This PDF file contains the editorial “Guest Editorial: Elaine Cherry's Legacy to All Who Foster Excellence” for OE Vol. 29 Issue 05

This PDF file contains the editorial “Book Rvw: Practical Laser Safety, Second Edition," by D.C. Winburn for OE Vol. 29 Issue 05

This PDF file contains the editorial “Book Rvw: The Fabry-Perot Interferometer: History, Theory, Practice andApplications” for OE Vol. 29 Issue 05

We present a tutorial review of recent developments in restoring images that are degraded by both blur and noise. We consider three fundamental aspects of digital image restoration: modeling, identification algorithms, and restoration algorithms. An overview of modeling the degradations, and certain properties of images are given first. We then survey the methods that identify these models. Image restoration algorithms are surveyed in two categories: general algorithms and specialized algorithms. We briefly discuss present and future research topics in the field. Our emphasis here is on fundamental concepts and ideas rather than mathematical details.

When a photographic imaging system has a focus error or a motion blur, it is not possible to correct the error by conventional optical printing. Digital image processing offers a chance to restore the blurred images by reversing the blurring processes under certain constraints. Although restoration algorithms vary in detail, they all need a model of the image-blurring process. This paper reviews image-blur models of each individual component in an image chain consisting of camera, film, and scanner. The models include image blur caused by focus error, camera shutter, object motion, light scattering in the film emulsion, film interimage effect, scanner optics, and scanner aperture. The emphasis is on mathematical simplicity and practical usefulness.

A number of different algorithms have recently been proposed to identify the image and blur model parameters from an image that is degraded by blur and noise. This paper gives an overview of the developments in image and blur identification under a unifying maximum likelihood framework. In fact, we show that various recently published image and blur identification algorithms are different implementations of the same maximum likelihood estimator resulting from different modeling assumptions and/or considerations about the computational complexity. The use of the maximum likelihood estimation in image and blur identification is illustrated by numerical examples.

In this paper, the problem of identifying the image and blur parameters and restoring a noisy blurred image is addressed. Specifying the blurring process by its point spread function (PSF), the blur identification problem is formulated as the maximum likelihood estimation (MLE) of the PSF. Modeling the original image and the additive noise as zeromean Gaussian processes, the MLE of their covariance matrices is also computed. An iterative approach, called the EM (expectation-maximization) algorithm, is used to find the maximum likelihood estimates ofthe relevant unknown parameters. In applying the EM algorithm, the original image is chosen to be part of the complete data; its estimate is computed in the E-step of the EM iterations and represents the restored image. Two algorithms for identification/restoration, based on two different choices of complete data, are derived and compared. Simultaneous blur identification and restoration is performed by the first algorithm, while the identification of the blur results from a separate minimization in the second algorithm. Experiments with simulated and photographically blurred images are shown.

Regularization is an effective method for obtaining satisfactory solutions to image restoration problems. The application of regularization necessitates a choice of the regularization parameter as well as the stabilizing functional. For most problems of interest, the best choices are not known a priori. We present a method for obtaining optimal estimates of the regularization parameter and stabilizing functional directly from the degraded image data. The method of generalized cross-validation (GCV) is used to generate the estimates. Implementation of GCV requires the computation of the system eigenvalues. Certain assumptions are made regarding the structure of the degradation so that the GCV criterion can be implemented efficiently. Furthermore, the assumptions on the matrix structure allow the regularization operator eigenvalues to be expressed as simple parametric functions. By choosing an appropriate structure for the regularization operator, we use the GCV criterion to estimate optimal parameters of the regularization operator and thus the stabilizing functional. Experimental results are presented that show the ability of GCV to give extremely reliable estimates for the regularization parameter and operator. By allowing both the degree and the manner of smoothing to be determined from the data, GCV-based regularization yields solutions that would otherwise be unattainable without a priori information.

Most image formation systems involve a built-in nonlinear transformation.Furthermore, many image processing applications involve longtailednoise processes, which introduce outliers in the gray-level distributionof the image. The performance of conventional restoration algorithmsis highly degraded by such noise processes. A robust-design approachaims to eliminate the effects of outliers in the performance of thealgorithm. The concept of M-estimation leads to robust estimators, which,however, require iterative evaluation. Such estimators have been employedin recursive image restoration, mainly for the robust parameteridentification. In this paper, the exploitation of M-estimators in iterativenonlinear restoration algorithms is considered. These algorithms are designedto optimize the robust least-squares-error criterion, which limitsthe effects of large residuals through an influence function. The robustversion of the steepest-descent (SD) algorithm is introduced. This algorithmovercomes the linear search required by the basic SD techniquethrough a particular procedure for the selection of the iteration parameter.Based on the convergence analysis of this algorithm a new influencefunction, the redescending nonzero function, is introduced. Structuralmodifications of the robust SD technique result in two hybrid SD algorithms,which achieve fast convergence without degrading the quality ofthe final estimate. The algorithms introduced can be easily implementedthrough Fourier transform techniques. The convergence properties of therobust algorithms are rigorously studied through their singular-valueanalysis and the global convergence theorem. The application of thesealgorithms on image restoration problems is demonstrated through examples.

Every iterative restoration algorithm requires some performance measure to determine the termination point for the iteration. Typically, the termination rule is based on a measure of the residual or the change in the solution from one iteration to the next. A better rule would be one that terminates the iteration when the distance between the original undistorted image and restored image is minimized in some sense. In this paper, we present a performance measure that estimates this distance without prior knowledge of the original image. Our measure relies on evaluating a spectral filter function at each step of the iteration. These functions describe the solution at each step in terms of the singular value decomposition of the system matrix. As such, spectral filter functions provide valuable insight into the behavior of an iteration as well as a means of defining a termination rule. We develop a general technique for determining the spectral filter functions for a given iteration, which we demonstrate by applying it to a linear iterative image restoration algorithm.

A nonlinear adaptive filtering approach is applied to the problem of estimating images corrupted by multiplicative noise. With a pointwise digital model of multiplicative noise, the problem of obtaining a minimum mean squared error estimate of a true image from an observed noisy image and the statistical properties of the degrading noise field is posed. The approach presented in this paper is to perform image estimation with locally adaptive estimating operators that display the property of multiplicative superposition. Optimal point and multipoint adaptive estimating operators are formulated. Image processing algorithms are developed based on the calculation of local statistics estimates, Taylor series approximations to optimal formulas, and the normalization of data by local multiplicative sample means. Both adaptive and edge-adaptive implementations are investigated. The image processing results of single-point and two-point algorithms are given; the resulting images are noise-smoothed both visually and in mean squared error. The image processing algorithms are compared with other known methods of multiplicative filtering and estimation such as adaptive homomorphic point filtering.

A new estimation criterion called the minimum-error minimum correlation (MEMC) criterion is implemented in conjunction with an adaptive windowing technique. The image statistics are calculated within the window, the dimensions of which vary depending on the local statistics. The MEMC estimator produces sharper and hence visually more pleasing restorations, while the adaptive windows tend to isolate regions of the image that are locally stationary. Since most of the error after restoration is in the vicinity of edges, a postprocessing step of filtering along the edges is applied. These image restorations are compared to both fixed and adaptive minimum mean squared error estimators.

Color images are two-dimensional, three-channel stochastic signals. Their statistical characterization is quite different from that of black and white images (single-channel, two-dimensional signals). This paper investigates the extension of the concept of ordering and order statistics filtering to multichannel and color images. Marginal order statistics is the most straightforward extension to multichannel filtering. Marginal median filtering is the same as median filtering each channel independently. However, its statistical analysis is not a simple extension of the analysis of the single-channel case. The statistical analysis of marginal order filtering for color images is provided.

Techniques are developed for signal reconstruction from sampIes of the bispectrum of a 1-D or 2-D signal. The phase and magnitude of the Fourier transform of a signal as well as those of the bispectrum are expanded into a series using a set of appropriate basis functions. Reconstruction is achieved by equating coefficients of like terms of these two expansions. The algorithms derive their speed by using a restricted portion of the bispectrum space and by being based on the use of the fast Fourier transform. In addition to detailed theoretical derivations, results of computer simulation are provided.

Iterative data refinement (IDR) is a general procedure for estimating data that would have been collected by an ideal measuring device from data that were collected by an actual measuring device. We show that IDR is general enough to encompass such well-accepted image recovery methods as the Gerchberg-Saxton algorithm and the error reduction and hybrid input-output methods of Fienup. The generalizations provided by IDR give new insights into the nature of such algorithms and, in particular, allow us to introduce the notion of relaxation into them. Along similar lines, DR provides a common framework within which new algorithms can be developed for improved magnetic resonance imaging (MRI). We apply the approach of IDR to a specific problem in MRI, namely, to the correction of spatially dependent blurs due to transverse relaxation. The algorithm is designed to reconstruct T2-weighted spin density images with improved spatial resolution. The practical computational significance of using the IDR approach is illustrated by the reconstruction of mathematical phantoms. We find that overrelaxation of the algorithm can improve computational speed by up to a factor of five.

Cone-beam tomography is the science of forming images by inverting three-dimensional divergent cone-beam ray-sum data sets. The impetus for its application is its three-dimensional data collection abilities, which result in (1) significant reduction in the time needed to collect a sufficient number of data to produce a three-dimensional image and (2) elimination of the inaccuracy due to misalignment of cross sectional images. On the other hand, the divergence of cone-beam data has hindered its application. Because of the divergence, the theory that has been developed for fan-beam and parallel two- and three-dimensional tomography does not provide a totally adequate means for analyzing or inverting cone-beam data. Consider the following: In practice, as the data are collected, the vertex of the cone is movedalong some path about the object. Which paths, if any, provide enough information to make an inversion possible? Suppose by some means enough information has been obtained. How does one derive an exact formula for inverting this data? To answer these questions a new theory that takes into account the three dimensional divergence of cone-beam data needs to be developed. In 1985, a paper was published in which several advances in the theory of cone-beam tomography were made. A tutorial review of the results given in the 1985 paper [B. D. Smith, "Image reconstruction from cone-beam projections: necessary and sufficient conditions and reconstruction methods," IEEE Trans. Med Imag. MI-4, 14-28 (1985)] will be given here. This review will include the advances that have been made since that time. Additionally, a brief review of the contributions made by a number of other researchers will be given.

Tomographic reconstruction from incomplete data is required in many fields, including medical imaging, sonar, and radar. In this paper, we present a new reconstruction algorithm for limited-angle tomography, a problem that occurs when projections are missing over a range of angles. The approach uses a variational formulation that incorporates the Ludwig- Helgason consistency conditions, measurement noise statistics, and a sinogram smoothness condition. Optimal restored sinograms, therefore, satisfy an associated Euler-Lagrange partial differential equation, which we solve on a lattice using a primal-dual optimization procedure. Object estimates are then reconstructed using convolution backprojection applied to the restored sinogram. We present results of simulations that illustrate the performance of the algorithm and discuss directions for further research.

This paper addresses the problem of imaging from (inverting) the data collected by a physical or synthesized array. A scattering model describing the source/object interaction is presented. The inversion based on this model is a computationally manageable method that incorporates the radiation pattern of each element on the array at the transmit and receive modes. This study indicates that a physical array and its synthesized counterpart possess the same resolution despite the fact that the synthesized array's signal subspace is a subset of the much larger signal subspace for the physical array.

The Fourier transform is the standard image reconstruction technique used in magnetic resonance imaging (MRI), and it is an integral part of the inverse scattering formalism in ultrasound (US) imaging. Unfortunately, artifacts such as Gibbs ringing induced by a finite sampling window or systematic errors in phase may significantly impede the interpretation of the resulting Fourier transform images. Further, when only a few parameters are needed to characterize the object function, it is likely not to be the best technique for optimal signal-to-noise. In this paper, the application of a parameter estimation reconstruction scheme using a priori constraints to remove Gibbs ringing and improve resolution and signalto- noise is presented for MRI and US. Projection onto convex set theory is also used to regenerate uncollected data in partial Fourier imaging, and model constraints are used to correct motion artifacts in MRI. These methods are found not to require unrealistic values of the signal-to-noise ratio and are likely to prove practical in future applications.

The quality of the images from positron emission tomography (PET) is regularly degraded by radial streak artifacts. In many cases this is attributed to data undersampling and the inexactitude of the models used in the image reconstruction. Because this noise cannot be easily isolated in either spatial or spectral domains, classical noise filtering techniques have proven inadequate. As an alternative, we propose a new filtering method, which is implemented as follows. The image is converted from the original rectangular x-y coordinate system into a polar p-O coordinate system. This causes the radial noise to map into well-defined horizontal streaks. After a polar transformation, radial noise is mapped around one of the axes of the Fourier spectrum in the corresponding o-m space. We have modeled radial noise using both rectangular and polar representations and have shown both analytically and through computer simulation that these streaks exhibit consistent mapping patterns. A class of filters is designed accordingly and applied to the image. The filtered image is then converted back to the rectangular coordinate system. This method has been tested using both synthetic and real data. It has shown little or no sensitivity to variations in the image data. Visual inspection of the filtered images show clear enhancement over the original images. Extreme levels of radial noise were removed with no apparent loss of sharpness around dominant features in the image