Perfect blind restoration of images blurred by multiple filters: theory and efficient algorithms

A Deconvolutional Deblurring Algorithm Based on Dual-Channel Images

Aiming at the motion blur restoration of large-scale dual-channel space-variant images, this paper proposes a dual-channel image deblurring method based on the idea of block aggregation, by studying imaging principles and existing algorithms. The study first analyzed the model of dual-channel space-variant imaging, reconstructed the kernel estimation process using the side prior information from the correlation of the two-channel images, and then used a clustering algorithm to classify kernels and restore the images. In the kernel estimation process, the study proposed two kinds of regularization terms. One is based on image correlation, and the other is based on the information from another channel input. In the image restoration process, the mean-shift clustering algorithm was used to calculate the block image kernel weights and reconstruct the final restored image according to the weights. As the experimental section shows, the restoration effect of this algorithm was better than that of the other compared algorithms.

Restoration of fast moving objects

If an object is photographed at motion in front of a static background, the object will be blurred while the background sharp and partially occluded by the object. The goal is to recover the object appearance from such blurred image. We adopt the image formation model for fast moving objects and consider objects undergoing 2D translation and rotation. For this scenario we formulate the estimation of the object shape, appearance, and motion from a single image and known background as a constrained optimization problem with appropriate regularization terms. Both similarities and differences with blind deconvolution are discussed with the latter caused mainly by the coupling of the object appearance and shape in the acquisition model. Necessary conditions for solution uniqueness are derived and a numerical solution based on the alternating direction method of multipliers is presented. The proposed method is evaluated on a new dataset.

Robust, blind multichannel image identification and restoration using stack decoder

In this study, the authors introduce new solutions and improvements to the multi‐channel blind image deconvolution problem. More precisely, authors’ contributions are threefold: (i) At first, a simplified version of the existing cross‐relation method for blind system identification is proposed; but most importantly, the authors incorporate into the channel estimation cost function a sparsity constraint to deal with the challenging issue of channel order overestimation errors; (ii) then, once the channel identification is achieved, a new image restoration method based on the stack decoding algorithm is introduced; and (iii) finally, a refining approach using an ‘all‐at‐once’ optimisation technique with an improved mixed norm regularisation is considered. The performance of the proposed approach was evaluated using several numerical simulations. Blind system identification and image restoration tasks were evaluated with respect to several criteria: numerical complexity, robustness to noise effects and channel order estimation. The results obtained are promising and highlight the effectiveness of the proposed approach.

Multichannel myopic deconvolution in underwater acoustic channels via low-rank recovery

This paper presents a technique for solving the multichannel blind deconvolution problem. The authors observe the convolution of a single (unknown) source with K different (unknown) channel responses; from these channel outputs, the authors want to estimate both the source and the channel responses. The authors show how this classical signal processing problem can be viewed as solving a system of bilinear equations, and in turn can be recast as recovering a rank-1 matrix from a set of linear observations. Results of prior studies in the area of low-rank matrix recovery have identified effective convex relaxations for problems of this type and efficient, scalable heuristic solvers that enable these techniques to work with thousands of unknown variables. The authors show how a priori information about the channels can be used to build a linear model for the channels, which in turn makes solving these systems of equations well-posed. This study demonstrates the robustness of this methodology to measurement noises and parametrization errors of the channel impulse responses with several stylized and shallow water acoustic channel simulations. The performance of this methodology is also verified experimentally using shipping noise recorded on short bottom-mounted vertical line arrays.

Scattering correction through a space-variant blind deconvolution algorithm

Scattering within biological samples limits the imaging depth and the resolution in microscopy. We present a prior and regularization approach for blind deconvolution algorithms to correct the influence of scattering to increase the imaging depth and resolution. The effect of the prior is demonstrated on a three-dimensional image stack of a zebrafish embryo captured with a selective plane illumination microscope. Blind deconvolution algorithms model the recorded image as a convolution between the distribution of fluorophores and a point spread function (PSF). Our prior uses image information from adjacent z-planes to estimate the unknown blur in tissue. The increased size of the PSF due to the cascading effect of scattering in deeper tissue is accounted for by a depth adaptive regularizer model. In a zebrafish sample, we were able to extend the point in depth, where scattering has a significant effect on the image quality by around 30???m.

A subspace-based coil combination method for phased-array magnetic resonance imaging

PURPOSE

Coil-by-coil reconstruction methods are followed by coil combination to obtain a single image representing a spin density map. Typical coil combination methods, such as square-root sum-of-squares and adaptive coil combining, yield images that exhibit spatially varying modulation of image intensity. Existing practice is to first combine coils according to a signal-to-noise criterion, then postprocess to correct intensity inhomogeneity. If inhomogeneity is severe, however, intensity correction methods can yield poor results. The purpose of this article is to present an alternative optimality criterion for coil combination; the resulting procedure yields reduced intensity inhomogeneity while preserving contrast.

THEORY AND METHODS

A minimum mean squared error criterion is adopted for combining coils via a subspace decomposition. Techniques are compared using both simulated and in vivo data.

RESULTS

Experimental results for simulated and in vivo data demonstrate lower bias, higher signal-to-noise ratio (about 7×) and contrast-to-noise ratio (about 2×), compared to existing coil combination techniques.

CONCLUSION

The proposed coil combination method is noniterative and does not require estimation of coil sensitivity maps or image mask; the method is particularly suited to cases where intensity inhomogeneity is too severe for existing approaches.

ESPIRiT--an eigenvalue approach to autocalibrating parallel MRI: where SENSE meets GRAPPA

PURPOSE

Parallel imaging allows the reconstruction of images from undersampled multicoil data. The two main approaches are: SENSE, which explicitly uses coil sensitivities, and GRAPPA, which makes use of learned correlations in k-space. The purpose of this work is to clarify their relationship and to develop and evaluate an improved algorithm.

THEORY AND METHODS

A theoretical analysis shows: (1) The correlations in k-space are encoded in the null space of a calibration matrix. (2) Both approaches restrict the solution to a subspace spanned by the sensitivities. (3) The sensitivities appear as the main eigenvector of a reconstruction operator computed from the null space. The basic assumptions and the quality of the sensitivity maps are evaluated in experimental examples. The appearance of additional eigenvectors motivates an extended SENSE reconstruction with multiple maps, which is compared to existing methods.

RESULTS

The existence of a null space and the high quality of the extracted sensitivities are confirmed. The extended reconstruction combines all advantages of SENSE with robustness to certain errors similar to GRAPPA.

CONCLUSION

In this article the gap between both approaches is finally bridged. A new autocalibration technique combines the benefits of both.

Image reconstruction from phased-array data based on multichannel blind deconvolution

In this paper we consider image reconstruction from fully sampled multichannel phased array MRI data without knowledge of the coil sensitivities. To overcome the non-uniformity of the conventional sum-of-square reconstruction, a new framework based on multichannel blind deconvolution (MBD) is developed for joint estimation of the image function and the sensitivity functions in image domain. The proposed approach addresses the non-uniqueness of the MBD problem by exploiting the smoothness of both functions in the image domain through regularization. Results using simulation, phantom and in vivo experiments demonstrate that the reconstructions by the proposed algorithm are more uniform than those by the existing methods.

High-dimensional camera shake removal with given depth map

Camera motion blur is drastically nonuniform for large depth-range scenes, and the nonuniformity caused by camera translation is depth dependent but not the case for camera rotations. To restore the blurry images of large-depth-range scenes deteriorated by arbitrary camera motion, we build an image blur model considering 6-degrees of freedom (DoF) of camera motion with a given scene depth map. To make this 6D depth-aware model tractable, we propose a novel parametrization strategy to reduce the number of variables and an effective method to estimate high-dimensional camera motion as well. The number of variables is reduced by temporal sampling motion function, which describes the 6-DoF camera motion by sampling the camera trajectory uniformly in time domain. To effectively estimate the high-dimensional camera motion parameters, we construct the probabilistic motion density function (PMDF) to describe the probability distribution of camera poses during exposure, and apply it as a unified constraint to guide the convergence of the iterative deblurring algorithm. Specifically, PMDF is computed through a back projection from 2D local blur kernels to 6D camera motion parameter space and robust voting. We conduct a series of experiments on both synthetic and real captured data, and validate that our method achieves better performance than existing uniform methods and nonuniform methods on large-depth-range scenes.

Unified blind method for multi-image super-resolution and single/multi-image blur deconvolution

This paper presents, for the first time, a unified blind method for multi-image super-resolution (MISR or SR), single-image blur deconvolution (SIBD), and multi-image blur deconvolution (MIBD) of low-resolution (LR) images degraded by linear space-invariant (LSI) blur, aliasing, and additive white Gaussian noise (AWGN). The proposed approach is based on alternating minimization (AM) of a new cost function with respect to the unknown high-resolution (HR) image and blurs. The regularization term for the HR image is based upon the Huber-Markov random field (HMRF) model, which is a type of variational integral that exploits the piecewise smooth nature of the HR image. The blur estimation process is supported by an edge-emphasizing smoothing operation, which improves the quality of blur estimates by enhancing strong soft edges toward step edges, while filtering out weak structures. The parameters are updated gradually so that the number of salient edges used for blur estimation increases at each iteration. For better performance, the blur estimation is done in the filter domain rather than the pixel domain, i.e., using the gradients of the LR and HR images. The regularization term for the blur is Gaussian (L2 norm), which allows for fast noniterative optimization in the frequency domain. We accelerate the processing time of SR reconstruction by separating the upsampling and registration processes from the optimization procedure. Simulation results on both synthetic and real-life images (from a novel computational imager) confirm the robustness and effectiveness of the proposed method.

Robust multichannel blind deconvolution via fast alternating minimization

Blind deconvolution, which comprises simultaneous blur and image estimations, is a strongly ill-posed problem. It is by now well known that if multiple images of the same scene are acquired, this multichannel (MC) blind deconvolution problem is better posed and allows blur estimation directly from the degraded images. We improve the MC idea by adding robustness to noise and stability in the case of large blurs or if the blur size is vastly overestimated. We formulate blind deconvolution as an l(1) -regularized optimization problem and seek a solution by alternately optimizing with respect to the image and with respect to blurs. Each optimization step is converted to a constrained problem by variable splitting and then is addressed with an augmented Lagrangian method, which permits simple and fast implementation in the Fourier domain. The rapid convergence of the proposed method is illustrated on synthetically blurred data. Applicability is also demonstrated on the deconvolution of real photos taken by a digital camera.

Commutability of blur and affine warping in super-resolution with application to joint estimation of triple-coupled variables

This paper proposes a new approach to the image blind super-resolution (BSR) problem in the case of affine interframe motion. Although the tasks of image registration, blur identification, and high-resolution (HR) image reconstruction are coupled in the imaging process, when dealing with nonisometric interframe motion or without the exact knowledge of the blurring process, classic SR techniques generally have to tackle them (maybe in some combinations) separately. The main difficulty is that state-of-the-art deconvolution methods cannot be straightforwardly generalized to cope with the space-variant motion. We prove that the operators of affine warping and blur commute with some additional transforms and derive an equivalent form of the BSR observation model. Using this equivalent form, we develop an iterative algorithm to jointly estimate the triple-coupled variables, i.e., the motion parameters, blur kernels, and HR image. Experiments on synthetic and real-life images illustrate the performance of the proposed technique in modeling the space-variant degradation process and restoring local textures.

Fourier-domain multichannel autofocus for synthetic aperture radar

Synthetic aperture radar (SAR) imaging suffers from image focus degradation in the presence of phase errors in the received signal due to unknown platform motion or signal propagation delays. We present a new autofocus algorithm, termed Fourier-domain multichannel autofocus (FMCA), that is derived under a linear algebraic framework, allowing the SAR image to be focused in a noniterative fashion. Motivated by the mutichannel autofocus (MCA) approach, the proposed autofocus algorithm invokes the assumption of a low-return region, which generally is provided within the antenna sidelobes. Unlike MCA, FMCA works with the collected polar Fourier data directly and is capable of accommodating wide-angle monostatic SAR and bistatic SAR scenarios. Most previous SAR autofocus algorithms rely on the prior assumption that radar's range of look angles is small so that the phase errors can be modeled as varying along only one dimension in the collected Fourier data. And, in some cases, implicit assumptions are made regarding the SAR scene. Performance of such autofocus algorithms degrades if the assumptions are not satisfied. The proposed algorithm has the advantage that it does not require prior assumptions about the range of look angles, nor characteristics of the scene.

Classification-based adaptive filtering for multiframe blind image restoration

In this paper, the blind restoration of a scene is investigated, when multiple degraded (blurred and noisy) acquisitions are available. An adaptive filtering technique is proposed, where the distorted images are filtered, classified and then fused based upon the classification decisions. Finite normal-density mixture (FNM) models are used to model the filtered outputs at each iteration. For simplicity, fixed number of Gaussian components (classes) is, initially, considered for each degraded frame and the selection of the optimal number of classes is performed according to the global relative entropy criterion. However, there exist cases where dynamically varying FNM models should be considered, where the optimal number of classes is selected according to the Akaike information criterion. The iterative application of classification and fusion, followed by optimal adaptive filtering, converges to a global enhanced representation of the original scene in only a few iterations. The proposed restoration method does not require knowledge of the point-spread-function support size or exact alignment of the acquired frames. Simulation results on synthetic and real data, using both fixed and dynamically varying FNM models, demonstrate its efficiency under both noisy and noise-free conditions.

Fast splitting algorithm for multiframe total variation blind video deconvolution

We consider the recovery of degraded videos without complete knowledge about the degradation. A spatially shift-invariant but temporally shift-varying video formation model is used. This leads to a simple multiframe degradation model that relates each original video frame with multiple observed frames and point spread functions (PSFs). We propose a variational method that simultaneously reconstructs each video frame and the associated PSFs from the corresponding observed frames. Total variation (TV) regularization is used on both the video frames and the PSFs to further reduce the ill-posedness and to better preserve edges. In order to make TV minimization practical for video sequences, we propose an efficient splitting method that generalizes some recent fast single-image TV minimization methods to the multiframe case. Both synthetic and real videos are used to show the performance of the proposed method.

A new look to multichannel blind image deconvolution

The aim of this paper is to propose a new look to MBID, examine some known approaches, and provide a new MC method for restoring blurred and noisy images. First, the direct image restoration problem is briefly revisited. Then a new method based on inverse filtering for perfect image restoration in the noiseless case is proposed. The noisy case is addressed by introducing a regularization term into the objective function in order to avoid noise amplification. Second, the filter identification problem is considered in the MC context. A new robust solution to estimate the degradation matrix filter is then derived and used in conjunction with a total variation approach to restore the original image. Simulation results and performance evaluations using recent image quality metrics are provided to assess the effectiveness of the proposed methods.

MCA: a multichannel approach to SAR autofocus

We present a new noniterative approach to synthetic aperture radar (SAR) autofocus, termed the multichannel autofocus (MCA) algorithm. The key in the approach is to exploit the multichannel redundancy of the defocusing operation to create a linear subspace, where the unknown perfectly focused image resides, expressed in terms of a known basis formed from the given defocused image. A unique solution for the perfectly focused image is then directly determined through a linear algebraic formulation by invoking an additional image support condition. The MCA approach is found to be computationally efficient and robust and does not require prior assumptions about the SAR scene used in existing methods. In addition, the vector-space formulation of MCA allows sharpness metric optimization to be easily incorporated within the restoration framework as a regularization term. We present experimental results characterizing the performance of MCA in comparison with conventional autofocus methods and discuss the practical implementation of the technique.

Single channel exact 3-D blind image deconvolution from cylindrically symmetric blur kernel

The rotational symmetry of point spread function (PSF) is common for many imaging systems such as optical microscopes, cameras, astigmatism corrected electron microscopes, and etc. In 2-D deconvolution problem, we showed that an exact PSF estimation from a single measured image on a flat background is possible by exploiting the circular symmetry. This paper extends the idea to 3-D blind deconvolution problem. More specifically, we show that the exact recovery of 3-D PSF is possible from a single set of z-stack images if the PSF of the microscope has a cylindrical symmetry and there exists enough working distance to cover the volumetric sample from top to bottom. The resulting algorithm allows an accurate computational optical sectioning of biological specimen from z-stack images using brightfield or fluorescence microscopes without separate PSF measurements. Experimental results confirm our theory.

Space-variant restoration of images degraded by camera motion blur

We examine the problem of restoration from multiple images degraded by camera motion blur. We consider scenes with significant depth variations resulting in space-variant blur. The proposed algorithm can be applied if the camera moves along an arbitrary curve parallel to the image plane, without any rotations. The knowledge of camera trajectory and camera parameters is not necessary. At the input, the user selects a region where depth variations are negligible. The algorithm belongs to the group of variational methods that estimate simultaneously a sharp image and a depth map, based on the minimization of a cost functional. To initialize the minimization, it uses an auxiliary window-based depth estimation algorithm. Feasibility of the algorithm is demonstrated by three experiments with real images.

Texas two-step: a framework for optimal multi-input single-output deconvolution

Multi-input single-output deconvolution (MISO-D) aims to extract a deblurred estimate of a target signal from several blurred and noisy observations. This paper develops a new two step framework--Texas Two-Step--to solve MISO-D problems with known blurs. Texas Two-Step first reduces the MISO-D problem to a related single-input single-output deconvolution (SISO-D) problem by invoking the concept of sufficient statistics (SSs) and then solves the simpler SISO-D problem using an appropriate technique. The two-step framework enables new MISO-D techniques (both optimal and suboptimal) based on the rich suite of existing SISO-D techniques. In fact, the properties of SSs imply that a MISO-D algorithm is mean-squared-error optimal if and only if it can be rearranged to conform to the Texas Two-Step framework. Using this insight, we construct new wavelet- and curvelet-based MISO-D algorithms with asymptotically optimal performance. Simulated and real data experiments verify that the framework is indeed effective.

A unified approach to superresolution and multichannel blind deconvolution

This paper presents a new approach to the blind deconvolution and superresolution problem of multiple degraded low-resolution frames of the original scene. We do not assume any prior information about the shape of degradation blurs. The proposed approach consists of building a regularized energy function and minimizing it with respect to the original image and blurs, where regularization is carried out in both the image and blur domains. The image regularization based on variational principles maintains stable performance under severe noise corruption. The blur regularization guarantees consistency of the solution by exploiting differences among the acquired low-resolution images. Several experiments on synthetic and real data illustrate the robustness and utilization of the proposed technique in real applications.

Multidimensional multichannel FIR deconvolution using Gröbner bases

We present a new method for general multidimensional multichannel deconvolution with finite impulse response (FIR) convolution and deconvolution filters using Gröbner 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 Gröbner 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 Gröbner 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.

Multiframe blind deconvolution of heavily blurred astronomical images

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.

Blind multichannel reconstruction of high-resolution images using wavelet fusion

We developed an approach to the blind multichannel reconstruction of high-resolution images. This approach is based on breaking the image reconstruction problem into three consecutive steps: a blind multichannel restoration, a wavelet-based image fusion, and a maximum entropy image interpolation. The blind restoration step depends on estimating the two-dimensional (2-D) greatest common divisor (GCD) between each observation and a combinational image generated by a weighted averaging process of the available observations. The purpose of generating this combinational image is to get a new image with a higher signal-to-noise ratio and a blurring operator that is a coprime with all the blurring operators of the available observations. The 2-D GCD is then estimated between the new image and each observation, and thus the effect of noise on the estimation process is reduced. The multiple outputs of the restoration step are then applied to the image fusion step, which is based on wavelets. The objective of this step is to integrate the data obtained from each observation into a single image, which is then interpolated to give an enhanced resolution image. A maximum entropy algorithm is derived and used in interpolating the resulting image from the fusion step. Results show that the suggested blind image reconstruction approach succeeds in estimating a high-resolution image from noisy blurred observations in the case of relatively coprime unknown blurring operators. The required computation time of the suggested approach is moderate.

Multichannel blind deconvolution of spatially misaligned images

Existing multichannel blind restoration techniques assume perfect spatial alignment of channels, correct estimation of blur size, and are prone to noise. We developed an alternating minimization scheme based on a maximum a posteriori estimation with a priori distribution of blurs derived from the multichannel framework and a priori distribution of original images defined by the variational integral. This stochastic approach enables us to recover the blurs and the original image from channels severely corrupted by noise. We observe that the exact knowledge of the blur size is not necessary, and we prove that translation misregistration up to a certain extent can be automatically removed in the restoration process.

Multivariate MIMO FIR inverses

The problem of computing exact finite impulse response (FIR) inverses for multivariate multiple-input multiple-output (MIMO) FIR systems is considered. Necessary and sufficient conditions for invertibility are given, along with computation techniques. Random systems and structured systems are defined. Necessary and sufficient conditions for the almost sure invertibility of structured random systems are derived. Bounds on the orders of the inverse filters are computed.

Multichannel blind iterative image restoration

Blind image deconvolution is required in many applications of microscopy imaging, remote sensing, and astronomical imaging. Unfortunately in a single-channel framework, serious conceptual and numerical problems are often encountered. Very recently, an eigenvector-based method (EVAM) was proposed for a multichannel framework which determines perfectly convolution masks in a noise-free environment if channel disparity, called co-primeness, is satisfied. We propose a novel iterative algorithm based on recent anisotropic denoising techniques of total variation and a Mumford-Shah functional with the EVAM restoration condition included. A linearization scheme of half-quadratic regularization together with a cell-centered finite difference discretization scheme is used in the algorithm and provides a unified approach to the solution of total variation or Mumford-Shah. The algorithm performs well even on very noisy images and does not require an exact estimation of mask orders. We demonstrate capabilities of the algorithm on synthetic data. Finally, the algorithm is applied to defocused images taken with a digital camera and to data from astronomical ground-based observations of the Sun.

Multichannel blind image deconvolution using the Bussgang algorithm: spatial and multiresolution approaches

This work extends the Bussgang blind equalization algorithm to the multichannel case with application to image deconvolution problems. We address the restoration of images with poor spatial correlation as well as strongly correlated (natural) images. The spatial nonlinearity employed in the final estimation step of the Bussgang algorithm is developed according to the minimum mean square error criterion in the case of spatially uncorrelated images. For spatially correlated images, the nonlinearity design is rather conducted using a particular wavelet decomposition that, detecting lines, edges, and higher order structures, carries out a task analogous to those of the (preattentive) stage of the human visual system. Experimental results pertaining to restoration of motion blurred text images, out-of-focus spiky images, and blurred natural images are reported.

On eigenstructure-based direct multichannel blind image restoration

Existing eigenstructure-based direct multichannel blind image restoration techniques include nullspace-based and direct deconvolver estimation techniques. The nullspace-based approach can be formulated as an optimization problem. We show that this formulation implies a new subspace-based approach that uses matrix operations. This new approach has the same advantages as the nullspace-based one but requires less computational complexity. Under some mild conditions, its complexity is equal to that of the FFT. Furthermore, the relation among the nullspace-based approach, the direct deconvolver estimation and the new subspace-based approach is studied.

Efficient generalized cross-validation with applications to parametric image restoration and resolution enhancement

In many image restoration/resolution enhancement applications, the blurring process, i.e., point spread function (PSF) of the imaging system, is not known or is known only to within a set of parameters. We estimate these PSF parameters for this ill-posed class of inverse problem from raw data, along with the regularization parameters required to stabilize the solution, using the generalized cross-validation method (GCV). We propose efficient approximation techniques based on the Lanczos algorithm and Gauss quadrature theory, reducing the computational complexity of the GCV. Data-driven PSF and regularization parameter estimation experiments with synthetic and real image sequences are presented to demonstrate the effectiveness and robustness of our method.

Exact image deconvolution from multiple FIR blurs

We address the problem of restoring an image from its noisy convolutions with two or more blur functions (channels). Deconvolution from multiple blurs is, in general, better conditioned than from a single blur, and can be performed without regularization for moderate noise levels. We characterize the problem of missing data at the image boundaries, and show that perfect reconstruction is impossible (even in the no-noise case) almost surely unless there are at least three channels. Conversely, when there are at least three channels, we show that perfect reconstruction is not only possible almost surely in the absence of noise, but also that it can be accomplished by finite impulse response (FIR) filtering. Such FIR reconstruction is vastly more efficient computationally than the least-squares solution, and is suitable for low noise levels. Even in the high-noise case, the estimates obtained by FIR filtering provide useful starting points for iterative least-squares algorithms. We present results on the minimum possible sizes of such deconvolver filters. We derive expressions for the mean-square errors in the FIR reconstructions, and show that performance comparable to that of the least-squares reconstruction may be obtained with relatively small deconvolver filters. Finally, we demonstrate the FIR reconstruction on synthetic and real data.