Noncausal Models Research Articles

A versatile algorithm for two-dimensional symmetric noncausal modeling

In this work a novel algorithm is presented for the efficient two-dimensional (2-D) symmetric noncausal finite impulse response (FIR) filtering and autoregressive (AR) modeling. Symmetric filter masks of general boundaries are allowed. The proposed algorithm offers the greatest maneuverability in the 2-D index space in a computationally efficient way. This flexibility can be taken advantage of if the shape of the 2-D mask is not a priori known and has to be dynamically configured.

Noncausal all-pole modeling of voiced speech

This paper introduces noncausal all-pole models that are capable of efficiently capturing both the magnitude and phase information of voiced speech. It is shown that noncausal all-pole filter models are better able to match both magnitude and phase information and are particularly appropriate for voiced speech due to the nature of the glottal excitation. By modeling speech in the frequency domain, the standard difficulties that occur when using noncausal all-pole filters are avoided. Several algorithms for determining the model parameters based on frequency-domain information and the masking effects of the ear are described. Our work suggests that high-quality voiced speech can be produced using a 14th-order noncausal all-pole model.

Noncausal predictive image codec

The paper describes a lossy image codec that uses a noncausal (or bilateral) prediction model coupled with vector quantization. The noncausal prediction model is an alternative to the causal (or unilateral) model that is commonly used in differential pulse code modulation (DPCM) and other codecs with a predictive component. We show how to obtain a recursive implementation of the noncausal image model without compromising its optimality and how to apply this in coding in much the same way as a causal predictor. We report experimental compression results that demonstrate the superiority of using a noncausal model based predictor over using traditional causal predictors. The codec is shown to produce high-quality compressed images at low bit rates such as 0.375 b/pixel. This quality is contrasted with the degraded images that are produced at the same bit rates by codecs using causal predictors or standard discrete cosine transform/Joint Photographic Experts Group-based (DCT/JPEG-based) algorithms.

Estimation of stochastic model for random moving image by means of noncausal model

AbstractA new method for estimating a stochastic model of a random moving image is proposed. Assuming that the spatiotemporal spectrum of a moving image is separable in space and time, a stochastic image model is obtained by combining a noncausal model in two‐dimensional space and a causal AR model in time. Model estimation uses the method of minimizing whiteness for the noncausal spatial model and the method of “sequential orthogonalization” for the causal temporal model. the separable spectrum can be readily applied to a drifting moving model. Applying the spatial filter to a drifting moving image, the moving image is decomposed into a set of uncorrelated time series along the drifting time axis; this allows the AR models to be estimated separately. It is confirmed by computer simulation that the correct model can be estimated by this method. On the other hand, in the model estimation for a real, natural moving image, spectral separability is not always valid and, therefore, the output time series of the spatial filter is not rigorously uncorrelated. For such a real moving image, the method devised is an extended AR model which takes into account the spatially correlated data for AR model estimation. For estimating an extended AR model, the method of quasisequential orthogonalization is developed by modifying the sequential orthogonalization. It is successfully applied to processing a real moving image taken from a video recording.

Image modeling using inverse filtering criteria with application to textures

Statistical approaches to image modeling have largely relied upon random models that characterize the 2-D process in terms of its first- and second-order statistics, and therefore cannot completely capture phase properties of random fields that are non-Gaussian. This constrains the parameters of noncausal image models to be symmetric and, therefore, the underlying random field to be spatially reversible. Research indicates that this assumption may not be always valid for texture images. In this paper, higher- than second-order statistics are used to derive and implement two classes of inverse filtering criteria for parameter estimation of asymmetric noncausal autoregressive moving-average (ARMA) image models with known orders. Contrary to existing approaches, FIR inverse filters are employed and image models with zeros on the unit bicircle can be handled. One of the criteria defines the smallest set of cumulant lags necessary for identifiability of these models to date, Consistency of these estimators is established, and their performance is evaluated with Monte Carlo simulations as well as texture classification and synthesis experiments.

Extended Pole Placement Method With Noncausal Reference Model for Digital Servo-Control

The extended pole placement (EPP) method extends the well-known pole placement method to digital servo-control with known control inputs. A servo-control system designed using the EPP method achieves an accurate command response through the use of a noncausal reference model and an accurate approximation of nonminimum-phase zeros. The phase error of the command response between the desired output and the actual output may be made arbitrarily small, while its gain extends uniformly to a selectable bandwidth. A selectable feedforward bandwidth is important in controlling flexible structures where unmodeled resonances might be excited by rough feedforward inputs. Experimental results on a machine tool slide show that a controller designed by the EPP method yields better servo performance for cornering and contour tracking tasks than an E-filter-enhanced zero phase error tracking controller.

Simple versus complex models: evaluation, accuracy, and combining.

"This paper argues that it is premature to decide whether simple forecasting models in demography are more (or less) accurate than complex models and whether causal models are more (or less) accurate than noncausal models. It is also too early to say under what conditions one type of model can outperform another. The paper also questions the wisdom of searching for a single best model or approach. It suggests that combining forecasts may improve accuracy." (SUMMARY IN FRE)

Linear Hysteretic Damping and the Hilbert Transform

Experimental observations have shown that the dissipation per cycle in many materials does not depend on the deformation frequency over a wide frequency range. A linear model used frequently to represent this type of mechanical behavior is the concept of linear hysteretic damping. Also referred to as structural damping and complex stiffness in the literature, this noncausal model is characterized by storage and loss moduli independent of frequency. In the present paper, a consistent time-domain representation for linear hysteretic damping is presented using the Hilbert transform. This time-domain representation is a mathematically correct way to replace the complex-stiffness parameters or frequency-dependent damping coefficients commonly used in differential equations that model the dynamics of structures with linear hysteretic damping. A technique is proposed for the computation of the response of structures containing linear hysteretic elements in the time domain; the convergence of the iterative technique is analyzed; and simple numerical examples are developed to illustrate the application of the method.

Texture classification using noncausal hidden Markov models

This paper addresses the problem of using noncausal hidden Markov models (HMMs) for texture classification. In noncausal models, the state of each pixel may be dependent on its neighbors in all directions. New algorithms are given to learn the parameters of a noncausal HMM of a texture and to classify it into one of several learned categories. Texture classification results using these algorithms are provided.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Noncausal nonminimum phase ARMA modeling of non-Gaussian processes

A method is presented for the estimation of the parameters of a noncausal nonminimum phase ARMA model for non-Gaussian random processes. Using certain higher order cepstra slices, the Fourier phases of two intermediate sequences (h/sub min/(n) and h/sub max/(n)) can be computed, where h/sub min/(n) is composed of the minimum phase parts of the AR and MA models, and h/sub max/(n) of the corresponding maximum phase parts. Under the condition that there are no zero-pole cancellations in the ARMA model, these two sequences can be estimated from their phases only, and lead to the reconstruction of the AR and MA parameters, within a scalar and a time shift. The AR and MA orders do not have to be estimated separately, but they are by product of the parameter estimation procedure. Through simulations it is shown that, unlike existing methods, the estimation procedure is fairly robust if a small order mismatch occurs. Since the robustness of the method in the presence of additive noise depends on the accuracy of the estimated phases of h/sub min/(n) and h/sub max/(n), the phase errors due to finite length data are studied and their statistics are derived. >

Recursive-in-order least-squares parameter estimation algorithm for 2-D noncausal Gaussian Markov random field model

We present in this paper a recursive-in-order least-squares (LS) algorithm to compute efficiently the parameters of a 2-D Gaussian Markov random field (GMRF) model. The algorithm is based on the fact that the least-squares estimation of the parameters of a 2-D noncausal GMRF model is consistent and the coefficient matrix in the normal equation has near-to-block-Toeplitz structure. Hence, it has a Levinson-like form for the updating of model parameters by introducing auxiliary variables. Moreover, this paper proposes the concept ofrecursive path for 2-D recursive-in-order algorithms, and points out that there exists a tradeoff between fast computation of the parameters and accurate choice of model support; a compromise recursive path is then suggested where the orders change alternately in two directions. The computational complexity of the developed algorithm is analyzed, and the results show that the algorithm is more efficient when either the image size or the model support is larger. It is found that the total number of multiplications (mps) involved in the new algorithm is only about 14% of that in the conventional LS method when the image size is 512 × 512 and the neighbor set of the model is a 17 × 17 window. Computer simulation results using the recursive-in-order algorithm developed in this paper and the conventional LS method are given to verify the correctness of the new algorithm.

Consistency of modified LS estimation method for identifying 2-D noncausal SAR model parameters

Least squares (LS) and maximum likelihood (ML) are the two main methods for parameter estimation of two-dimensional (2D) noncausal simultaneous autoregressive (SAR) models. ML is asymptotically consistent and unbiased but computationally unattractive. On the other hand, conventional LS is computationally efficient but does not produce accurate parameter estimates for noncausal models. Recently, Zhao-Yu (1993) proposed a modified LS estimation method and was shown to be unbiased. In this paper we prove that, under certain assumptions, the method introduced by Zhao-Yu is also consistent. >

Lossless compression of multispectral image data

While spatial correlations are adequately exploited by standard lossless image compression techniques, little success has been attained in exploiting spectral correlations when dealing with multispectral image data. The authors present some new lossless image compression techniques that capture spectral correlations as well as spatial correlation in a simple and elegant manner. The schemes are based on the notion of a prediction tree, which defines a noncausal prediction model for an image. The authors present a backward adaptive technique and a forward adaptive technique. They then give a computationally efficient way of approximating the backward adaptive technique. The approximation gives good results and is extremely easy to compute. Simulation results show that for high spectral resolution images, significant savings can be made by using spectral correlations in addition to spatial correlations. Furthermore, the increase in complexity incurred in order to make these gains is minimal. >

Synthesis of “narrow-band” random moving image via computer

This paper presents a simple generating algorithm for a moving random image by generalizing the algorithm for a stationary random image in a previous work. Although the algorithm is applicable to a more general case, special emphasis is placed on the “narrow-band” random image with two or more spectral peaks which models a rough sea surface with strong periodic correlation. Such a “narrow-band” image can be given as the real or imaginary part of a complex-valued random image described by a noncausal model. A moving random image which is a spatio-temporal random field and a moving image drifting in a certain direction can be generated by a modified simple algorithm; and several examples are given. Generation of such a stationary or moving random image with given statistical properties is useful in the study of various image processing.

Noncausal Gauss Markov random fields: parameter structure and estimation

The parameter structure of noncausal homogeneous Gauss Markov random fields (GMRF) defined on finite lattices is studied. For first-order (nearest neighbor) and a special class of second-order fields, a complete characterization of the parameter space and a fast implementation of the maximum likelihood estimator of the field parameters are provided. For general higher order fields, tight bounds for the parameter space are presented and an efficient procedure for ML estimation is described. Experimental results illustrate the application of the approach presented and the viability of the present method in fitting noncausal models to 2-D data. >

Symmetric noncausal spatial models for 2-D signals with applications in stochastic texture modeling

In many one-dimensional (1-D) and two-dimensional (2-D) digital signal processing applications, auto-regressive (AR) models are very useful and powerful tools. Most of the development work done so far in 2-D AR modeling was limited to causal models. Recently, noncausal models have generated a great deal of interest because these models are a more natural choice for many applications in image processing. In this paper, we generalize the 1-D problems of noncausal linear-phase signal mdoeling and system modeling to their 2-D counterparts. The purpose of this paper is twofold. First, for homogeneous random fields, we introduce and investigate the 2-D symmetric (zero-phase) noncausal AR signal and system modeling problems. We then develop two computationally efficient algorithms for the determination of model parameters. Finally, we investigate an application in stochastic texture modeling and provide experimental results.

Blind equalisation of multilevel PAM data for nonminimum phase channels via second- and fourth-order cumulants

This paper introduces a new blind equalisation algorithm for the pulse amplitude modulation (PAM) data transmitted through nonminimum phase (NMP) channels. The algorithm itself is based on a noncausal AR model of communication channels and the second- and fourth-order cumulants of the received data series, where only the diagonal slices of cumulants are used. The AR parameters are adjusted at each sample by using a successive over-relaxation (SOR) scheme, a variety of the ordinary LMS scheme, but with a faster convergence rate and a greater robustness to the selection of the ‘step-size’ in iterations. Computer simulations are implemented for both linear time-invariant (LTI) and linear time-variant (LTV) NMP channels, and the results show that the algorithm proposed in this paper has a fast convergence rate and a potential capability to track the LTV NMP channels.

Analysis of identified 2-D noncausal models

There are two approaches to the identification of noncausal autoregressive systems in two dimensions differing in the assumed noise model. For both approaches, the maximum likelihood estimator formulated in the frequency domain is presented. The Fisher information matrix is evaluated and found to be the sum of a block-Toeplitz and a block-Hankel matrix. The variance of the parameters, however, cannot be used for comparison of the two approaches, so the variance in the frequency domain is evaluated, assuming that the true system in each case can be described by a model of that type, possibly high-order. In particular, the variance of the spectrum estimate is derived. If the number of parameters tends to infinity, it is shown that the two approaches give the same spectrum estimate variance. The question of which set of true spectra can be described by the respective approaches is discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

An unbiased and computationally efficient LS estimation method for identifying parameters of 2D noncausal SAR models

An unbiased and computationally efficient modified least squares (LS) estimation method for identifying parameters of two-dimensional noncausal simultaneous autoregressive models is presented. Some intuitive and mathematical proof of the unbiasedness of the method are given, and a recursive in-order fast algorithm to implement it is introduced. Computer simulation results are given to sustain the theoretical analysis. Both the theoretical analysis and the computer simulation show that the method possesses much higher estimation accuracy and lower computational complexity than the conventional LS estimation method. Compared to the approximate maximum-likelihood method of Kashyap and Chellappa (1983), the scheme is much faster, has the same estimation accuracy, and is parallelizable. >

Spatially recursive algorithms for adaptive estimation of two-dimensional non-causal and non-stationary simultaneous autoregressive model parameters

Using the shift-invariance properties (Zhao and He 1988), two fast spatially recursive algorithms for adaptive estimation of two-dimensional (2D) non-causal and non-stationary simultaneous autoregressive (SAR)model parameters are presented. One is based on the conventional least-squares (LS) criterion, and the other producing unbiased estimates of the parameters is based on a modified LS criterion. The multiplicative operation numbers of the algorithms are 15m3/2 + 16m and 16m3/2 + 21m PR (per recursion) respectively, where m is the number of the estimated parameters. This makes a considerable reduction in the computational burden in contrast with the algorithms now in existence. The tracking performances of the algorithms are evaluated by computer simulations which show that both the algorithms track the variations of the parameters, but the unbiased one possesses much high estimation accuracy as expected.