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The statistical model for describing the refractive index of the troposphere based on nested semi-Markov processes is proposed. The possibility of describing the statistics of the refractive index in season by locally Gaussian model is shown. First considered the use of atomic functions (AF) to describe the statistics of the refractive index, a new approach for testing hypotheses about the form of the random variable, based on the use of inverse functions is proposed.

An important factor when realizing image capture using electronic means (photographic or video camera), is the lighting source type present at the capture moment. If the lighting is deficient, the information present in the image is hidden for the Human Vision System (HVS), such that the color and detail characteristics introduce values with low luminance. As a solution for this lighting problem, the Multiscale-Retinex (MSR) algorithm is proposed, based on the understanding of how the HVS interprets and adapts the perception of colors. This solution can be used as a preprocessing stage to correct the lack of an adequate lighting source like the standard illuminant D 65 (light source with similar characteristics to midday sunlight). The MSR algorithm increase the chromatic content of the image, this improves the objects visibility in the scene. The lack effects of lighting are minimized and its magnitude content evaluates the resulting chromaticity vector to characterize the MSR.

The possibility of using finite atomic functions of Kravchenko-Rvachev for description of the laws of distribution of the refractive index of the troposphere, the intensity of the scattering from the sea, seasonal behavior unit RCS of land areas with vegetation covering, as well as the spectrum of electromagnetic spikes of lithospheric origin is considered.

In this report the theory of spectral estimation of signals on based of generalized two-dimensional (2D) Kravchenko-Kotel’nikov-Levitan theorems is considered. In the first part the 2D weight functions with support area of the complex shape on the basis of R-functions are constructed. In the second part the 2D Kravchenko-Kotel’nikov-Levitan theorems are formulated. In the third part it’s application for spectral estimation, filtering and correction of 2D signals and images. As the examples the problems of noise reduction, scaling and also spectral properties improvement of images are submited. The numerical experiment and physical analysis of 2D signal processing results are show efficiency of new 2D kernels construction of sampling theorems.

In this paper, we suggest and substantiate constructions of a new class of WA-systems of Kravchenko‐ Rvachev functions [1, 2] based on atomic functions (AFs) and using the ideas and results of [1‐7]. In the first part, we describe an algorithm for constructing WA-systems of functions, and in the second part, we perform a computational study of a new family of complex wavelets based on AFs and possessing a number of useful properties. These wavelets and their Fourier images are determined by analytical relations.

Wavelets are widely applied in modern science and technology. Construction of wavelets based on new family of atomic functions cha, n is presented. This family also allows unified examination of wavelets based on atomic functions up(x), ha(x) and Ξn(x). The construction of orthogonal Kravchenko wavelets (also known as WA-systems) is based on the Riesz basis and Daubechies theorems.

A review of modern optimal and quasi-optimal algorithms for radiometric signal processing, which provide images with a high spatial resolution is given. It is shown that the high resolution is achieved with ultra-wideband or multi-band time or space-time signal processing. Analytical expressions for the limiting error of image estimation, ambiguity functions and potential fluctuation sensitivity of systems designed according to the algorithms are derived.

This paper presents results of construction of the new class of weight and WA-systems of Kravchenko-Kaiser functions. It has some efficiency in a variety of physical applications. The work consists of three parts. First, on the basis of the theory of atomic functions (AF) [1-10] and Kaiser [10] windows the Kravchenko-Kaiser weight functions are constructed. Then, the new construction of analytical WA-systems of functions is proposed and their advantages are shown. In the final part, the applications to problems of weight averaging the difference frequency are discussed.

Different hardware implementations of designed automatic 2D to 3D video color conversion employing 2D video and sequence are presented. The analyzed frameworks include together processing of neighboring frames using the following blocks: CIELa*b* color space conversion, wavelet transform (WT) with edge detection using HF wavelet sub-bands (HF, LH and HH) or pyramidal scheme, color segmentation via k-means on a*b* color plane, up-sampling in wavelet case, disparity map (DM) estimation, adaptive post-filtering, and finally, the anaglyph 3D scene generation. The SSIM and QBP criteria are applied in order to compare the performance of the proposed frameworks against other 3D computation techniques. The designed techniques has been implemented on DSP TMS320DM648, Matlab’s Simulink module over a PC with Windows 7, and using graphic card (NVIDIA Quadro K2000) demonstrating that the proposed approach can be applied in real-time processing mode.

A new concept for solving statistical synthesis problems of radiometric devices and systems (RDS) is developed. The reasonability of using ultra-wideband (UWB) RDSs is substantiated. The basic data of mathematical apparatus of V-transforms employed when processing space-time UWB fields and their statistical characteristics is given. A wide class of UWB RDS statistical synthesis problems (compensation, zero-type, modified modulation type, modulation compensation type, multi-antenna systems including cross-correlation-compensation systems and aperture synthesis systems) of stationary and scanning types are solved. The appropriateness of using the Kravchenko weighting functions for refinement of primary radiometric images in radiometric scanning devices is substantiated.

Connection between atomic functions and Thue-Morse sequence and more general self-similar sequences is discussed. This connection improves the atomic functions theory and study of properties of Thue-Morse-like sequences and makes advantages in numerical methods, digital signal processing and many other applications of atomic functions and Thue-Morse sequences.

The efficiency of navigation, radiolocation and communication largely depends on the propagation of radio waves, which can be determined by the state of atmospheric refraction. Therefore, studying spatial and temporal distributions of the refractive index N and creating its statistical model are of great interest to predictions of the range of radio systems, as well as estimates of the accuracy of measuring the coordinates of GNSS. To experimentally study how the propagational conditions of radio waves influence the accuracy of measuring the coordinates, measuring points are set up in Kharkov, Poltava, Sumy, equipped with receiving equipments design Ltd. "Navis-Ukraine", measuring errors of estimating the coordinates around the clock. Measurements are accumulated by data of regular meteorological stations located in these cities, from which we calculated the values of the refractive index of troposphere.

On the basis of the atomic functions ha(t) the Kravchenko-Kotelnikov interpolation series for restoration of correlation functions in problems of stochastic processes modeling is offered.

The report consists of two parts. In the first part the new designs of kernels of the generalized sampling series are proposed. This approach ensures high interpolation properties and good time-and-frequency localization. In the second part the weight functions for improvement of convergence of the generalized sampling series are suggested.

In the report new weight functions with low side-lobe level, based on convolutions of atomic functions are presented. There are effective algorithms for their computation. To increase coherent gain the window is truncated with an additional one. This approach allows us to obtain windows with good properties. The degree of cut of the function determines the relation between side-lobes and coherent gain. Usually the best windows are obtained with the cut near the effective support of function.

Integral non-parametric estimations of the probability density and its derivatives [5–8] found on the basis of atomic functions (AF) [1–4] and of results of works [5–7] are presented. This approach enables us to obtain smoother estimations of the probability density, which improves the efficiency of solving the problems of image classification and recognition. The mathematical apparatus of the integral nonparametric statistics enables us to estimate more precisely the sequence characteristics without a priori parametric information.

In this report the new constructions of digital filters are proposed. They are based on the generalized Kravchenko-Kotelnikov sampling theorem. Efficiency of the proposed low- and high- frequency filters is investigated for an example of digital processing of signals.

An efficient algorithm to generate three-dimensional (3D) video sequences is presented in this work. The algorithm is based on a disparity map computation and an anaglyph synthesis. The disparity map was first estimated by employing the wavelet atomic functions technique at several decomposition levels in processing a 2D video sequence. Then, we used an anaglyph synthesis to apply the disparity map in a 3D video sequence reconstruction. Compared with the other disparity map computation techniques such as optical flow, stereo matching, wavelets, etc., the proposed approach produces a better performance according to the commonly used metrics (structural similarity and quantity of bad pixels). The hardware implementation for the proposed algorithm and the other techniques are also presented to justify the possibility of real-time visualization for 3D color video sequences.

In this work on the basis of the theory atomic and R-functions the new class of analytical two-dimensional Kravchenko-Rvachev wavelets (2DWA KR) with basic areas of complex shape is constructed. Numerical experiment and the physical analysis of modeling results confirms that constructed 2DWA KR can effectively be applied in problems of the interferential effects analysis with a complex configuration, signal filtrations and image recognition.

Nonparametric estimations of probability density function (PDF) [1–3] and its derivatives are considered. New constructions of weight functions (WF) [4, 5] with the compact support are offered and proved, allowing to build admissible estimations, both the density of probability and its derivatives 1st and 2nd orders. The numerical experiment and the physical analysis of its results confirm efficiency of nonparametric characteristics estimation of stochastic process.

In this report a general theory of phase-locked loop (PLL) for N-dimensional UWBsignals (for example radar and optical signals) is given. On the first step a discretization method for solving differential equations of the PLL systems is presented. It is shown that there are a lot of zones of synchronous in discretized PLL and a technical realization of this device (in PLL with samples). A spectral theory of analysis of oscillations in PLL with samples is given. It is shown a lot of different physical properties in PLL with samples. The effect of shape of the input signals (harmonic signals, splines and atomic functions) at the border zones of operating modes being entered into PLL with samples is examined.

Nonlinear Schrodinger equation is used to describe wave processes in many areas of physics [1–4] such as quantum mechanics, nonlinear optics, theory of solitons, biophysics. There are many different methods and numerical algorithms to approximation solutions of nonlinear Schrodinger equations [5] from simple giving satisfactory results over a sufficiently large number of operations to complex. At present solutions of boundary value problems of mathematical physics [6–12] are widely used theory of atomic functions (AF). This solution is represent as a sum of shift-scaled AF families. AFs are infinitely differentiable functions and solutions of functional differential equations (FDE). They stay an intermediate position between the splines and trigonometric and algebraic polynomials. They are smoother than splines but less smooth than the polynomial. In this work we construct AFs shift-contraction of which constitutes the solution of Schrodinger equations. On the first stage the operator method [4] solution of FDE is considered. On the second step AF building of N-variables generated by the Schrodinger equation is carried out.

The problem of development of controlled radiation patterns (RP) with given form using antenna arrays (AA) with a limited number of elements is relevant [1]. One of the main tasks in this way is the synthesis of AA with the required form of RP (with remaining parameters for a given antenna) with a minimum number of elements. In this work the method of synthesis of AA makes it simple to determine the location of the elements, as well as the magnitude of the currents in the radiators to ensure receipt of the RP, close enough to be given. Methods of approximation of RP, based on the theory of atomic functions (AF) [1–5] are considered. It is shown that generalized on AF series interpolation better approximate the RP to given. The generalization of this approach (Kravchenko-Kotel'nikov theorem) on 2D-case in the approximation of signals with complex geometry of the emitters is presented.

Phase locked loop (PLL) systems used in the problems of generating high precision oscillation (frequency synthesis) [1,2,4–6], the coherent filtering of analog and digital signals [1], the optimal reception of digital signals [1], the organization of parallel computations in multiprocessor systems [3], the generation, and processing of ultra wideband (chaotic) signals [7], etc.

In the real world, there are a variety of applications of high resolution (HR) images in remote sensing, video frame freezing, medicine, robot artificial viewing, military information acquisition, etc. Because of the high cost and physical limitations of the acquisition hardware, the low-resolution (LR) images are used frequently. So, super-resolution (SR) restoration is an emerged solution permitting to form one or a set of HR images from a sequence of LR images. The proposed SR framework takes into account the spatial and spectral WT pixel information reconstructing different video and texture nature, presenting good performance in terms of objective (PSNR, MAE, NCD) criteria and visual subjective perception, employing the Wavelets based on atomic functions (WAF). Statistical simulations have demonstrated the effectiveness of the novel approach. The real time digital processing has been implemented on DSP of Texas Instruments TMS320DM642, demonstrating the effectiveness of the reconstruction of SR images in real time processing mode, and justifying this in the video sequences of different nature, pixel resolution and motion behavior.

The work focuses on the problem of determination of cutoff frequency of waveguides and resonators of a complex shape. The problem is sold by method of R‐functions. This approach has a lot of advantages, it possesses geometric flexibility, broad capabilities of numerical realization as for the production of the variation problem and for the selection of basis functions system as well. As basis functions the polynomials (trigonometrical, power, Tchebyshev of I and II types, Legendre, Gegenbauer) or local functions (atomic functions, splines) are used. The contrastive analysis of approximate boundary value problem solving is carried out in accordance to the basis functions system selected.

In this article, the approach based on the orthogonal Kravchenko wavelets $\{\widetilde{h_{a}(t)}\}$ is proposed. There is shown that obtained structures have some advantages in comparison with spectral wave analysis of ultra wideband (UWB) signals that are widely used in the remote sensing. This approach based on application of wavelets as spectral kernels is considered in the problems of digital UWB signal processing. In communication theory, the signals are represented in the form of linear combination of elementary functions. Application of spectral analysis of UWB signals in basis of digital functions in comparison with the spectral harmonious analysis gives certain advantages which are defined by the physical nature of the signal representation. Optimal processing in spectral area in comparison with the time possesses has advantages on application of numerical algorithms. The physical characteristics and analysis of numerical experiment confirm efficiency of new wavelets in the spectral estimation and digital UWB signal processing.

We present a new 3D Fuzzy Directional (3D-FD) algorithm for the denoising of video colour sequences corrupted by impulsive noise. The proposed approach consists of the estimations of movement levels, noise in the neighborhood video frames, permitting to preserve the edges, fine details and chromaticity characteristics in video sequences. Experimental results show that the noise in these sequences can be efficiently removed by the proposed 3D-FD filter, and that the method outperforms other state of the art filters of comparable complexity on video sequences. Finally, hardware requirements are evaluated permitting real time implementation on DSP EVM DM642.

Mammography is considered the most effective method for early detection of the breast cancer. However, it is difficult for radiologists to detect microcalcification (MC) clusters and camouflages masses. The mammograms (MG) images were decomposed into several subimages using Wavelet transform (WT) based on classical and novel class Wavelets using atomic function for reducing the volume of data in the classification stage. Various regions of interest (ROIs) in the MG images were selected where input data for multilayer artificial neural network (ANN) type classifier are formed applying the WT. We used different patterns to classify the normal, MC, spiculated and circumscribed masses ROIs. The detection performance has been evaluated on MG images from the Mammographic Image Analysis Society (MIAS) database. The proposed classification scheme was shown good performance in detecting the MC clusters and masses with acceptable classification.

Remote sensing systems, such as SAR usually apply FM signals to resolve nearly placed targets (objects) and improve SNR. Main drawbacks in the pulse compression of FM radar signal that it can add the range side-lobes in reflectivity measurements. Using weighting window processing in time domain it is possible to decrease significantly the side-lobe level (SLL) of output radar signal that permits to resolve small or low power targets those are masked by powerful ones. There are usually used classical windows such as Hamming, Hanning, Blackman-Harris, KaiserBessel, Dolph-Chebyshev, Gauss, etc. in window processing. Additionally to classical ones in here we also use a novel class of windows based on atomic functions (AF) theory. For comparison of simulation and experimental results we applied the standard parameters, such as coefficient of amplification, maximum level of side-lobe, width of main lobe, etc. In this paper we also proposed to implement the compression-windowing model on a hardware level employing Field Programmable Gate Array (FPGA) that offers some benefits like instantaneous implementation, dynamic reconfiguration, design, and field programmability. It has been investigated the pulse compression design on FPGA applying classical and novel window technique to reduce the SLL in absence and presence of noise. The paper presents simulated and experimental examples of detection of small or nearly placed targets in the imaging radar Paper also presents the experimental hardware results of windowing in FM radar demonstrating resolution of the several targets for classical rectangular, Hamming, Kaiser-Bessel, and some novel ones: Up(x)

On the base of the theory of R-functions, new weighting functions (windows) for synthesis of multi-frequency fractal antenna arrays are proposed and justified. Two-dimensional arrays are studied and the band-switching scheme is developed. Some types of ring antenna arrays with low sidelobe levels are considered. Analysis of the results of numerical experiments shows the benefit of such an approach, which is in the fact that the proper choice of the Kravchenko optimal weighting function (windows) allows one to synthesize a fractal antenna array with the required number of bands and sidelobe level.

This paper describes the use of fractal arrangements and atomic functions for the design of planar antenna arrays. Spiral initiators were used to play the chaos game, which generated our fractal array. Once these antenna arrays were produced, the resulting radiated field was calculated.

The generalized method of eigenoscillations (GME), as applied to solving a wide class of internal and external diffraction problems, is a further development of the method of eigenfrequencies. Some modifications of the GME are known, namely, the k-method (the eigenfrequency method), the e-method (eigenvalue in the equation), the w-method (eigenvalues in the impedance boundary condition), the p-method (eigenvalues in the conjugation condition), and the s-method (eigenvalues in the infinity condition). From the point of view of numerical analysis, the GME is based on a solution of some auxiliary eigenvalue problem in a complex-shaped domain. Here, the unknown solution is expanded in a series with respect to a proper system of basis functions, whose undetermined coefficients are found by one of variational or projective methods (Ritz, Bubnov-Galerkin, the least squares, etc.). The basis functions should satisfy definite requirements, in particularly, boundary conditions of the original problem (the latter requirement is obligatory for the Dirichlet boundary conditions). The choice of such functions is a complicated problem in the case of an arbitrarily shaped domain and it can be efficiently solved with the help of the theory of R-functions. In the report, the procedure of the combined use of the GME and the R-function method (RFM) is described and results of numerical experiments are presented.