Restoration in the presence of unknown spatially varying blur

The last two decades brought significant progress in the development of efficient methods for classical deconvolution and super-resolution problems in both the single and multi-image scenarios. Most of these methods work with blurs modelled by convolution, which assumes that the properties of blur are the same 2 Image Restoration: Fundamentals and Recent Advances in the whole image (space-invariant blur). Unfortunately, in practice, the blur is typically spatially variant. The most common types of space-variant blur are defocus, optical aberrations and motion blur caused by either camera motion or motion of objects. Extension of deconvolution methods to spatially varying blur is not straightforward. What makes such restoration problems more challenging than in the case of the space-invariant blur, is a much higher number of unknowns that must be estimated. Consequently, the solution is ill-posed and requires additional constrains that must be chosen depending on the type of blur we wish to suppress. The requirement to remove only certain types of blur while keeping others is surprisingly common. A typical example is the removal of motion blur from portrait pictures taken in low-light conditions while keeping a small depth of focus. Similarly, it is usually desirable to remove optical aberrations but we may wish to preserve motion blur conveying the sensation of speed. In the last few years, despite the complexity of space-variant deblurring, we can observe an increasing effort in this direction of research, including difficult problems such as the blur dependent on the depth of scene or several independently moving objects. In this chapter, we give an overview of the current state of the art in the space-variant restoration, addressing the latest results in both deblurring and super-resolution. The chapter is divided into two main parts. In Sec. 1.2, we describe mathematical models used for spatially varying blur and basic restoration approaches connected with these models. Our purpose is not to give a complete survey of all known algorithms, instead we just briefly outline the models and point out interesting papers that appeared in the last several years to indicate the current trends in the research of space-variant restoration. Among others, we introduce models describing the blur caused by camera motion by three-dimensional kernels analogous to those used in standard deconvolution algorithms and models used for complex scenes consisting of several independently moving objects. The second part of this chapter (Sec. 1.3) details a new approach to space-variant super-resolution for …