Blind deconvolution of ultrasound sequences using nonparametric local polynomial estimates of the pulse

The problem of reconstructing the reflectivity of a biological tissue is examined by means of blind deconvolution of the echo ultrasound signals. It is shown that the quality of the reconstruction procedure can be significantly improved when initially the ultrasonic pulse is accurately estimated. A new approach to the estimation of the ultrasound pulse echo sequences is proposed, using local polynomial approximation, which is closely related to the wavelet transform theory. This approach can be viewed as a modification of homomorphic deconvolution, by using bases different from the Fourier basis of the space of square-integrable functions L2. The bases used here are the orthogonal compactly supported wavelet bases. It is shown that the locality of the estimate can be extremely useful in number of cases of practical interest, resulting in estimates with smaller root-mean squared (rms) errors, as compared with estimates employing the Fourier basis. This approach is applied to ultrasound signals, for estimation of the ultrasound pulse log-spectrum from the log-spectrum of radio-frequency (RF) sequences. It is shown, conceptually and experimentally, that the proposed approach can provide robust and rapidly computed estimates of the ultrasound pulses from the RF-sequences, as obtained in the process of tissue scanning. The pulse phase was recovered using the minimum-phase assumption, which was found to hold for the transducers in use. The obtained pulse estimates are used for the deconvolution of the RF-sequences, which result in stable estimates of the tissue reflectivity function, fairly independent of the properties of the imaging system. Simulated data, data obtained from several phantoms and from in vitro experiments have been processed and the results seem to be quite promising.     

Generalized quadratic minimization and blind multichanneldeconvolution

This paper deals with the problem of quadratic minimization subject to linear equality constraints. Unlike the standard formulation, we assume the most general case of a possibly singular quadratic form. We explain that the existing formal solution to this problem has several drawbacks. Our new approach is free from most of these drawbacks. In particular, it has a simple interpretation and is relatively easy to implement. The practical importance of this result lies in its numerous applications: filter design, spectral analysis, direction finding, and blind deconvolution of multiple FIR channels. Here, we focus on the blind deconvolution application for which we present a novel solution with enhanced performance     

A fast thresholded landweber algorithm for wavelet-regularized multidimensional deconvolution.

We present a fast variational deconvolution algorithm that minimizes a quadratic data term subject to a regularization on the l(1)-norm of the wavelet coefficients of the solution. Previously available methods have essentially consisted in alternating between a Landweber iteration and a wavelet-domain soft-thresholding operation. While having the advantage of simplicity, they are known to converge slowly. By expressing the cost functional in a Shannon wavelet basis, we are able to decompose the problem into a series of subband-dependent minimizations. In particular, this allows for larger (subband-dependent) step sizes and threshold levels than the previous method. This improves the convergence properties of the algorithm significantly. We demonstrate a speed-up of one order of magnitude in practical situations. This makes wavelet-regularized deconvolution more widely accessible, even for applications with a strong limitation on computational complexity. We present promising results in 3-D deconvolution microscopy, where the size of typical data sets does not permit more than a few tens of iterations.     

Multichannel seismic deconvolution

Deals with Bayesian estimation of 2D stratified structures from echosounding signals. This problem is of interest in seismic exploration, but also for nondestructive testing or medical imaging. The proposed approach consists of a multichannel Bayesian deconvolution method of the 2D reflectivity based upon a theoretically sound prior stochastic model. The Markov-Bernoulli random field representation introduced by Idier et al. (1993) is used to model the geometric properties of the reflectivity, and emphasis is placed on representation of the amplitudes and on deconvolution algorithms. It is shown that the algorithmic structure and computational complexity of the proposed multichannel methods are similar to those of single-channel B-G deconvolution procedures, but that explicit modeling of the stratified structure results in significantly better performances. Simulation results and examples of real-data processing illustrate the performances and the practicality of the multichannel approach     

Wavelet estimation

An important problem in seismic exploration is the estimation of and correction for the seismic wavelet. A seismic signal may be modeled as a convolutional model with the wavelet as one component. The wavelet propagated by the seismic energy source is complicated by transmission and recording filters. Some filters in the system can be deterministically defined while others are more conjectural. The estimation of the wavelet is useful in two major ways. Borehole measurements are used to model the surface seismograms. The wavelet used in the model needs to match that of the seismogram to correlate the two measurements. Conversely, the estimated wavelet can be used to design inverse filters which make the seismogram approach the borehole measures. Some well-known methods for estimation of the wavelet are based on assumptions about the wavelet or the earth reflectivity. Examples of the methods indicate success on some data even though each makes different assumptions. The methods serve to point out basic problems in reliably estimating the wavelet from the seismogram. Basic problems include noise, band-limiting, nonstationarity, uncertain theoretical models, assumption failure, and widely diverse geological sequences of the earth. Quality control or evaluation of the performance of an estimation algorithm is a nontrivial problem. The estimation of the wavelet from a seismic recording remains an area of challenging research and importance in exploration for hydrocarbons.     

On homomorphic deconvolution of bandpass signals

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     

Domain decomposition for variational optical-flow computation.

We present an approach to parallel variational optical-flow computation by using an arbitrary partition of the image plane and iteratively solving related local variational problems associated with each subdomain. The approach is particularly suited for implementations on PC clusters because interprocess communication is minimized by restricting the exchange of data to a lower dimensional interface. Our mathematical formulation supports various generalizations to linear/nonlinear convex variational approaches, three-dimensional image sequences, spatiotemporal regularization, and unstructured geometries and triangulations. Results concerning the effects of interface preconditioning, as well as runtime and communication volume measurements on a PC cluster, are presented. Our approach provides a major step toward real-time two-dimensional image processing using off-the-shelf PC hardware and facilitates the efficient application of variational approaches to large-scale image processing problems.     

Simultaneous wavelet estimation and deconvolution of reflectionseismic signals

The problem of simultaneous wavelet estimation and deconvolution is investigated with a Bayesian approach under the assumption that the reflectivity obeys a Bernoulli-Gaussian distribution. Unknown quantities, including the seismic wavelet, the reflection sequence, and the statistical parameters of reflection sequence and noise are all treated as realizations of random variables endowed with suitable prior distributions. Instead of deterministic procedures that can be quite computationally burdensome, a simple Monte Carlo method, called Gibbs sampler, is employed to produce random samples iteratively from the joint posterior distribution of the unknowns. Modifications are made in the Gibbs sampler to overcome the ambiguity problems inherent in seismic deconvolution. Simple averages of the random samples are used to approximate the minimum mean-squared error (MMSE) estimates of the unknowns. Numerical examples are given to demonstrate the performance of the method     

Minimum-Variance and Maximum-Likelihood Deconvolution for Noncausal Channel Models

This paper extends the previous works of Mendel and his students on the subject of deconvolution from causal channel (wavelet) models to noncausal channel models. Noncausal wavelets occur, for example, in seismic data processing when a land vibrator is used to excite the Earth. Minimum-variance and maximum-likelihood deconvolution algorithms are developed herein for symmetrical and/or nonsymmetrical time-invariant wavelets that are excited by stationary and/or nonstationary white noise inputs. Minimum-variance deconvolution algorithms for a noncausal wavelet turn out to be quite different than those for a causal wavelet; however, maximum-likelihood deconvolution algorithms for a noncausal wavelet, which involve event detection and amplitude restoration, are essentially the same as those for a causal wavelet. Examples are provided that illustrate the performance of the different deconvolution algorithms.     

Robust adaptive beamforming using worst-case performance optimization: a solution to the signal mismatch problem

Adaptive beamforming methods are known to degrade if some of underlying assumptions on the environment, sources, or sensor array become violated. In particular, if the desired signal is present in training snapshots, the adaptive array performance may be quite sensitive even to slight mismatches between the presumed and actual signal steering vectors (spatial signatures). Such mismatches can occur as a result of environmental nonstationarities, look direction errors, imperfect array calibration, distorted antenna shape, as well as distortions caused by medium inhomogeneities, near-far mismatch, source spreading, and local scattering. The similar type of performance degradation can occur when the signal steering vector is known exactly but the training sample size is small. In this paper, we develop a new approach to robust adaptive beamforming in the presence of an arbitrary unknown signal steering vector mismatch. Our approach is based on the optimization of worst-case performance. It turns out that the natural formulation of this adaptive beamforming problem involves minimization of a quadratic function subject to infinitely many nonconvex quadratic constraints. We show that this (originally intractable) problem can be reformulated in a convex form as the so-called second-order cone (SOC) program and solved efficiently (in polynomial time) using the well-established interior point method. It is also shown that the proposed technique can be interpreted in terms of diagonal loading where the optimal value of the diagonal loading factor is computed based on the known level of uncertainty of the signal steering vector. Computer simulations with several frequently encountered types of signal steering vector mismatches show better performance of our robust beamformer as compared with existing adaptive beamforming algorithms.     

One-dimensional normal-incidence inversion: A solution procedure for band-limited and noisy data

In this paper we present a one-dimensional normal-incidence inversion procedure for reflection seismic data. A lossless layered system is considered which is characterized by reflection coefficients and traveltimes. A priori knowledge for the unknown parameters, in the form of statistics, is incorporated into a nonuniform layered system, and a maximum a posteriori estimation procedure is used for the estimation of the system's unknown parameters (i.e., we assume a random reflector model) from noisy and band-limited data. Our solution to the inverse problem includes a downward continuation procedure for estimation of the states of the system. The state sequences are composed of overlapping wavelets. We show that estimation of the unknown parameters of a layer is equivalent to estimation of the amplitude and detection of the time delay of the first wavelet in the upgoing state sequence of the layer. A suboptimal maximum-likelihood deconvolution procedure is employed to perform estimation and detection. The most desirable features of the proposed algorithm are its layer-recursive structure and its ability to process noisy and band-limited data.     

Synthetic aperture radio telescopes

Next-generation radio telescopes will be much larger, more sensitive, have a much larger observation bandwidth, and will be capable of pointing multiple beams simultaneously. Obtaining the sensitivity, resolution, and dynamic range supported by the receivers requires the development of new signal processing techniques for array and atmospheric calibration as well as new imaging techniques that are both more accurate and computationally efficient since data volumes will be much larger. This article provides an overview of existing image formation techniques and outlines some of the directions needed for information extraction from future radio telescopes. We describe the imaging process from measurement equation until deconvolution, both as a Fourier inversion problem and as an array processing estimation problem. The latter formulation enables the development of more advanced techniques based on state-of-the-art array processing. We also demonstrate the techniques on simulated and measured radio telescope data.     

Sparse Poisson noisy image deblurring

Deblurring noisy Poisson images has recently been a subject of an increasing amount of works in many areas such as astronomy and biological imaging. In this paper, we focus on confocal microscopy, which is a very popular technique for 3-D imaging of biological living specimens that gives images with a very good resolution (several hundreds of nanometers), although degraded by both blur and Poisson noise. Deconvolution methods have been proposed to reduce these degradations, and in this paper, we focus on techniques that promote the introduction of an explicit prior on the solution. One difficulty of these techniques is to set the value of the parameter, which weights the tradeoff between the data term and the regularizing term. Only few works have been devoted to the research of an automatic selection of this regularizing parameter when considering Poisson noise; therefore, it is often set manually such that it gives the best visual results. We present here two recent methods to estimate this regularizing parameter, and we first propose an improvement of these estimators, which takes advantage of confocal images. Following these estimators, we secondly propose to express the problem of the deconvolution of Poisson noisy images as the minimization of a new constrained problem. The proposed constrained formulation is well suited to this application domain since it is directly expressed using the antilog likelihood of the Poisson distribution and therefore does not require any approximation. We show how to solve the unconstrained and constrained problems using the recent alternating-direction technique, and we present results on synthetic and real data using well-known priors, such as total variation and wavelet transforms. Among these wavelet transforms, we specially focus on the dual-tree complex wavelet transform and on the dictionary composed of curvelets and an undecimated wavelet transform.     

Solution of inverse problems in image processing by waveletexpansion

We describe a wavelet-based approach to linear inverse problems in image processing. In this approach, both the images and the linear operator to be inverted are represented by wavelet expansions, leading to a multiresolution sparse matrix representation of the inverse problem. The constraints for a regularized solution are enforced through wavelet expansion coefficients. A unique feature of the wavelet approach is a general and consistent scheme for representing an operator in different resolutions, an important problem in multigrid/multiresolution processing. This and the sparseness of the representation induce a multigrid algorithm. The proposed approach was tested on image restoration problems and produced good results.     

Development of a neural network based algorithm for rainfallestimation from radar observations

Rainfall estimation based on radar measurements has been an important topic in radar meteorology for more than four decades. This research problem has been addressed using two approaches, namely a) parametric estimates using reflectivity-rainfall relation (Z-R relation) or equations using multiparameter radar measurements such as reflectivity, differential reflectivity, and specific propagation phase, and b) relations obtained by matching probability distribution functions of radar based estimates and ground observations of rainfall. In this paper the authors introduce a neural network based approach to address this problem by taking into account the three-dimensional (3D) structure of precipitation. A three-layer perceptron neural network is developed for rainfall estimation from radar measurements. The neural network is trained using the radar measurements as the input and the ground raingage measurements as the target output. The neural network based estimates are evaluated using data collected during the Convection and Precipitation Electrification (CaPE) experiment conducted over central Florida in 1991. The results of the evaluation show that the neural network can be successfully applied to obtain rainfall estimates on the ground based on radar observations. The rainfall estimates obtained from neural network are shown to be better than those obtained from several existing techniques. The neural network based rainfall estimate offers an alternate approach to the rainfall estimation problem, and it can be implemented easily in operational weather radar systems     

Wavelet maxima reconstruction by conjugate gradient

The problem of reconstructing a signal from its wavelet transform modulus maxima representation can be formulated as minimising a quadratic term using the conjugate gradient technique. It is shown that this formulation obtains the same result as the accelerated frame algorithm which starts from the frame theory. A method to avoid super-resolution which is inherent in this problem is proposed, based on the measure of the regularity of the signal in the wavelet domain     

Wavelet thresholding techniques for power spectrum estimation

Estimation of the power spectrum S(f) of a stationary random process can be viewed as a nonparametric statistical estimation problem. We introduce a nonparametric approach based on a wavelet representation for the logarithm of the unknown S(f). This approach offers the ability to capture statistically significant components of ln S(f) at different resolution levels and guarantees nonnegativity of the spectrum estimator. The spectrum estimation problem is set up as a problem of inference on the wavelet coefficients of a signal corrupted by additive non-Gaussian noise. We propose a wavelet thresholding technique to solve this problem under specified noise/resolution tradeoffs and show that the wavelet coefficients of the additive noise may be treated as independent random variables. The thresholds are computed using a saddle-point approximation to the distribution of the noise coefficients     

A new SURE approach to image denoising: interscale orthonormal wavelet thresholding.

This paper introduces a new approach to orthonormal wavelet image denoising. Instead of postulating a statistical model for the wavelet coefficients, we directly parametrize the denoising process as a sum of elementary nonlinear processes with unknown weights. We then minimize an estimate of the mean square error between the clean image and the denoised one. The key point is that we have at our disposal a very accurate, statistically unbiased, MSE estimate--Stein's unbiased risk estimate--that depends on the noisy image alone, not on the clean one. Like the MSE, this estimate is quadratic in the unknown weights, and its minimization amounts to solving a linear system of equations. The existence of this a priori estimate makes it unnecessary to devise a specific statistical model for the wavelet coefficients. Instead, and contrary to the custom in the literature, these coefficients are not considered random anymore. We describe an interscale orthonormal wavelet thresholding algorithm based on this new approach and show its near-optimal performance--both regarding quality and CPU requirement--by comparing it with the results of three state-of-the-art nonredundant denoising algorithms on a large set of test images. An interesting fallout of this study is the development of a new, group-delay-based, parent-child prediction in a wavelet dyadic tree.