Algebraic methods for constructing blur-invariant operators and their applications

Image acquisition devices are always subject to physical limitations that often manifest as distortions in the appearance of the captured image. The most common types of distortions can be divided into two categories: geometric and radiometric distortions. Examples of the latter ones are: changes in brightness, contrast, or illumination, sensor noise and blur. Since image blur can have many different causes, it is usually not convenient and also computationally expensive to develop ad hoc algorithms to correct each specific type of blur. Instead, it is often possible to extract a blurinvariant representation of the image, and utilize such information to make algorithms that are insensitive to blur. The work presented here mainly focuses on developing techniques for the extraction and the application of blur-invariant operators. This thesis contains several contributions. First, we propose a generalized framework based on group theory to constructively generate complete blurinvariants. We construct novel operators that are invariant to a large family of blurs occurring in real scenarios: namely, those blurs that can be modeled by a convolution with a point-spread function having rotational symmetry, or combined rotational and axial symmetry. A second important contribution is represented by the utilization of such operators to develop an algorithm for blur-invariant translational image registration. This algorithm is experimentally demonstrated to be more robust than other state-of-the-art registration techniques. The blurinvariant registration algorithm is then used as pre-processing steps to several restoration methods based on image fusion, like depth-of-field extension, and multi-channel blind deconvolution. All the described techniques are then re-interpreted as a particular instance of Wiener deconvolution filtering. Thus, the third main contribution is the generalization of the blurinvariants and the registration techniques to color images, by using respectively a representation of color images based on quaternions, and the quaternion Wiener filter. This leads to the development of a blur-and-noise-robust registration algorithm for color images. We observe experimentally a significant increase in performance in both color texture recognition, and in blurred color image registration.