No abstract

Image deblurring is an important topic in imaging science. In this review, we consider together fluorescence microscopy and optical/infrared astronomy because of two common features: in both cases the imaging system can be described, with a sufficiently good approximation, by a convolution operator, whose kernel is the so-called point-spread function (PSF); moreover, the data are affected by photon noise, described by a Poisson process. This statistical property of the noise, that is common also to emission tomography, is the basis of maximum likelihood and Bayesian approaches introduced in the mid eighties. From then on, a huge amount of literature has been produced on these topics. This review is a tutorial and a review of a relevant part of this literature, including some of our previous contributions. We discuss the mathematical modeling of the process of image formation and detection, and we introduce the so-called Bayesian paradigm that provides the basis of the statistical treatment of the problem. Next, we describe and discuss the most frequently used algorithms as well as other approaches based on a different description of the Poisson noise. We conclude with a review of other topics related to image deblurring such as boundary effect correction, space-variant PSFs, super-resolution, blind deconvolution and multiple-image deconvolution.

Several methods based on different image models have been proposed and developed for image denoising. Some of them, such as total variation (TV) and wavelet thresholding, are based on the assumption of additive Gaussian noise. Recently the TV approach has been extended to the case of Poisson noise, a model describing the effect of photon counting in applications such as emission tomography, microscopy and astronomy. For the removal of this kind of noise we consider an approach based on a constrained optimization problem, with an objective function describing TV and other edge-preserving regularizations of the Kullback-Leibler divergence. We introduce a new discrepancy principle for the choice of the regularization parameter, which is justified by the statistical properties of the Poisson noise. For solving the optimization problem we propose a particular form of a general scaled gradient projection (SGP) method, recently introduced for image deblurring. We derive the form of the scaling from a decomposition of the gradient of the regularization functional into a positive and a negative part. The beneficial effect of the scaling is proved by means of numerical simulations, showing that the performance of the proposed form of SGP is superior to that of the most efficient gradient projection methods. An extended numerical analysis of the dependence of the solution on the regularization parameter is also performed to test the effectiveness of the proposed discrepancy principle.

In applications of imaging science, such as emission tomography, fluorescence microscopy and optical/infrared astronomy, image intensity is measured via the counting of incident particles (photons, γ-rays, etc). Fluctuations in the emission-counting process can be described by modeling the data as realizations of Poisson random variables (Poisson data). A maximum-likelihood approach for image reconstruction from Poisson data was proposed in the mid-1980s. Since the consequent maximization problem is, in general, ill-conditioned, various kinds of regularizations were introduced in the framework of the so-called Bayesian paradigm. A modification of the well-known Tikhonov regularization strategy results in the data-fidelity function being a generalized Kullback-Leibler divergence. Then a relevant issue is to find rules for selecting a proper value of the regularization parameter. In this paper we propose a criterion, nicknamed discrepancy principle for Poisson data, that applies to both denoising and deblurring problems and fits quite naturally the statistical properties of the data. The main purpose of the paper is to establish conditions, on the data and the imaging matrix, ensuring that the proposed criterion does actually provide a unique value of the regularization parameter for various classes of regularization functions. A few numerical experiments are performed to demonstrate its effectiveness. More extensive numerical analysis and comparison with other proposed criteria will be the object of future work.

We present two wide-field (%5 0 ; 3A5), diffraction-limited (k=D ' 0B5 at 10 m), broadband 10 and 20 m images of the Orion Nebula, plus six 7-13 m narrowband (k=Ák ' 1) images of the BN/ KL complex taken at the 3.8 m UKIRT telescope with the MPIA MAX camera. The wide-field images, centered on the Trapezium and BN/ KL regions, are mosaics of 35 00 ; 35 00 frames obtained with standard chopping and nodding techniques and reconstructed using a new restoration method developed for this project. They show the filamentary structure of the dust emission from the walls of the H ii region and reveal a new remarkable group of arclike structures %1 0 to the south of the Trapezium. The morphology of the Ney-Allen Nebula, produced by wind-wind interaction in the vicinity of the Trapezium stars, suggests a complex kinematical structure at the center of the cluster. We find indications that one of the most massive members of the cluster, the B0.5 V star 1 Ori D, is surrounded by a photoevaporated circumstellar disk. Among the four historic Trapezium OB stars, this is the only one without a binary companion, suggesting that stellar multiplicity and the presence of massive circumstellar disks may be mutually exclusive. In what concerns the BN / KL complex, we find evidence for extended optically thin silicate emission on top of the deep 10 m absorption feature. Assuming a simple two-component model, we map with '0B5 spatial resolution the foreground optical depth, color temperature, and mid-IR luminosity of the embedded sources. We resolve a conspicuous point source at the location of the IRc2-A knot, approximately 0B5 north of the deeply embedded H ii region ''I.'' We analyze the spectral profile of the 10 m silicate absorption feature and find indication for grain crystallization in the harsh nebular environment. In the OMC-1 South region, we detect several point sources and discuss their association with the mass-loss phenomenology observed at optical and millimeter wavelengths. Finally, we list the position and photometry of 177 point sources, the large majority of which are detected for the first time in the mid-IR. Twenty-two of them lack a counterpart at shorter wavelengths and are therefore candidates for deeply embedded protostars. The comparison of photometric data obtained at two different epochs reveals that source variability at 10 m is present up to a level of %1 mag on a timescale of $2 yr. With the possible exception of a pair of OB stars, all point sources detected at shorter wavelengths display 10 m emission well above the photospheric level, which we attribute to disk circumstellar emission. The recent model of Robberto et al. provides the simplest explanation for the observed mid-IR excess.