Blind Image Deconvolution: Problem Formulation and Existing Approaches (2017)

Rafael Molina Departamento de Ciensas de la Computacion e I. A., Universidad de Granada, Granada, Spain e-mail: rms@decsai.ugr.esImages are ubiquitous and indispensable in science and everyday life. Mirroring the abilities of our own human visual system, it is natural to displayobservations of the world in graphical form. Images are obtained in areas ranging from everyday photography to astronomy, remote sensing, medical imaging, and microscopy. In each case, there is an underlying object or scene we wish to observe; the original or true image is the ideal representation of the observed scene. Yet the observation process is never perfect: there is uncertainty in the mea-surements, occurring as blur, noise, and other degradations in the recorded images. Digital image restoration aims to recover an estimate of the original image from the degraded observations. The key to being able to solve this ill-posed inverse problem is proper incorporation of prior knowledge about the original image into the restoration process. Classical image restoration seeks an estimate of the true image assumingthe blur is known. In contrast, blind image restoration tackles the much more diﬃcult, but realistic, problem where the degradation is unknown. In general, the degradation is nonlinear (including, for example, saturation and quantization) and spatially varying (non-uniform motion, imperfect optics); however, for most of the work, it is assumed that the observed image is the output of a Linear Spatially Invariant (LSI) system to which noise is added. Therefore it becomes a Blind Deconvolution (BD) problem, with the unknown blur represented as a Point Spread Function (PSF). Classical restoration has matured since its inception, in the context of spaceexploration in the 1960s, and numerous techniques can be found in the literature (for recent reviews see [1, 2]). These diﬀer primarily in the prior information about the image they include to perform the restoration task. The earliest algorithms to tackle the BD problem appeared as long ago as the mid-1970s [3,4], and attempted to identify known patterns in the blur; a small but dedicated eﬀort followed through the late 1980s (see for instance [5-9]), and a resurgence was seen in the 1990s (see the earlier reviews in [10, 11]). Since then, the area has been extensively explored by the signal processing, astronomical, and optics communities. Many of the BD algorithms have their roots in estimation theory, linear algebra, and numerical analysis. An important question one may ask is why is BD useful? Could we notsimply use a better observation procedure in the ﬁrst place? Perhaps, but there always exist physical limits, such as photonic noise, diﬀraction, or an observation channel outside of our control, and often images must be captured in suboptimal conditions. Also there are existing images of unique events that cannot be retaken that we would like to be able to recover (for instance with forensics or archive footage); furthermore in these cases it is often infeasible to measure properties of the imaging system directly. Another reason is that of cost. High-quality optics and sensing equipment are expensive. However, processing power is abundant today and opens the door to the application of increasingly sophisticated models. Thus BD represents a valuable tool that can be used for improving image quality without requiring complicated calibrations of the real-time image acquisition and processing system (i.e., in medical imaging, video conferencing, space exploration, x-ray imaging, bio-imaging, and so on). The BD problem is encountered in many diﬀerent technical areas, such asastronomical imaging [12, 13], remote sensing [14], microscopy [15], medical imaging [16], optics [17,18], photography [19,20], superresolution applications [21], and motion tracking applications [22], among others. For example, astronomical imaging is one of the primary applications of BDalgorithms [12,13]. Ground-based imaging systems are subject to blurring due to the rapidly changing index of refractions of the atmosphere. Extraterrestrial observations of the Earth and the planets are degraded by motion blur as a result of slow camera shutter speeds relative to the rapid spacecraft motion.