimators of dsω(t)/dts, s = 0,1,2,…, m − 1, are obtained for the frequency with bounded m-derivative. Simulation results are presented.

A spatially adaptive nonparametric regression image deblurring

We propose a novel nonparametric regression method for deblurring noisy images. The method is based on the local polynomial approximation (LPA) of the image and the paradigm of intersecting confidence intervals (ICI) that is applied to define the adaptive varying scales (window sizes) of the LPA estimators. The LPA-ICI algorithm is nonlinear and spatially adaptive with respect to smoothness and irregularities of the image corrupted by additive noise. Multiresolution wavelet algorithms produce estimates which are combined from different scale projections. In contrast to them, the proposed ICI algorithm gives a varying scale adaptive estimate defining a single best scale for each pixel. In the new algorithm, the actual filtering is performed in signal domain while frequency domain Fourier transform operations are applied only for calculation of convolutions. The regularized inverse and Wiener inverse filters serve as deblurring operators used jointly with the LPA-design directional kernel filters. Experiments demonstrate the state-of-art performance of the new estimators which visually and quantitatively outperform some of the best existing methods.

Shape-adaptive DCT for denoising and image reconstruction

The shape-adaptive DCT (SA-DCT) can be computed on a support of arbitrary shape, but retains a computational complexity comparable to that of the usual separable block DCT. Despite the near-optimal decorrelation and energy compaction properties, application of the SA-DCT has been rather limited, targeted nearly exclusively to video compression. It has been recently proposed by the authors8 to employ the SA-DCT for still image denoising. We use the SA-DCT in conjunction with the directional LPA-ICI technique, which defines the shape of the transforms support in a pointwise adaptive manner. The thresholded or modified SA-DCT coefficients are used to reconstruct a local estimate of the signal within the adaptive-shape support. Since supports corresponding to different points are in general overlapping, the local estimates are averaged together using adaptive weights that depend on the regions statistics. In this paper we further develop this novel approach and extend it to more general restoration problems, with particular emphasis on image deconvolution. Simulation experiments show a state-of-the-art quality of the final estimate, both in terms of objective criteria and visual appearance. Thanks to the adaptive support, reconstructed edges are clean, and no unpleasant ringing artifacts are introduced by the fitted transform.

Discrete-time local polynomial approximation of the instantaneous frequency

The local polynomial approximation (LPA) of the time-varying phase is used to develop a new form of the Fourier transform and the local polynomial periodogram (LPP) as an estimator of the instantaneous frequency (IF) /spl Omega/(t) of a harmonic complex-valued signal. The LPP is interpreted as a time-frequency energy distribution over the t-(/spl Omega/(t), /spl Omega//sup 1/(t)),...,/spl Omega//sup m-1/(t) space, where m is a degree of the LPA. The variance and bias of the estimate are studied for the short- and long-time asymptotic behavior of the IF estimates. In particular, it is shown that the optimal asymptotic mean squared errors of the estimates of /spl Omega//sup k-1/(t) have orders O(N/sup -(2k+1)/) and O(N/sup -/2(m-k+1)/2m+3), k=1.2,...,m, respectively, for a polynomial /spl Omega/(t) of the degree m-1 and arbitrary smooth /spl Omega/(t) with a bounded mth derivative.