Abstract: We describe some numerical techniques for the total variation image restoration method, namely a primal-dual linearization for the Euler-Lagrange equations and some preconditioning issues. We also highlight extension of this technique to color images, blind deconvolution and the staircasing effect.

Abstract: It is quite common in the statistical literature on nonparametric deconvolution to assume that the error density is perfectly known. Since this seems to be unrealistic in many practical applications, we study the effect of estimating the unknown error density. We derive minimax rates of convergence and propose a modification of the usual kernel-based estimation scheme, which takes the uncertainty about the error density into account. A simulation study quantifies the possible gains by this new method in finite sample situations.

Abstract: Truncated Volterra expansions model nonlinear systems encountered with satellite communications, magnetic recording channels, and physiological processes. A general approach for blind deconvolution of single-input multiple-output Volterra finite impulse response (FIR) systems is presented. It is shown that such nonlinear systems can be blindly equalized using only linear FIR filters. The approach requires that the Volterra kernels satisfy a certain coprimeness condition and that the input possesses a minimal persistence-of-excitation order. No other special conditions are imposed on the kernel transfer functions or on the input signal, which may be deterministic or random with unknown statistics. The proposed algorithms are corroborated with simulation examples.

Abstract: Two contrasts for the problem of multichannel blind deconvolution have been given and theoretically studied by Comon [1996]. The maximization of these criteria allows us to solve the problem of multi-input/multi-output (MIMO) blind deconvolution. In this paper, we show that many other contrast functions may be considered. The two aforementioned criteria are proved to be included in the wide class of contrast functions, which is here defined through simple conditions.

Abstract: This paper compares the performance of seven different cepstrum-based methods for radial blind deconvolution of medical ultrasound images. The first is the generalized cepstrum method. The second is the spectral root cepstrum method. These methods have received little attention so far. The last five methods are all based on the complex cepstrum, but different computational techniques in the spatial and frequency domain are employed. Using in vivo radio frequency data from a clinical scanner, the generalized cepstrum method gave the best images closely followed by the complex cepstrum using phase unwrapping or polynomial rooting. The complex cepstrum method using higher-order statistics was ranked as low as number five. These results are an important guideline for selecting a specific cepstrum-based radial deconvolution method for implementation in ultrasound scanners.

Abstract: Advanced nondestructive testing techniques use a laser to generate ultrasonic waves at the surface of a test material. An air-coupled transducer receives the ultrasound that is the convolution of the signal leaving the test material and the distortion function. Blind deconvolution methods are applied to estimate the signal leaving the material.

Abstract: The separation of multiple signals from their superposition recorded at several sensors is addressed. The methods employ polyspectra of the sensor data in order to extract the unknown signals and estimate the finite impulse response (FIR) coupling systems via a linear equation based algorithm. The procedure is useful for multichannel blind deconvolution of colored input signals with (possibly) overlapping spectra. An extension of the main algorithm, which can be applied for quasiperiodic signal separation, is also given. Simulation results corroborate the applicability of the algorithm.

Abstract: High-resolution multipath parameter estimates can be obtained through various deconvolution procedures, all of which-in the limit-rely on some form of inverse filtering. Although deconvolution in a multipath environment free from Doppler is well understood and well documented, this is not true for the case when motion of the multipath components relative to the receiver imposes a Doppler shift on the transmitted probing signal. This paper describes the effect of Doppler on a broad class of deconvolution methods by studying the effect of Doppler on the output of an inverse filter. It is shown that in the presence of Doppler, the deconvolution outputs are comprised chiefly of two signal-related functions, one of which may be designed in such a way as to be free from the range-Doppler coupling effects inherent in correlation processing. Knowledge of these two functions provides insight into the signal design issues relevant to deconvolution-based multipath parameter estimation systems and is useful in designing appropriate constraints and post-processing algorithms that may lead to an accurate extraction of the Doppler and delay parameters of the multipath channel. These results are applied to two known deconvolution methods: the method of projection onto convex sets (POCS) and the method of least squares (LS).

Abstract: A regularized method of spectral curve deconvolution is proposed. This method is based on three fundamental principles: the regularized method of solving the convolution equation; the use, instead of the apodization function, of the digital low-pass filter, which permits exact knowledge of its characteristics; and the use of the Fourier transform modulus of the spectrum being treated for obtaining a priori information about the frequency characteristics of the solution and noise, required for determination of the optimum parameters of the regularizing operator. The regularized method of deconvolution permits the acquisition of an approximately stable solution for the deconvolution problem of spectral curves, which moves toward an exact solution with the decrease of the experimental spectrum error. Examples are given of the application of the regularized method of deconvolution to simulated and experimental IR spectra. A conclusion about the expediency of using the given method for resolution enhancement in complex spectra is made.

Abstract: We explore the relationships between the two related tasks of blind deconvolution and blind source separation. The maximum likelihood method for parameter estimation is shown to provide a unifying framework for deriving blind deconvolution and blind source separation algorithms. To illustrate the structural relationships between the two tasks, we consider the problem of blind source separation under circulant mixing conditions, derive iterative algorithms for its solution, and then relate these algorithms to recently-proposed for blind deconvolution techniques. The results of these various studies suggests the potential benefits that can be obtained from the cross-fertilization of these two fields.

Abstract: In this paper we describe a time domain algorithm for determining the influence function from the measured input and output signals of the system. The deconvolution, which is a highly unstable inverse problem with measurement errors, is an important step for obtaining the system's influence function that provides insight about flow regimes normally masked by the time-dependent input signal. The algorithms presented for deconvolution in the literature are generally based on data reduction, with the exception of constrained deconvolution methods. We propose a constrained least-squares deconvolution method to reconstruct the influence func tion from noisy data. The constraints are the lower bounds on the first few lags of the normalized autocorrelation coefficients of the influence function. The lower bounds may represent known or desirable smoothness properties of the function. By choosing the constraint values larger, a smoother deconvolution can be obtained. We also impose an energy constraint on the derivative of the recon structed signal for further regularization.

Abstract: Starting from maximizing information flow through a nonlinear neuron Bell and Sejnowski (see Neural Computation, vol.7, no.6, p.1004-34, 1995) derived adaptation equations for blind deconvolution using an FIR filter. We derive a simpler form of the adaptation and apply it to more complex filter structures, such as recursive filters. As an application, we study blind echo cancellation for speech signals. We also present a method that avoids whitening the signals in the procedure.

Abstract: We present a simple efficient local unsupervised learning algorithm for online adaptive multichannel blind deconvolution and separation of i.i.d. sources. Under mild conditions, there exits a stable inverse system so that the source signals can be exactly recovered from their convolutive mixtures. Based on the existence of the inverse filter, we construct a two-stage neural network which consists of blind equalization and source separation. In the blind equalization stage, we employ anti-Hebbian learning in the temporal domain for decorrelation. For blind separation, we can apply any existing algorithms. Extensive computer simulations confirm the validity and high performance of our proposed learning algorithm.

Abstract: A new blind deconvolution method is proposed for recovering an unknown source signal, which is observed through two unknown channels characterized by non-minimum phase impulse response filters. Conventional methods cannot estimate the non-minimum phase parts. Our method is based on computing the eigenvector corresponding to the smallest eigenvalue of the input correlation matrix and using a cost function to determine the order of the impulse response filter model. Multi-channel inverse filtering with the estimated impulse responses is used to recover the unknown source signal. Sub-band processing is also used to reduce the complexity of dealing with long impulse responses such as room impulse responses. Computer simulation shows that the effectiveness of our method.

Abstract: This paper describes a technique for the blind deconvolution of extended objects such as the Hubble Space Telescope (HST), scanning electron and 3D fluorescence microscope images. The blind deconvolution mechanism is based on the Richardson-Lucy (1972, 1974) algorithm and alternates between deconvolution of the image and point spread function (PSF). This form of iterative blind deconvolution differs from that typically employed in that multiple PSF iterations are performed after each image iteration. The initial estimate for the PSF is the autocorrelation of the blurred image and the edges of the image are windowed to minimise wrap around artifacts. Acceleration techniques are employed to speed restoration and results from real HST, electron microscope and 3D fluorescence images are presented.

Abstract: Techniques for three-dimensional (3D) light microscopy of a wide variety of biological specimens are rapidly maturing. The next advances are to improve image contrast, make image resolution as isotropic as possible, and perform quantitative analysis. This article discusses methods of automatic image analysis and blind deconvolution to compensate for the microscope's point-spread function (impulse response). Special emphasis is given to quantitative analysis based on adaptive segmentation, the results of which can be used to quantify and trace individual structures, and to montage high-resolution fields as well as the results of image analysis on these fields into a wide-area view with an associated computer database representation. We exploit the specific advantages accruing from the 3D nature of the data to achieve new capabilities not possible with 2D imaging. Fundamental and practical challenges in this area and our progress to date including automated segmentation, cell counting, neuron tracing, mosaic synthesis, “blind” deconvolution of large 3D images, and high-speed computation are presented. © 1997 John Wiley & Sons, Inc. Int J Imaging Syst Technol, 8, 240–254, 1997

Abstract: Image quality of single-photon emission computerized tomography (SPECT) is essentially determined by the count sensitivity of the detector and the geometry of the collimator. The authors introduce a novel geometry for a dual-head SPECT imaging system (Park Medical Systems Inc., Lachine, Quebec) to improve the count sensitivity. The imaging system is equipped with a coded aperture and a parallel hole collimators. The camera head equipped with the coded aperture collimator is used to acquire count-rich projection images, and the other head with the parallel hole collimator is used to acquire high resolution projection images. To further improve the image resolution of the projections, two expectation-maximization-type blind deconvolution (EMBD) algorithms were derived. The algorithms were evaluated using elliptical cylindrical rod phantom and human hand data. The coded aperture and parallel hole projections were acquired simultaneously using the dual-head SPECT imaging system. The acquired coded aperture projections were decoded using the standard uniformly redundant array (URA) decoding technique to yield decoded projections. The projections acquired from the parallel hole collimator head were incorporated into the decoded images in the EMBD restoration process. Using the coded aperture collimator, the count sensitivity was markedly increased, 8-fold for the phantom data and approximately 20-fold for the hand data, as compared to the parallel hole collimator. Also, image quality of the SPECT projections was significantly improved when the EMBD algorithms were implemented on the projections. Thus, the dual-head SPECT imaging system associated with the EM-type blind deconvolution algorithms may be a preferred system for low-count SPECT imaging.

Abstract: The resolution of the data from many instruments can be improved, or the rate of data collection can be increased for the same final resolution, by applying to the data reliable algorithms for smoothing and deconvolution. Iterative methods which were formerly impractical can easily be applied on a small computer. An ingenious linear algorithm for deconvolution of one-dimensional data (van Cittert, 1931) gave much better results when Jansson (1963) introduced a relaxation function which ensured the results remained positive. Gold (1964) derived by a matrix approach a nonlinear algorithm which used a different method of comparison, but Xu et al. showed 30 years later that it is a special van Cittert algorithm with a variable relaxation function. Tests of Gold's method show that it is reliable and much faster than Jansson's algorithm, converging in 20 iterations or fewer. If a microprobe beam spot is to a good approximation square or rectangular a 2-D image can be smoothed or deconvolved in the X and Y directions independently, and the Gold algorithm has proved suitable for the deconvolution stage. Almost all smoothing methods will broaden narrow peaks, but an exception is the linear iterative method of Morrison (1962), which reduces any structure narrower than the resolution function. The negative feedback step used in the deconvolution algorithms is not possible in a smoothing algorithm. The method suffers from a halting problem, since it smoothes during early iterations but eventually reproduces the original data. This can be prevented by introducing a relaxation function which is unity for the first iteration but decreases rapidly with succeeding iterations.

Abstract: A new l/sub 1/ optimal deconvolution filter design approach for systems with uncertain (or unknown)-but-bounded inputs and external noises is proposed. The purpose of this deconvolution filter is to minimize the peak gain from the input signal and noise to the error by the viewpoint of the time domain. The solution consists of two steps. In the first step, the l/sub 1/ norm minimization problem is transferred to an equivalent A-norm minimization problem, and the minimum value of the peak gain is calculated. In the second step, based on the minimum peak gain, the l/sub 1/ optimal deconvolution filter is constructed by solving a set of constrained linear equations. Some techniques of inner-outer factorization, polynominal spectral factorization, linear programming, and some optimization theorems found in a book by Luenberger are applied to treat the l/sub 1/ optimal deconvolution filter design problem. Although the analysis of the algorithm seems complicated, the calculation of the proposed design algorithm for actual systems is simple. Finally, one numerical example is given to illustrate the proposed design approach. Several simulation results have confirmed that the proposed l/sub 1/ optimal deconvolution filter has more robustness than the l/sub 2/ optimal deconvolution filter under uncertain driving signals and noises.

Abstract: The total variational (TV) regularization method was first proposed for gray scale images and was extended for vector valued images. In this work, we apply the TV regularization method to solve the multichannel image deconvolution problem. The motivation for regularizing with the TV norm is that it is extremely effective for recovering edges of images. In this paper, a fast iterative method is developed to solve the deconvolution problem. Our method involves solving linear systems and the conjugate gradient method is applied in which Fourier transform type preconditioners are used to speed up the convergence rate. Numerical experiments demonstrate the effectiveness of the TV regularization method. In this paper, we present some preliminary results on multichannel blind deconvolution with TV regularization.

Abstract: In equalization and deconvolution tasks, the correlated nature of the input signal slows the convergence speeds of stochastic gradient adaptive filters In this paper, we present two simple algorithms that employ the equalizer as a prewhitening filter to effectively and iteratively decorrelate the input signal within the gradient updates These algorithms provide quasi-Newton convergence locally about the optimum coefficient solution for deconvolution and equalization tasks Simulations indicate that the algorithms have excellent adaptation properties both for supervised and unsupervised (blind) adaptation criteria

Abstract: A convolution may be represented as x(.)=r(.)* w(.). The goal of deconvolution is to extract r(.) and w(.) from a knowledge of x(.) and it finds numerous applications in digital signal processing. Of practical interest in oil exploration is the case where w(.) is a seismic pressure wavelet, x(.) is the observed seismic response, and r(.) is the reflectivity of the Earth. A number of procedures have been proposed, including predictive, deterministic, and homomorphic deconvolution. Homomorphic deconvolution has been found to be particularly efficient for those cases where x(.) is known to be fullband. This paper presents a robust constructive procedure for efficient homomorphic deconvolution for those cases where x(.) is a bandpass signal. Extensive comparisons with other methods for deconvolving bandpass signals on measured seismic data traces (including the Novaya Zemlya event) illustrate the improvement in the deconvolution.