Abstract: Since their introduction in a classic paper by Rudin, Osher and Fatemi [695], total variation minimizing models have become one of the most popular and successful methodology for image restoration. More recently, there has been a resurgence of interest and exciting new developments, some extending the applicabilities to impainting, blind deconvolution and vector-valued images, while others offer improvements in better preservation of contrast, geometry and textures, in ameliorating the staircasing effect, and in exploiting the multiscale nature of the models. In addition, new computational methods have been proposed with improved computational speed and robustness. We shall review some of these recent developments.

Abstract: We present a method based on Tikhonov regularization for solving one-dimensional inverse tomography problems that arise in combustion applications. In this technique, Tikhonov regularization transforms the ill-conditioned set of equations generated by onion-peeling deconvolution into a well-conditioned set that is less susceptible to measurement errors that arise in experimental settings. The performance of this method is compared to that of onion-peeling and Abel three-point deconvolution by solving for a known field variable distribution from projected data contaminated with an artificially generated error. The results show that Tikhonov deconvolution provides a more accurate field distribution than onion-peeling and Abel three-point deconvolution and is more stable than the other two methods as the distance between projected data points decreases.

Abstract: Image restoration and segmentation are both classical problems, that are known to be difficult and have attracted major research efforts. This paper shows that the two problems are tightly coupled and can be successfully solved together. Mutual support of image restoration and segmentation processes within a joint variational framework is theoretically motivated, and validated by successful experimental results. The proposed variational method integrates semi-blind image deconvolution (parametric blur-kernel), and Mumford-Shah segmentation. The functional is formulated using the /spl Gamma/-convergence approximation and is iteratively optimized via the alternate minimization method. While the major novelty of this work is in the unified treatment of the semi-blind restoration and segmentation problems, the important special case of known blur is also considered and promising results are obtained.

Abstract: In this study, a frequency domain formulation of continuous loop averaging deconvolution (CLAD) of overlapping evoked potentials is developed and applied for the extraction of transient responses from recordings obtained at high stimulation rates. This formulation allows for a faster execution of CLAD by using fast Fourier transform algorithms. The frequency characteristics of the deconvolution filter depends exclusively on the stimulus sequence and determines whether the noncoherent noise is amplified or attenuated in different frequencies. A formula for calculating the signal-to-noise ratio (SNR) achieved by the deconvolution process is developed. The newly developed theory and the methodology is applied to the extraction of the auditory brainstem and middle latency responses using various sequences. The effects of the sequence used and the number of sweeps averaged in ongoing acquisition on SNR are examined by using single sweep recordings. The results verify the deconvolution theory and the methodology and show its limitations. Depending on the frequency characteristics of the sequence, the deconvolution process can amplify or attenuate the EEG noise. Proper selection of the stimulus sequence can increase the SNR enhancement obtained with conventional averaging.

Abstract: Standard Euler deconvolution is applied to potential-field functions that are homogeneous and harmonic. Homogeneity is necessary to satisfy the Euler deconvolution equation itself, whereas harmonicity is required to compute the vertical derivative from data collected on a horizontal plane, according to potential-field theory. The analytic signal modulus of a potential field is a homogeneous function but is not a harmonic function. Hence, the vertical derivative of the analytic signal is incorrect when computed by the usual techniques for harmonic functions and so also is the consequent Euler deconvolution. We show that the resulting errors primarily affect the structural index and that the estimated values are always notably lower than the correct ones. The consequences of this error in the structural index are equally important whether the structural index is given as input (as in standard Euler deconvolution) or represents an unknown to be solved for. The analysis of a case history confirms serious errors in the estimation of structural index if the vertical derivative of the analytic signal is computed as for harmonic functions. We suggest computing the first vertical derivative of the analytic signal modulus, taking into account its nonharmonicity, by using a simple finite-difference algorithm. When the vertical derivative of the analytic signal is computed by finite differences, the depth to source and the structural index consistent with known source parameters are, in fact, obtained.

Abstract: Digital images have been used in growing number of applications from law enforcement and surveillance, to medical diagnosis and consumer photography. With such widespread popularity and the presence of low-cost image editing softwares, the integrity of image content can no longer be taken for granted. In this paper, we propose a novel technique based on blind deconvolution to verify image authenticity. We consider the direct output images of a camera as authentic, and introduce algorithms to detect further processing such as tampering applied to the image. Our proposed method is based on the observation that many tampering operations can be approximated as a combination of linear and non-linear components. We model the linear part of the tampering process as a filter, and obtain its coefficients using blind deconvolution. These estimated coefficients are then used to identify possible manipulations. We demonstrate the effectiveness of the proposed image authentication technique and compare our results with existing works.

Abstract: Mathematical Background.- Fundamentals of Statistical Signal Processing.- SISO Blind Equalization Algorithms.- MIMO Blind Equalization Algorithms.- Applications of MIMO Blind Equalization Algorithms.- Two-Dimensional Blind Deconvolution Algorithms.- Applications of Two-Dimensional Blind Deconvolution Algorithms.

Abstract: To remove the axial sidelobes from 4Pi images, deconvolution forms an integral part of 4Pi microscopy. As a result of its high axial resolution, the 4Pi point spread function (PSF) is particularly susceptible to imperfect optical conditions within the sample. This is typically observed as a shift in the position of the maxima under the PSF envelope. A significantly varying phase shift renders deconvolution procedures based on a spatially invariant PSF essentially useless. We present a technique for computing the forward transformation in the case of a varying phase at a computational expense of the same order of magnitude as that of the shift invariant case, a method for the estimation of PSF phase from an acquired image, and a deconvolution procedure built on these techniques.

Abstract: In this study a general formula for the signal to noise ratio (SNR) of the maximum length sequence (MLS) deconvolution averaging is developed using the frequency domain framework of the generalized continuous loop averaging deconvolution procedure [Ozdamar and Bohorquez, J. Acoust. Soc. Am. 119, 429-438 (2006)]. This formulation takes advantage of the well known equivalency of energies in the time and frequency domains (Parseval's theorem) to show that in MLS deconvolution, SNR increases with the square root of half of the number of stimuli in the sweep. This increase is less than that of conventional averaging which is the square root of the number of sweeps averaged. Unlike arbitrary stimulus sequences that can attenuate or amplify phase unlocked noise depending on the frequency characteristics, the MLS deconvolution attenuates noise in all frequencies consistently. Furthermore, MLS and its zero-padded variations present optimal attenuation of noise at all frequencies yet they present a highly jittered stimulus sequence. In real recordings of evoked potentials, the time advantage gained by noise attenuation could be lost by the signal amplitude attenuation due to neural adaptation at high stimulus rates.

Abstract: A new method for two-dimensional deconvolution of medical ultrasonic images is presented. The spatial resolution of the deconvolved images is much higher compared to the common images of the fundamental and second harmonic. The deconvolution also results in a more distinct speckle pattern. Unlike the most published deconvolution algorithms for ultrasonic images, the presented technique can be implemented using currently available hardware in real-time imaging, with a rate up to 50 frames per second. This makes it attractive for application in the current ultrasound scanners. The algorithm is based on two-dimensional homomorphic deconvolution with simplified assumptions about the point spread function. Broadband radio frequency image data are deconvolved instead of common fundamental harmonic data. Thus, information of both the first and second harmonics is used. The method was validated on image data recorded from a tissue-mimicking phantom and on clinical image data

Abstract: Source and receiver responses must be equalized when their behavior or coupling changes with location within a given survey. Existing surface-consistent deconvolution techniques that account for these effects assume that common-midpoint (CMP) gathering is valid — the seismic trace is decomposed into a source function, a receiver response, a normal-incidence reflectivity term, and an offsetrelated component that is laterally shift invariant. As a result, the performance of existing surface-consistent deconvolution techniques is best when applied to primary reflection data only, since the offset dependency of ground roll and multiples varies laterally in media with lateral variations. We have developed an alternative method for surfaceconsistent deconvolution that is applicable to the entire seismic trace and is therefore essentially a raw-data preprocessing step. The method is based on reciprocity of the medium response. Assuming that conditions for applicability of reciprocity are met, we can attribute differences between normal and reciprocal recordings to the source and receiver perturbations. Contrary to existing surfaceconsistent deconvolution methods, this approach uses the full description of the wavefield and is therefore ideally suited for prestack processing. We have applied this technique to single-sensor data acquired in Manistee County, Michigan. At this site, nearsurface conditions vary, and this significantly affects data quality. The application of the new deconvolution procedure substantially improves S/N ratio on both prestack and poststack data, and these results compare favorably to those obtained using existing surface-consistent deconvolution techniques, since they require subjective data scaling to obtain acceptable results. The obtained source corrections are correlated to changes in near-surface conditions — in this case, to changes in water-saturation levels. We do not observe such a correlation for the receiver corrections, which vary rapidly along the spread. Finally, the receiver response does not agree with the generally accepted damped harmonic oscillator model. For frequencies below 100 Hz, the retrieved receiver variations are larger than predicted by this model, and we cannot explain the receiver response using a single resonant frequency for the geophone-ground coupling.

Abstract: In first pass magnetic resonance brain perfusion imaging, arterial input functions are used in the deconvolution of the observed contrast concentrations to obtain quantitative hemodynamic parameters. Ideally, arterial input functions should be measured in each imaged voxel to eliminate the effects of delay and dispersion of the contrast agent from the injection site. An approach based on iterative blind deconvolution with the Richardson-Lucy algorithm is proposed for the simultaneous estimation of voxel-specific arterial input functions and voxel-specific tissue residue functions. An extended contrast concentration model was used to separate the first pass bolus from additional recirculation and leakage signals. The extended model was evaluated using in vivo data. Computer simulations examined the feasibility of iterative blind deconvolution in perfusion imaging. Preliminary in vivo results from a patient with fibromuscular dysplasia showed territories with delayed/dispersed arterial input functions that coincided with the location of territories supplied by collateral circulation as described from the complete radiologic examination. Higher flow values and shorter mean transit times compared to conventional methods were obtained in these areas, suggesting that the effects of dispersion were minimized. The in vivo estimated arterial input functions visualized the patient's blood supply patterns as a function of time.

Abstract: This paper outlines an exciting new approach for carrying out blind seismic deconvolution. In this algorithm, overlapping source wavelets are modeled as amplitude-modulated sinusoids, and blind deconvolution is carried out by initially determining the seismogram's principle phase components. Once the principle phases are determined, a Rao-Blackwellized particle filter (RBPF) is utilized to separate the corresponding overlapping source wavelets. This deconvolution technique is referred to as principle phase decomposition (PPD). The PPD technique makes use of the fact that in reflection seismology the discrete convolution operation can be represented as the summation of several source wavelets of differing arrival times. In this algorithm, a jump Markov linear Gaussian system (JMLGS) is defined where changes (jumps) in the state-space system and measurement equations are due to the occurrences and losses of overlapping source wavelet events. The RBPF obtains optimal estimates of the possible overlapping source wavelets by individually weighting and subsequently summing a bank of Kalman filters (KFs). These KFs are specified and updated by samples drawn from a Markov chain distribution that defines the probability of the overlapping source wavelets that compose the JMLGS. In addition, hidden Markov model filters are utilized for refining the principle phase estimates

Abstract: In contrast to the classical spread spectrum or data encryption methods, we propose an array redundancy-based approach for wireless transmissions with inherent low probability of interception (LPI). The redundancy of transmit antenna arrays introduces some degrees of freedom for deliberate signal randomization, based on which, diversity is exploited to randomize the eavesdropper's signal. LPI is analyzed by proving the indeterminacy of eavesdroppers' blind deconvolution. Extensive simulations and preliminary experiments are conducted to demonstrate the proposed method.

Abstract: The spectroscopic data recorded by a dispersion spectrophotometer are usually degraded by the response function of the instrument. To improve the resolving power, double or triple cascade spectrophotometers and narrow slits have been employed, but the total flux of the radiation available decreases accordingly, resulting in a lower signal-to-noise ratio (SNR) and a longer measurement time. However, the spectral resolution can be improved by mathematically removing the effect of the instrument response function. A high-order statistical Gauss–Newton algorithm is proposed to blindly deconvolve the measured spectroscopic data. The true spectrum and the instrument response function are estimated simultaneously. Experiments on artificial and real measured spectroscopic data demonstrate the feasibility of this method.

Abstract: The APEX method is a noniterative direct blind deconvolution technique that can sharpen certain kinds of high-resolution images in quasi real time. The method is predicated on a restricted class of blurs, in the form of 2-D radially symmetric, bell-shaped, heavy-tailed, probability density functions. Not all images can be usefully enhanced with the APEX method. However, the method is found effective on a broad class of galaxy images, including Hubble space telescope advanced camera for surveys (ACS) color imagery. APEX-detected optical transfer functions that successfully sharpen these images are very far from Gaussian, and of a type seldom found in the imaging literature. Several examples are given where significantly sharper and visually striking reconstructions are obtained, with sharpening confirmed by the tripling or quadrupling of image gradient norms.

Abstract: Digital images often suffer from point spreading or blurring from both known and unknown filters or point spread functions The sources of degradation can be lens point spreading, misfocus, motion, and scattering in case of x-ray images or atmospheric turbulence Therefore a digital image can suffer blurring from a single or an combination of various point spread functions, for example many images suffer from lens out of focus blur because of manufacturing limitations or satellite/aerial images suffer from lens focus and atmospheric turbulence etc The obvious requirement of an imaging system is to reproduce an image that is as close to original as possible Most existing image restoration methods uses blind deconvolution and deblurring methods that require good knowledge about both the signal and the filter and the performance depends on the amount of prior information regarding the blurring function and the signal Often an iterative procedure is required for estimating the blurring function such as Richardson-Lucy method and is computational complex and expensive and sometime instable This paper presents a blind image restoration method based on techniques of blind signal separation (BSS) in combination with the genetic algorithm for parameters optimization The method is not only simple but also requires little priori knowledge regarding the signal and the blurring function

Abstract: This work presents the development, validation and application of a novel deconvolution method based on B-splines for analyzing variable-rate reservoir performance data. Variable-rate deconvolution is a mathematically unstable problem which has been under investigation by many researchers over the last 35 years. While many deconvolution methods have been developed, few of these methods perform well in practice - and the importance of variable-rate deconvolution is increasing due to applications of permanent downhole gauges and large-scale processing/analysis of production data. Under these circumstances, our objective is to create a robust and practical tool which can tolerate reasonable variability and relatively large errors in rate and pressure data without generating instability in the deconvolution process. We propose representing the derivative of unknown unit rate drawdown pressure as a weighted sum of Bsplines (with logarithmically distributed knots). We then apply the convolution theorem in the Laplace domain with the input rate and obtain the sensitivities of the pressure response with respect to individual B-splines after numerical inversion of the Laplace transform. The sensitivity matrix is then used in a regularized least-squares procedure to obtain the unknown coefficients of the B-spline representation of the unit rate response or the well testing pressure derivative function. We have also implemented a physically sound regularization scheme into our deconvolution procedure for handling higher levels of noise and systematic errors. We validate our method with synthetic examples generated with and without errors. The new method can recover the unit rate drawdown pressure response and its derivative to a considerable extent, even when high levels of noise are present in both the rate and pressure observations. We also demonstrate the use of regularization and provide examples of under and over-regularization, and we discuss procedures for ensuring proper regularization. Upon validation, we then demonstrate our deconvolution method using a variety of field cases. Ultimately, the results of our new variable-rate deconvolution technique suggest that this technique has a broad applicability in pressure transient/production data analysis. The goal of this thesis is to demonstrate that the combined approach of B-splines, Laplace domain convolution, least-squares error reduction, and regularization are innovative and robust; therefore, the proposed technique has potential utility in the analysis and interpretation of reservoir performance data.

Abstract: In this paper, a new blind single-input single-output (SISO) deconvolution method based on the minimization of the mutual information rate of the deconvolved output is proposed. The method works in the frequency domain and requires estimation of the signal probability density function. Thus, the algorithm uses higher order statistics (except for Gaussian source) and allows non-minimum-phase filter estimation. In practice, the criterion contains a regularization term for limiting noise amplification as in Wiener filtering. The score function estimation, which represents a key point of the algorithm, is detailed, and the most robust estimate is selected. Finally, experiments point to the relevance of the proposed algorithm: 1) any filter, minimum phase or not, can be estimated and 2) on actual data (underwater explosions, seismovolcanic phenomena), this deconvolution algorithm provides good results with a better tradeoff between deconvolution quality and noise amplification than existing methods.

Abstract: Publisher Summary Deconvolution is a data processing technique that is very widely used in science and engineering. Any microscope image of a fluorescent specimen can, in principle, be deconvolved after acquisition in order to improve contrast and resolution. The most common application in biology is for deblurring images acquired as three-dimensional (3D) image stacks using a wide-field fluorescence microscope, where each image includes considerable out-of-focus light or blur originating from regions of the specimen. Deconvolution and confocal imaging are by no means mutually exclusive. Confocal imaging can nearly always benefit from the improvements in contrast, signal-to-noise ratio, and resolution afforded by restorative deconvolution methods. Deconvolution, particularly the more advanced approaches, is implemented through a software package where the algorithm is generally assisted by data correction before and noise reduction steps both before and between iterations. With a given deconvolution package, the quality of results obtained will depend foremost upon the quality of raw image data and upon the accuracy of PSF data.

Abstract: We present a new method for general multidimensional multichannel deconvolution with finite impulse response (FIR) convolution and deconvolution filters using Grobner bases. Previous work formulates the problem of multichannel FIR deconvolution as the construction of a left inverse of the convolution matrix, which is solved by numerical linear algebra. However, this approach requires the prior information of the support of deconvolution filters. Using algebraic geometry and Grobner bases, we find necessary and sufficient conditions for the existence of exact deconvolution FIR filters and propose simple algorithms to find these deconvolution filters. The main contribution of our work is to extend the previous Grobner basis results on multidimensional multichannel deconvolution for polynomial or causal filters to general FIR filters. The proposed algorithms obtain a set of FIR deconvolution filters with a small number of nonzero coefficients (a desirable feature in the impulsive noise environment) and do not require the prior information of the support. Moreover, we provide a complete characterization of all exact deconvolution FIR filters, from which good FIR deconvolution filters under the additive white noise environment are found. Simulation results show that our approaches achieve good results under different noise settings

Abstract: A multichannel blind deconvolution algorithm that incorporates the maximum-likelihood image restoration by several estimates of the differently blurred point-spread function (PSF) into the Ayers-Dainty iterative algorithm is proposed. The algorithm uses no restrictions on the image and the PSFs except for the assumption that they are positive. The algorithm employs no cost functions, input parameters, a priori probability distributions, or the analytically specified transfer functions. The iterative algorithm permits its application in the presence of different kinds of distortion. The work presents results of digital modeling and the results of processing real telescope data from several satellites. The proof of convergence of the algorithm to the positive estimates of object and the PSFs is given. The convergence of the Ayers-Dainty algorithm with a single processed frame is not obvious in the general case; therefore it is useful to have confidence in its convergence in a multiframe case. The dependence of convergence on the number of processed frames is discussed. Formulas for evaluating the quality of the algorithm performance on each iteration and the rule of stopping its work in accordance with this quality are proposed. A method of building the monotonically converging subsequence of the image estimates of all the images obtained in the iterative process is also proposed.

Abstract: Our contribution deals with blind deconvolution of sparse spike trains. More precisely, we examine the problem in the Markov chain Monte-Carlo (MCMC) framework, where the unknown spike train is modeled as a Bernoulli-Gaussian process. In this context, we point out that time-shift and scale ambiguities jeopardize the robustness of basic MCMC methods, in quite a similar manner to the label switching effect studied by Stephens (2000) in mixture model identification. Finally, we propose proper modifications of the MCMC approach, in the same spirit as Stephens' contribution.

Abstract: In this paper we present a novel method for multiframe blind deblurring of noisy images. It is based on minimization of the energy criterion produced in the frequency domain using a recursive gradient-projection algorithm. For filtering and regularization we use the local polynomial approximation (LPA) of both the image and blur operators, and paradigm of the intersection of confidence intervals (ICI) applied for selection adaptively varying scales (window sizes) of LPA. The LPA-ICI algorithm is nonlinear and spatially-adaptive with respect to the smoothness and irregularities of the image and blur operators. Simulation experiments demonstrate efficiency and good performance of the proposed deconvolution technique.

Abstract: Performances of iterative blind deconvolution methods for motion-blurred images are usually reduced depending on the accuracy of the required initial guess of the blur. We examine this dependency, and a two-stage restoration procedure is proposed: First we perform a direct technique with a single straight-forward process to produce a rough initial estimate of the blur, and then an iterative technique is employed to refine the blur estimate. Two common iterative techniques (the expectation-maximization and the Richardson-Lucy methods) are examined here and implemented in the combined direct-iterative modification for a variety of motion blur types. Results show that the combined method significantly improves the reliability of the deconvolution process.