Abstract: The constant modulus algorithm (CMA) is recognized as the most widely used algorithm in blind channel equalization practice. However, the CMA cost function exhibits local minima, which often leads to ill-convergence. This paper proposes a concurrent equalizer, in which a least mean square (LMS) equalizer operates cooperatively with a CMA equalizer, controlled through a non-linear link that depends on the system a priory state. Simulation results using M-QAM signalling have shown that the concurrent equalizer presents a much lower sensitivity to local minima than the CMA approach.

Abstract: Finding a good algorithm for the deconvolution of pressure and flow rate data is one of the long-standing problems in well test analysis. In this paper we give a survey of methods which have been suggested in the past 40 years, and develop a new formulation in terms of the logarithmof the response function. The main advantage of this nonlinear encoding over prior methods is that it does not require explicit sign constraints. Moreover we introduce a new error model which accounts for errors in both pressure and rate data; here the rates can be cumulative or continously measured. In this formulation, deconvolution is equivalent to a separable nonlinear Total Least Squares problem for which standard algorithms exist. Preliminary numerical results with both simulated and field data suggest that the method is capable of producing smooth, interpretable reservoir response estimates from data contaminated with errors of up to 10% in rates, provided a careful choice of weight and regularization parameters is made. Introduction The purpose of well test analysis is to determine geological properties of hydrocarbon or water reservoirs from measurements of wellbore pressure and production rate over time. This involves three steps: 1. Estimating the reservoir responsefrom the data, 2. matching the shape of the response function against a library of type curves to identify a suitable reservoir model, and 3. fitting the parameters of this model to the data. Here the second step is mainly qualitative, while the first and third steps are entirely quantitative. It is solely the first step with which we shall be concerned in this paper. As a function of time, the pressure drop is the convolution product of rate and reservoir response; this is the content of Duhamel’s principle. Thus, estimating the reservoir response essentially amounts to inverting a convolution integral, and is therefore an instance of a widely encountered mathematical problem called deconvolution. In the simple case of a single flow period with constant rate, the response function can be obtained (up to scale) as the derivative of the pressure drop with respect to the logarithm of time. The standard method of obtaining response estimates is therefore to perform numerical differentiation on the pressure data. 4 This method has a number of serious limitations: Numerical differentiation has the effect of amplifying measurement errors with which the data are always contaminated due to limited gauge accuracy as well as external sources of noise. The effect of gauge resolution and the radius of investigation of a well test have been studied in a number of publications; see Daungkaew, Hollaender and Gringarten7 and references there. The method extends to multirate tests, where the derivative is formed with respect to the superposition time.4 However, this extension is justified not so much by rigorous error analysis as by experimental evidence that it faithfully preserves the “visual shape” of model features for a set of standard type curves. Moreover, even this extension cannot give response estimates beyond the longest flow period with constant rate, which reduces the radius of investigation even further. In the narrower sense, the term deconvolution refers to the variable rateproblem only; this includes the case with multiple constant flow periods. The variable rate problem has received scant, but recurring attention over the last 40 years by researchers both in Petroleum and Water Resources Engineering; see2, 6, 12 and references there. In the next section we shall briefly sketch the main developments and summarize their merits and shortcomings. As for the latter ones, the two methods 2 T. VON SCHROETER, F. HOLLAENDER, A. C. GRINGARTEN SPE 71574 which were tested with simulated data seem to share a strong sensitivity to data uncertainty once the error level reaches the order of 5% in the pressure or 1% in the rates, which in practice appears to be a fairly common level of uncertainty, at least as far as flow rates are concerned. Results obtained from such noisy signals are often affected by oscillations which can render them uninterpretable in the worst case. In the following two sections we develop a new method which contains two novel ideas. The first of these is an error measure for deconvolution that accounts for uncertainties not only in the pressure, but also in the rate data, which are usually much less accurately measured. The resulting formulation is what is known as a Total Least Squares (TLS) problem in the Numerical Analysis literature and as an Errors-In-Variables(EIV) problem in Statistics. TLS has become a standard approach in many disciplines, but its application to well test analysis seems to be new. The second idea concerns the way in which the solution space is chosen to reflect prior knowledge about the solution. This can be done implicitly by the way the solution space is parametrized, or explicitly in the form of constraints on the parameters. In the Petroleum Engineering literature, the explicit approach has a long history going back to the paper by Coats, Rapoport, McCord, and Drews6 who used sign constraints on the response and its first two derivatives. Later, Kuchuk, Carter and Ayestaran12 took up this approach in a least squares fashion. Another case of was considered by Baygün, Kuchuk and Arikan2 who used constraints on the autocorrelation sequence and the energy of the solution vector in order to keep the solution smooth and reduce oscillations. Our approach uses the implicit alternative instead. Its main idea is to encode the response function in a more natural way such that sign constraints are not necessary. This has the unwelcome consequence of rendering the problem nonlinear; however some of this complication is offset by the reduction in the number of constraints. In fact, our algorithm uses no constraints at all, but just a single regularizing function which is chosen as the “energy” of the solution, i.e. the sum of squares of its derivatives on the interpolation intervals. We explain our method in considerable detail and report preliminary numerical experiments with both simulated and field data. These experiments suggest that the method is capable of producing smooth, interpretable reservoir response estimates from data contaminated with fairly large errors. However, so far practical experience is still limited, and more comprehensive tests are needed. Further work is in progress. Deconvolution and well test analysis The foundation of well test analysis is Duhamel’s principle, which states that the pressure drop p at the wellbore is the convolution of the flow rate q and the reservoir impulse response g as functions of time: p p0 p = q g ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1) which is shorthand for

Abstract: The APEX method is an FFT-based direct blind deconvolution technique that can process complex high resolution imagery in seconds or minutes on current desktop platforms. The method is predicated on a restricted class of shift-invariant blurs that can be expressed as finite convolution products of two-dimensional radially symmetric Levy stable probability density functions. This class generalizes Gaussian and Lorentzian densities but excludes defocus and motion blurs. Not all images can be enhanced with the APEX method. However, it is shown that the method can be usefully applied to a wide variety of real blurred images, including astronomical, Landsat, and aerial images, MRI and PET brain scans, and scanning electron microscope images. APEX processing of these images enhances contrast and sharpens structural detail, leading to noticeable improvements in visual quality. The discussion includes a documented example of nonuniqueness, in which distinct point spread functions produce high-quality restorations ...

Abstract: In this paper, a novel algorithm for separating mixtures of multiple speech signals measured by multiple microphones in a room environment is proposed. The algorithm is a modification of an existing approach for density-based multichannel blind deconvolution using natural gradient adaptation. It employs linear predictors within the coefficient updates and produces separated speech signals whose autocorrelation properties can be arbitrarily specified. Stationary point analyses of the proposed method illustrate that, unlike multichannel blind deconvolution methods, the proposed algorithm maintains the spectral content of the original speech signals in the extracted outputs. Performance comparisons of the proposed method with existing techniques show its desirable properties in separating real-world speech mixtures.

Abstract: We discuss the blind deconvolution of multiple input/multiple output (MIMO) linear convolutional mixtures and propose a set of hierarchical criteria motivated by the maximum entropy principle. The proposed criteria are based on the constant-modulus (CM) criterion in order to guarantee that all minima achieve perfectly restoration of different sources. The approach is moreover robust to errors in channel order estimation. Practical implementation is addressed by a stochastic adaptive algorithm with a low computational cost. Complete convergence proofs, based on the characterization of all extrema, are provided. The efficiency of the proposed method is illustrated by numerical simulations.

Abstract: Three-dimensional ultrasound images are blurred by the ultrasound pulse through the convolution between the 3-D tissue signal and the 3-D pulse. The blurring reduces the spatial resolution of the 3-D ultrasound images and, consequently, their diagnostic value. This paper presents a method for 3-D blind homomorphic deconvolution of medical 3-D ultrasound images to improve their spatial resolution. The blind estimate of the 3-D pulse is necessary because the pulse changes in spatial extent and frequency composition as it passes through the tissues and because the pulse is not separable in its spatial dimensions. The method was tested on a 3-D image of a phantom with anechoic spheres of known size in a uniform diffuse scattering matrix. The spheres were clearly better defined and had volumes much closer to the true volume in the deconvolved image than in the original image.

Abstract: Ideally, traditional vibroseis processing produces a band-limited zero-phase Klauder wavelet through cross-correlation of the sweep with the recorded signal. An alternative wavelet processing method involves deconvolving the sweep from the recorded vibroseis trace. This deconvolution can be achieved through frequency-domain division. We compare and contrast the advantages and disadvantages of sweep deconvolution versus cross-correlation on synthetic and real data.

Abstract: We focus on image deconvolution and image reconstruction problems where a sought image is recovered from degraded observed data. The solution is defined to be the minimizer of an objective function combining a data-fidelity term and an edge-preserving, convex regularization term. Our objective is to speed up the calculation of the solution in a wide range of situations. To this end, we propose a method applying pertinent preconditioning to an adapted half-quadratic equivalent form of the objective function. The optimal solution is then found using an alternating minimization (AM) scheme. We focus specifically on Huber regularization. We exhibit the possibility of getting very fast calculations while preserving the edges in the solution. Preliminary numerical results are reported to illustrate the effectiveness of our method.

Abstract: Tugnait (1995) and Chi and Chen proposed multi-input multi-output inverse filter criteria (MIMO-IFC) using higher order statistics for blind deconvolution of MIMO linear time-invariant systems. This paper proposes three properties on the performance of the MIMO linear equalizer associated with MIMO-IFC for any signal-to-noise ratio, including (P1) perfect phase equalization property, (P2) a relation to MIMO minimum mean square error (MIMO-MMSE) equalizer, and (P3) a connection with the one obtained by MIMO super-exponential algorithm (MIMO-SEA) that usually converges fast but does not guarantee convergence for finite data. Based on (P2), a fast algorithm for computing the theoretically optimum MIMO equalizer is proposed. Moreover, based on (P3), a fast MIMO-IFC based algorithm with performance similar to that of the MIMO-SEA and with guaranteed convergence is proposed as well as its application to suppression of multiple access interference and intersymbol interference (ISI) for multiuser asynchronous DS/CDMA systems in multipath. Finally, some simulation results are presented to support the analytic results and the proposed algorithms.

Abstract: An algorithm is presented for the automated analysis of rotating probe multifrequency eddy current data obtained from nuclear power plant steam generator tubes (SGT). The algorithm consists of four steps, namely, a preprocessing stage for conditioning the data, a decision tree based feature extraction stage for identifying relevant features for analysis, a neural network based classification stage for identifying signals from various defect types and benign structures, and finally a blind deconvolution based characterization stage for accurately estimating the size and orientation of the detected defects. This algorithm is optimized to maximize the probability of detection (POD), while keeping the number of false alarms (PFA) at a minimum. Initial results presented in this paper look very promising and demonstrate the effectiveness of the proposed algorithm.

Abstract: This paper is devoted to blind deconvolution and blind separation problems. Blind deconvolution is the identification of a point spread function and an input signal from an observation of their convolution. Blind source separation is the recovery of a vector of input signals from a vector of observed signals, which are mixed by a linear (unknown) operator. We show that both problems are paradigms of nonlinear ill-posed problems. Consequently, regularization techniques have to be used for stable numerical reconstructions. In this paper we develop a rigorous convergence analysis for regularization techniques for the solution of blind deconvolution and blind separation problems. Convergence of regularized point spread functions and signals to a solution is established and a convergence rate result in dependence of the noise level is presented. Moreover, we prove convergence of the alternating minimization algorithm for the numerical solution of regularized blind deconvolution problems and present some numerical examples. Moreover, we show that many neural network approaches for blind inversion can be considered in the framework of regularization theory.

Abstract: A method for performing blind deconvolutions on degraded images and data has been developed. The technique uses a power law relation applied to the Fourier transform of the degraded data to extract a filter function. This filter function closely resembles the point-spread function of the system and can be used to restore and enhance higher-frequency content. The process is noniterative and requires only that the point-spread function be space invariant and the transfer function be real. The algorithm has been validated by direct comparisons by use of a pseudoinverse filter with known transfer functions.

Abstract: In this paper we address the problem of blind source extraction of a subset of "interesting" independent sources from a linear convolutive or instantaneous mixture The interesting sources are those which are independent and, in a certain sense, are sparse and far away from Gaussianity We show that in the low-noise limit and when none of the desired sources is Gaussian, the minimum entropy and cumulants based approaches can solve the problem These criteria, with roots in Blind Deconvolution and in Projection Pursuit, will be proposed here for the simultaneous blind extraction of a group of independent sources Then, we suggest simple algorithms which, working on the Stiefel manifold perform maximization of the proposed contrast functions

Abstract: SUMMARY This paper describes a series of innovations in the problem of deconvolving forward scattered P-to-S conversions. We introduce a theoretical foundation for a recently developed multichannel stacking technique and show that this process is equivalent to a spatial convolution of the incident wavefield with the discretely sampled set of station locations. We then show that deconvolution of the stacked data is a form of multichannel deconvolution with a spatially variable set of weights equal to those used in stacking. This result is independent of the particular deconvolution method that is used. A second innovation focuses on the design of deconvolution operators that correctly account for the loss of high frequency components of P-to-S conversions caused by differential attenuation of P and S waves. We describe two complimentary methods to implement this: (1) through the use of a regularization operator that penalizes high frequencies and increases with P-to-S lag time, or (2) through the use of a quelling operator. For the latter, we introduce the use of a t* operator that is applied to the deconvolution matrix operator. The t* operator progressively filters the vertical component seismogram with increasing P-to-S lag time and is based on an earth model of body wave attenuation. Both techniques produce progressively smoother solutions for increasing P-to-S lag times. The quelling approach has two advantages: (1) it is based on the physical principle that this solution is designed to address, and (2) it provides a unified inversion framework for the combination of stacking and deconvolution. This combination may be interpreted as a three-dimensional quelling (smoothing) operator that is applied to the full wavefield to stabilize the inversion. Application of this procedure to synthetic data shows that while the addition of a time dependent component to the deconvolution tends to decrease the frequency content of the solution, the amplitude of background ringing is reduced and the input model is reliably recovered. Further tests with data from the Lodore broad-band array in Colorado and Wyoming show significant improvement over conventional time domain methods. We image lateral variations in Moho continuity and reflectivity across the array, with significant improvement in resolution in the first 10 seconds of data.

Abstract: Following the approach described by A. V. Kryazhimskii and Yu. S. Osipov, we present a recursive algorithm which can be applied to solve the deconvolution problem of linear finite dimensional input output systems. The method gives an on line approximation of the unknown input, based on approximate samples of the output. Key features of this approach are the introduction of an associated singularly perturbed system and the use of a quasi canonical form due to Morse.

Abstract: A signal processing algorithm has been developed in which a filter function is extracted from degraded data through mathematical operations. The filter function can then be used to restore much of the degraded content of the data through use of any deconvolution algorithm. This process can be performed without prior knowledge of the detection system, a technique known as blind deconvolution. The extraction process, designated Self-deconvolving Data Reconstruction Algorithm (SeDDaRA), has been used successfully to restore digitized photographs, digitized acoustic waveforms, and other forms of data. The process is non-iterative, computationally efficient, and requires little user input. Implementation is straight-forward, allowing inclusion into all types of signal processing software and hardware. The novelty of the invention is the application of a power law and smoothing function to the degraded data in frequency space. Two methods for determining the value of the power law are discussed. The first method is by educated guess where the value is deemed a constant of frequency that ranges between zero and one. This approach requires no knowledge of the original data or the degradation and is quite effective. The second method compares the frequency spectrum of the degraded data to the spectrum of a signal with the desired frequency response. This approach produces a superior result, but requires additional processing.

Abstract: In this article deconvolution of ultrasonic pulse-echo data acquired from attenuative layered media is considered. The problem is divided in two subproblems: treating the sparse reflection sequence caused by the layered structure of the media and treating the frequency-dependent attenuation. The first subproblem is solved by means of joint maximum a posteriori estimation of the assumed zero mean, white, nonstationary reflection sequence and its corresponding sequence of unknown standard deviations. This approach leads to an algorithm that seeks minimum entropy solutions for the reflection sequence and therefore the algorithm serves as a novel link between the classical Wiener filter and methods for sparse or minimum entropy deconvolution. The second subproblem is solved by introducing a new signal processing-oriented, linear discrete-time model for frequency-dependent attenuation in isotropic and homogeneous media. The deconvolution algorithm is tested using simulated data and its performance for real normal incidence pulse-echo data from a composite material is also demonstrated. The results show that the algorithm, in combination with the attenuation model, yields estimates that reveal the internal structure of the composite and, thus, simplify the interpretation of the ultrasonic data.

Abstract: Independent Component Analysis is a new data analysis method, and its computation algorithms and applications have been widely studied recently. Most applications, however, are in the field of one-dimensional data analysis, such as sound data analysis, and few applications for two-dimensional data (e.g., image data) have been studied. In this paper, a new application of ICA for two-dimensional data analysis is given, namely, image restoration of blurred images. The proposed method can restore the original image without knowing the blurring process. © 2001 Scripta Technica, Electron Comm Jpn Pt 3, 84(12): 1–9, 2001

Abstract: This paper studies the so-called inverse filtering and deconvolution problem from different angles. To start with, both exact and almost deconvolution problems are formulated, and the necessary and sufficient conditions for their solvability are investigated. Exact and almost deconvolution problems seek filters that can estimate the unknown inputs of the given plant or system either exactly or almostly whatever may be the unintended or disturbance inputs such as measurement noise, external disturbances, and model uncertainties that act on the system. As such they require strong solvability conditions. To alleviate this, several optimal and suboptimal deconvolution problems are formulated and studied. These problems seek filters that can estimate the unknown inputs of the given system either exactly, almostly or optimally in the absence of unintended (disturbance) inputs, and on the other hand, in the presence of unintended (disturbance) inputs, they seek that the influence of such disturbances on the estimation error be as small as possible in a certain norm ($H_2$ or $H_\infty$) sense. Both continuous- and discrete-time systems are considered. For discrete-time systems, the counter parts of all the above problems when an $\ell$-step delay in estimation is present are introduced and studied. Next, we focus on the exact and almost deconvolution but this time when the uncertainties in plant dynamics can be structurally modeled by a $\Delta$-block as a feedback element to the nominally known plant dynamics. This is done either in the presence or absence of external disturbances.

Abstract: The analysis of acoustic emission signals has been widely applied to damage detection and damage characterization in composites. Features of acoustic emission signals, such as amplitude, frequency, and counts, are usually used to identify the type of a damage. Recently, time-frequency distribution techniques, such as the wavelet transform and the Choi-Williams distribution, have also been applied to characterize damage. A common feature of these approaches is that the analysis is on the acoustic emission signal itself. Nevertheless, this signal is not the wave source signal as it has been modulated by the signal transfer path. Real information on damage is actually hidden behind the signal. To reveal direct information on damage, a blind deconvolution method has been developed. It is a frequency domain method based on the cepstrum technique. With the method, the acoustic emission signal is demodulated, and information on the wave source can be revealed, and thus damage can be identified. This paper presents preliminary test data to assess the validity of the proposed methodology as a means of identifying specific damage modes in fiber-reinforced composites.

Abstract: In this paper, we present a new nonlinear receiver for the blind deconvolution of intersymbol interference (ISI) impaired data. The proposed receiver achieves fast identification of an unknown transmission channel using only one channel estimator and requiring the computation of only the second-order conditional statistics of the baud-rate sampled received signal and the knowledge of the transmitted constellation. The main novelty of the proposed approach is that the receiver accomplishes fast channel-identification by using soft-statistics. In particular, the receiver consists of a symbol-by-symbol maximum a posteriori (SbS-MAP) detector that feeds a nonlinear Kalman-like channel estimator with the soft statistics constituted by the a posteriori probabilities (APPs) of the state sequence of the ISI channel. Several numerical results confirm that the proposed blind detector achieves the identification of nonminimum phase channels with deep spectral notches within 300 symbols, even at low signal-to-noise ratios (SNRs). Furthermore, an attractive feature of the proposed blind channel estimator is that it directly estimates the discrete-time impulse response of the unknown channel so that, in principle, any equalization technique for known channels may be performed after channel identification has been achieved.

Abstract: A method for the processing of data of seismic traces for the geophysical interpretation of the earth's subsurface employs deconvolution (17, 21) based on fractionally integrated white noise. A generalized form of deconvolution that more accurately models the earth's reflectivity applies the steps of (1) estimating the process order of the fractionally integrated noise model; (2) computing one or both of two correction filters (6, 13) based on the fractionally integrated noise model; and (3) applying one or both of the filters to the deconvolution processing (17, 21) of the seismic data. The resulting graphical displays based on the deconvolution processing more accurately portray the reflectivity and provide improved wavelet compression and signal resolution, thereby aiding in, and improving the interpretation of the subsurface strata.

Abstract: We present novel algorithms for multichannel blind deconvolution under output whitening constraints. The algorithms are inspired by recently-developed procedures for gradient adaptive paraunitary filter banks. Several algorithms are developed, including one algorithm that successfully deconvolves mixtures of arbitrary non-zero kurtosis source signals. We provide detailed local stability analyses of the proposed methods to verify their capabilities. Simulations show that the methods successfully deconvolve spatio-temporal mixtures of statistically independent source signals.

Abstract: The analytically continued Fourier transform of a two-dimensional image vanishes to zero on a two-dimensional surface embedded in a four-dimensional space. This surface uniquely characteristics the image and is known as a 'zero sheet'. Since the manipulation of a function in four-dimensional space is cumbersome, the projections of zero sheets, known as 'zero tracks' are calculated. This knowledge of zero sheets can be extended to a number of practical applications including image processing. Image restoration can be realised without prior knowledge of the point spread function, i.e. blind deconvolution is possible even when only a single blurred image is given. If the blurred image concerned contains a point spread function with diagonal symmetry, the point zeros calculated row-wise and column-wise contain some similarities, which supports retrieval of both the true image and a point spread function. This novel scheme performs the separation effectively in the absence of contamination.