Nonsymmetric Half-plane Research Articles

On linear prediction for stationary random fields with nonsymmetrical half-plane past

An explicit autoregressive series representation for the best multi-step ahead linear predictor of stationary random fields with nonsymmetrical half-plane past (NSHP) is established. Necessary and sufficient condition for the mean square convergence of these series is given. Moreover, step recursive relations between the prediction coefficients for the infinite past predictor are provided, these relations are used to calculate explicitly the multi-step prediction coefficients. Some specific examples to validate the applicability of our relations are presented.

Lattice Structure Realization for The Design of 2-D Digital Allpass Filters With General Causality

This paper presents a lattice structure for the realization of two-dimensional (2-D) recursive digital allpass filters (DAFs) with general causality. We employ four basic lattice sections to realize 2-D recursive DAFs with wedge-shaped coefficient support region like a nonsymmetric half-plane (NSHP) support region. The theory and transfer functions of the realized 2-D lattice DAFs are derived. Some variations of the 2-D lattice structure are also presented. We use the Roesser state space model to verify the minimal realization of the proposed 2-D recursive lattice DAF. We present a least-squares design technique and a minimax design technique to solve the nonlinear optimization problems of the proposed 2-D lattice DAF structure. The novelty of the presented lattice structure is that it not only inherits the desirable attributes of 1-D Gray-Markel lattice allpass structure but also possesses the advantage of better performance over the existing 2-D lattice allpass structures. Finally, several design examples are provided for conducting illustration and comparison.

The Phase Characteristics for the Stability of 2-D Nonsymmetric Half-Plane Digital Allpass Filters

The phase-response property for the bounded-input bounded-output (BIBO) stability of a two-dimensional (2-D) causal recursive digital allpass filter (DAF) with nonsymmetric half-plane (NSHP) support region is presented. Both cases of filters with real coefficients and complex coefficients are explored. Moreover, the effect of the numerator polynomial of a 2-D NSHP DAF on stability is also considered. The presented phase-response property has several applications. A remarkable application is that it can be utilized to enforce stability for a 2-D NSHP DAF design by choosing an appropriate phase specification. The eigenfilter design of 2-D NSHP DAFs for this application is also presented. Design examples are given to show these applications and the stability of a 2-D NSHP DAF is also inspected by the 2-D spectral factorization-based algorithm and a zeros-examining method presented by Ekstrom and Woods for confirmation.

A novel algorithm for image denoising based on unscented Kalman filtering

This paper presents a noise removal algorithm based on unscented Kalman filtering in order to improve image quality. We first analysed the characteristics of the background noise, and then discussed the unscented Kalman filter UKF. After that, one-dimensional unscented Kalman filtering, and two-dimensional non-symmetric half plane NSHP support image model based on two-dimensional unscented Kalman filtering are introduced. Experimental results show that as an adaptive method, the algorithm reduces the noise while retaining the image details, and two-dimensional NSHP model performs better than one-dimensional UKF algorithm. Therefore, UKF together with its two-dimensional NSHP implementation have efficacy for noise removal of images.

Two-Dimensional Hidden Markov Model for Classification of Continuous-Valued Noisy Vector Fields

In this paper we present a statistical model with a nonsymmetric half-plane (NSHP) region of support for two-dimensional continuous-valued vector fields. It has the simplicity, efficiency, and ease of use of the well-known hidden Markov model (HMM) and associated Baum-Welch algorithms for time-series and other one-dimensional problems. At the same time it is able to learn textures on a two-dimensional field. We describe a fast approximate forward procedure for computation of the joint probability density function (pdf) of the vector field as well as an approximate Baum-Welch algorithm for parameter reestimation. Radar and sonar applications include classification of two-dimensional fields such as range versus azimuth or range versus aspect angle data wherein each data point in the field consists of a multi-dimensional feature vector. We test the method using synthetic textures.

Two-Channel Parallelogram QMF Banks Using 2-D NSHP Digital All-Pass Filters

This paper presents a novel structure for the analysis and synthesis filters of two-channel parallelogram quadrature mirror filter (PQMF) banks. Two-dimensional recursive digital all-pass filters (DAFs) with nonsymmetric half-plane (NSHP) support regions are used as the basic sections for building this structure. With an appropriate combination of the 2-D NSHP DAFs, the resultant analysis/synthesis filters possess a 2-D doubly complementary symmetry that facilitates the design and implementation of two-channel PQMF banks. Moreover, the proposed PQMF bank provides approximately linear-phase response without magnitude distortion. The design problem of the proposed PQMF bank is appropriately formulated to result in a minimization of the linear-phase error associated with the 2-D NSHP DAFs in the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$p$</tex></formula> th-norm <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$(L_{p})$</tex></formula> sense. Efficient design techniques are developed to solve the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$L_{p}$</tex> </formula> optimization problems. Simulation results for illustration and comparison are also provided.

Testing the Stability of 2-D Recursive QP, NSHP and General Digital Filters of Second Order

Several methods for testing stability of first quadrant quarter-plane two dimensional (2-D) recursive digital filters have been suggested in 1970's and 80's. Though Jury's row and column algorithms, row and column concatenation stability tests have been considered as highly efficient mapping methods. They still fall short of accuracy as they need infinite number of steps to conclude about the exact stability of the filters and also the computational time required is enormous. In this paper, we present procedurally very simple algebraic method requiring only two steps when applied to the second order 2-D quarter - plane filter. We extend the same method to the second order Non-Symmetric Half-plane (NSHP) filters. Enough examples are given for both these types of filters as well as some lower order general recursive 2-D digital filters. We applied our method to barely stable or barely unstable filter examples available in the literature and got the same decisions thus showing that our method is accurate enough.

Color spectral analysis for spatial structure characterization of textures in IHLS color space

In this article, a linear prediction model based approach for color texture characterization and classification in the improved hue luminance and saturation color space is presented. Pure chrominance structure information is used in addition to the normally used luminance structure information for color texture classification. Hue and saturation channels of a color image in IHLS color space are combined using a complex exponential to give a single channel which holds all the chrominance information of the image. Two dimensional complex multichannel versions of the non-symmetric half plane autoregressive model, the quarter plane autoregressive model and the Gauss Markov random field model are used to perform parametric power spectrum estimation of both luminance and the “combined chrominance” channels of the image. The accuracy and precision of these spectral estimates are proven quantitatively by performing tests on a large number of images. Spectral distance measures are calculated for the spectral information of luminance and chrominance channels individually as well as combined through a combination coefficient. Using these distance measures, color texture classification is done with k-nearest neighbor algorithm. Experimental results verify that the IHLS color space exhibits better performance than the RGB color space indicating the significance of using IHLS for such analysis. They also show that color texture characterization and percentage classification obtained by combined luminance and chrominance structure information is better than the color texture classification done using only the luminance structure information.

Maximum Likelihood Blind Image Separation Using Nonsymmetrical Half-Plane Markov Random Fields

This paper presents a maximum likelihood approach for blindly separating linear instantaneous mixtures of images. The spatial autocorrelation within each image is described using nonsymmetrical half-plane (NSHP) Markov random fields in order to simplify the joint probability density functions of the source images. A first implementation assuming stationary sources is presented. It is then extended to a more realistic nonstationary image model: two approaches, respectively based on blocking and kernel smoothing, are proposed to cope with the nonstationarity of the images. The estimation of the mixing matrix is performed using an iterative equivariant version of the Newton-Raphson algorithm. Moreover, score functions, required for the computation of the updating rule, are approximated at each iteration by parametric polynomial estimators. Results achieved with artificial mixtures of both artificial and real-world images, including an astrophysical application, clearly prove the high performance of our methods, as compared to classical algorithms.

Two-dimensional non-symmetric half-plane recursive doubly complementary digital filters

This paper deals with the problem of two-dimensional (2-D) recursive doubly complementary (DC) digital filter design. We present a structure of 2-D recursive DC filters by using 2-D non-symmetric half-plane (NSHP) digital all-pass filters (DAFs). The novelty of using 2-D NSHP recursive DAFs to construct a 2-D recursive DC filter is that the resulting 2-D NSHP recursive DC filter shows better performance than conventional 2-D recursive DC filter based on 2-D quarter-plane (QP) DAFs. Moreover, the proposed structure possesses a favorable 2-D doubly complementary half-band (DC-HB) property that allows about half of the 2-D NSHP recursive DAF's coefficients to be zero. This leads to considerable savings in computational burden. To deal with the stability of the 2-D NSHP recursive DC filter during the design process, we can either impose some necessary constraints on the phases of the 2-D NSHP recursive DAFs or utilize a penalty function involving a stability error computed by using a spectral factorization-based algorithm to ensure the stability of a designed 2-D NSHP recursive DC filter pair. Design of a diamond-shape decimation/interpolation filter is presented for illustration and comparison.

Stabilization of 2D NSHP Recursive Digital Filters with Guaranteed Stability Using PLSI Polynomials

Two-dimensional digital filters have gained wide acceptance in recent years. For recursive filters, nonsymmetric half-plane versions (also known as semicausal) are more general than quarter-plane versions (also known as causal) in approximating arbitrary magnitude characteristics. The major problem in designing two-dimensional recursive filters is to guarantee their stability with the expected magnitude response. In general, it is very difficult to take stability constraints into account during the stage of approximation. This is the reason why it is useful to develop techniques, by which stability problem can be separated from the approximation problem. In this way, at the end of approximation process, if the filter becomes unstable, there is a need for stabilization procedures that produce a stable filter with similar magnitude response as that of the unstable filter. This paper, demonstrates a stabilization procedure for a two-dimensional nonsymmetric half-plane recursive filters based on planar least squares inverse (PLSI) polynomials. The paper's findings prove that, a new way of form-preserving transformation can be used to obtain stable PLSI polynomials. Therefore obtaining PLSI polynomial is computationally less involved with the proposed form-preserving transformation as compared to existing methods, and the stability of the resulting filters is guaranteed.

Stabilization of 2D nonsymmetric half-plane digital filters using the concept of reciprocity

In this paper, the concept of reciprocity for two-dimensional (2D) quarter plane (QP) filters are extended to stabilize 2D nonsymmetric half-plane (NSHP) filters. In general the stabilization of NSHP filters is done by transforming NSHP arrays into QP arrays using mapping theorems. The concept of reciprocity is applied directly without any mapping to stabilize a class of 2D NSHP filters which are otherwise unstable when all singularities lie within the unit bicircle. It is shown that the magnitude spectrum of the stabilized 2D NSHP filter is identical to the magnitude spectrum of the original unstable filter. The proposed stabilization method is simple as compared to two other popular stabilization techniques in the literature, the double planar least squares inverse (DPLSI) and 2D discrete Hilbert transform (2D DHT).

Computation of the exact Cramer–Rao lower bound for the parameters of a nonsymmetric half-plane 2-D ARMA model

The Cramer-Rao lower bound (CRLB) that gives the minimal achievable variance/standard deviation for any unbiased estimator offers a useful tool for an assessment of the consistency of parameter estimation techniques. In this paper, a closed-form expression for the computation of the exact CRLB on unbiased estimates of the parameters of a two-dimensional (2-D) autoregressive moving average (ARMA) model with a nonsymmetric half-plane (NSHP) region of support is developed. The proposed formulation is mainly based on a matrix representation of 2-D real-valued discrete and homogeneous random field characterized by the NSHP ARMA model. Assuming that the random field is Gaussian, the covariance matrix of the NSHP ARMA random field is first expressed in terms of the model parameters. Then, using this matrix structure, a closed-form expression of the exact Fisher information matrix required for the CRLB computation of the NSHP ARMA model parameters is developed. Finally, the main formulas derived for the NSHP ARMA model are rearranged for its autoregressive and moving average counterparts, separately. Numerical simulations are included to demonstrate the behavior of the derived CRLB formulas.

Design Of 2-D Recursive Digital Filters Using Nonsymmetric Half-Plane Allpass Filters

A novel structure using recursive nonsymmetric half-plane (NSHP) digital allpass filters (DAFs) is presented for designing 2-D recursive digital filters. First, several important properties of 2-D recursive DAFs with NSHP support regions for filter coefficients are investigated. The stability of the 2-D recursive NSHP DAFs is guaranteed by using a spectral factorization-based algorithm. Then, the important characteristics regarding the proposed novel structure are discussed. The design problem of 2-D recursive digital filters using the novel structure is considered. We formulate the problem by forming an objective function consisting of the weighted sum of magnitude, group delay, and stability-related errors. A design technique using a trust-region Newton-conjugate gradient method in conjunction with the analytic derivatives of the objective function is presented to efficiently solve the resulting optimization problem. The novelty of the presented 2-D structure is that it possesses the advantage of better performance in designing a variety of 2-D recursive digital filters over existing 2-D filter structures. Finally, several design examples are provided for conducting illustration and comparison.

On the Parameter Estimation of 2-D Moving Average Random Fields

The problem of estimating the parameters of 2-D homogeneous moving average (MA) random fields only from output measurements is addressed. A novel computationally efficient algorithm for the estimation of the parameters of a minimum-phase 2-D MA model with a nonsymmetric half-plane (NSHP) region of support (ROS) is proposed. Using the 2-D spectral factorization, relationship between the NSHP MA model parameters and the cepstral coefficients of a 2-D MA random field is considered. Based on this relation, recursive equations are derived so as to estimate the NSHP MA model parameters. It is noteworthy that the proposed algorithm is practical, i.e., it does not require computationally complex processes namely fitting to a high-order autoregressive model, any initial estimates, nor matrix inversion. Performance analysis of the derived algorithm together with an existing method is given for comparison purposes. Index

Modeling 2-D AR Processes With Various Regions of Support

We show that there exists a causal 2-D linear process in the nonsymmetric half-plane having the same autocorrelations as a noncausal 2-D linear process in the whole-plane; this property is called the autocorrelation equivalence relation, and can be used for practical fitting and modeling of 2-D processes. Some causal 2-D autoregressive (AR) models with various regions of support are considered such as half-cross, half-diamond, quarter-plane square, half-square, half-hexagon, half-octagon, and half-circle. Considerations of parsimony in 2-D model fitting are then focused not only on the number of parameters in our model, but also most importantly on the optimal shape of the region of support. Their 2-D Yule-Walker equations are derived, and a computationally efficient order-recursive algorithm is proposed to solve them. The autocorrelation equivalence relation and the order-recursive algorithm are utilized to specify a noncausal 2-D AR process as well as its spectrum from a given realization of a random field

Computation of the exact cramer-rao lower bound for 2-D ARMA parameter Estimation-I: the quarter-plane case

A closed-form expression for computing the exact Cramer-Rao lower bound (CRLB) on unbiased estimates of the parameters of a two-dimensional (2-D) autoregressive moving average (ARMA) model is developed. The formulation is based on a matrix representation of 2-D homogeneous Gaussian random process that is generated uniformly from the related 2-D ARMA model. The formulas derived for the exact Fisher information matrix (FIM) are an explicit function of the 2-D ARMA parameters and are valid for real-valued homogeneous quarter-plane (QP) 2-D ARMA random fields, where data are propagated using only the past values. It is noteworthy that our approach is practical especially for quantifying the accuracy of 2-D ARMA parameter estimates realized with short data records. Computer simulations display the behavior of the derived CRLB expression for some QP causal 2-D ARMA processes, as a function of the number of data points. The extension of this algorithm for the nonsymmetric half-plane (NSHP) case will be presented in a subsequent paper

A Recursive Algorithm to Restore Images Based on Robust Estimation of NSHP Autoregressive Models

The objective of this article is to present a new image restoration algorithm. First, each pixel in the image is classified into k categories. Then we assume that the gray levels in each category follow a nonsymmetric half-plane (NSHP) autoregressive model. Robust estimation of theparameters of the model is considered to attenuate the effect of the image contamination on the parameters. In each iteration we will construct a new image using a robustified version of the residuals. The introduction of the classification techniques as a first step of the algorithm reducesconsiderably the number of parameters to estimate. Hence, the computational time is also reduced because the robust estimations of the parameters are solutions of nonlinear system of equations. Some applications are presented to real synthetic aperture radar (SAR) images to illustrate how ouralgorithm restores an image in practice.

The PLSI Method of Stabilizing Two-Dimensional Nonsymmetric Half-Plane Recursive Digital Filters

Two-dimensional (2D) recursive digital filters find applications in image processing as in medical X-ray processing. Nonsymmetric half-plane (NSHP) filters have definitely positive magnitude characteristics as opposed to quarter-plane (QP) filters. In this paper, we provide methods for stabilizing the given 2D NSHP polynomial by the planar least squares inverse (PLSI) method. We have proved in this paper that if the given 2D unstable NSHP polynomial and its PLSI are of the same degree, the PLSI polynomial is always stable, irrespective of whether the coefficients of the given polynomial have relationship among its coefficients or not. Examples are given for 2D first-order and second-order cases to prove our results. The generalization is done for theth order polynomial.

Model identification of a noncausal 2-D ar process using a causal 2-D AR model on the nonsymmetric half-plane

If a noncausal two-dimensional (2-D) autoregressive (AR) process is bi-causal, there exists a causal 2-D AR process on the nonsymmetric half-plane having the same autocorrelations as the noncausal 2-D AR process. A formula is presented to relate the AR coefficients of the noncausal 2-D AR process with those of the causal 2-D AR process on the nonsymmetric half plane. The 2-D Yule-Walker equations are derived for causal 2-D AR models on the nonsymmetric half plane. A computationally efficient order-recursive algorithm is proposed to solve the 2-D Yule-Walker equations. Using the autocorrelation equivalence relation and the order-recursive algorithm, we can easily identify a noncausal 2-D AR process from its autocorrelations.