Shift-variant (SV) image restoration requires knowledge of the point spread function (PSF) at each image location. If we have access to the imaging device, and the capture conditions are known, it may be possible to obtain the PSF field by pre-calibration. In many practical situations, though, we do not have access to this information. Moreover, the PSF field may change due to factors that are beyond our control (e.g., atmospheric turbulences, device temperature, vibrations, fog, relative movement of camera and objects). A given PSF field must then be estimated solely from the observed image(s) we want to restore.

The most common approach for estimating the blurring kernel from a single image consists of formulating a joint optimization problem, often based on statistical models of the image and the degradation, which is used to estimate both the uncorrupted image and the kernel. Typically, a starting guess is refined by alternating between estimating the image (assuming the kernel is known) and the PSF kernel (assuming the image is known). Apart from other potential problems like convergence and stability, adding such an outer loop to the estimation makes it especially heavy in computational terms, as each of these marginal estimations is, by itself, computationally expensive. To the best of the authors’ knowledge, the intrinsic complexity of these approaches has prevented their application to the general case of SV blur kernels (although special cases have been treated, like considering foreground and background layers, e.g.,[1]).

Furthermore, as blind restoration problems are intrinsically highly ill posed, stable solutions have been obtained mostly using some prior information about the kernel, adapting the problem solution to particular situations. Many methods benefit computationally from using a restricted (e.g., parametric) PSF model: Prior knowledge is applied in camera motion[2, 3], parametric models are used in defocus and lens aberration correction[4, 5], sparse characteristics are exploited for frosted glass[6], etc. In the specific case of PSF field estimation in astronomical images, several authors have studied orthogonal representations to characterize observed PSFs from stars[7, 8]. The underlying motivation is to provide a robust tool against noise, flexible and possibly adaptive, without imposing a narrow structure to the PSF field. They may also facilitate PSF field interpolation (see, e.g.,[9, 10]). In the data-adaptive case, the linear dimensionality reduction (by singular value decomposition (SVD) or principal component analysis) deserves to be mentioned, which connects directly to the idea of deformable filters[11] approached in the companion paper, Part I. Function bases are optimal in a least-square sense, and they are ranked in terms of energy contribution to the PSF field description, so they can be selected to filter measurement noise in the estimation in a way that can be easily automated with techniques like Generalized Cross Validation[12]. Not surprisingly, such dimensionality reduction techniques have previously been used to estimate the PSF field[13]. However, they have typically been used for special cases, such as for modeling particular devices (for instance, the Large Synoptic Survey Telescope[14] or the Advanced Camera for Surveys on Hubble space Telescope[15]), adapted to particular stars as the ideal PSF reference, or for modeling gravitational lenses[16, 17]. Hence, these approaches lack generality in the sense that they cannot be used as a general, “knowledge blind” tool for astronomical PSF field estimation. Only a few of the referred methods (like[18]) intend to apply their PSF estimation results to image deblurring, and none of them draw the clear connection between image restoration and PSF field estimation.

To alleviate the previously mentioned computational bottleneck, many techniques analyze the image in search of local features which provide information of the PSFs across the image, such as edges and corners[4]. These groups of pixels give clues which are used to compute point and line spread functions without user intervention. The idea of extracting *local* information about the PSF to build *local* PSF estimates is especially relevant for the SV blur case. Considering the lack of generality of the most common approaches outlined above, and taking the benefits of utilizing local estimates into account, we design here a novel approach to astronomical PSF field estimation. It is based on two simple observations relating to the fundamental characteristics of astronomical imagery, as opposed to typical photographic images.

First, astronomical images typically contain an abundance of stars, which can be modeled well as ideal point light sources. While only a narrow set of typical photographic images present enough repeated patterns, such as bright distant lights, to be used directly for characterizing the PSF field, the presence of stars in astronomical images allows for a particularly simple way to locally estimate the PSF, by weighted averaging, after sub-pixel localization. Note that, even under the smoothness assumption, obtaining a more or less dense set of local PSF estimates is not enough to fully characterize the blur, and a certain type of regularization and interpolation of that information to the rest of the image locations is necessary.

This is where our second observation comes in: we take advantage of the fact that the PSF field in astronomical images is usually simpler compared to typical photographic images, as there is no foreground–background structure. To a first approximation, PSFs result from the composition of the telescope and atmosphere PSF fields (for Earth-based observations), and they usually vary smoothly across the image field. Typical photographic images, on the other hand, may present a very rich variability in PSF field structures, due to differences in focus and/or relative speed of objects in the scene. Only in particular cases (e.g., camera movement and/or defocus, still long distance objects) one may expect to find a smooth PSF field. Furthermore, only in an even narrower set of cases, typical photographic images present enough repeated patterns (e.g., bright distant lights) to be used directly for characterizing the PSF field, similarly to the star field case.

Hence, our approach is twofold, and sequential. The first step consists of estimating the *most likely* local PSF for each block of a set of overlapping image blocks covering the whole image, according to a simple local image model. The second step extends and refines these local PSF estimates through linear dimensionality reduction, nonlinear filtering of outliers, and spatial interpolation. Whereas the first part is completely model-based, the second part is presented here, to a large extent, in an ad hoc way. This does not diminish its strong connection with the ideas presented in the companion paper, Part I, about using the concept of deformable kernels to deal with smooth PSF fields.^{a}

There is a third fundamental characteristic of astronomical images that needs to be mentioned. Astronomical images have a huge dynamic range compared to typical photographic images, and very often, the search for relevant information requires processing low-contrast details in a contrast-adaptive fashion. Therefore, traditional error measures, such as quadratic error, must fail to describe the quality of a restoration method for star fields. To evaluate the quality of a restoration method on this kind of images is, thus, challenging. Quantitatively assessing the quality of the deblurring results has therefore not been tried here.

The rest of the article is organized as follows. Section 2. presents a simple image model for SV blurred star fields to be used as a local approximation for image blocks. Then, Section 3. describes an alternate optimization algorithm for obtaining, through maximum likelihood, an estimate of the local PSFs. Section 4. describes a generic approach for estimating a smooth PSF field from a regular grid of local PSF estimates. A broad set of experiments, and their corresponding results, are described and discussed in Section 5. First (Subsection 5.1), objective measurements of the PSF field estimation quality are made by means of simulated star fields using two synthetic PSF fields. Then (Subsection 5.2), real astronomical images are analyzed to estimate their PSF field. These estimates are used for restoration of real images in Section 5.2a. Section 6. concludes the article.