Combined Invariants to Linear Filtering and Rotation

In the above model, h(x, y) is the point-spread function (PSF) of the system, n(x, y) is an additive random noise, a is a constant describing the overall change of contrast, τ stands for a transform of spatial coordinates due to projective imaging geometry and ∗ denotes 2D convolution. In many application areas, it is desirable to find a description of the original scene that does not depend on the imaging system without any prior knowledge of its parameters. Basically, there are two different approaches to this problem: image normalization or direct description by invariants. Image normalization consists of two major steps: geometric registration, that eliminates the impact of imaging geometry and transforms the image into some “standard” form, and blind deconvolution, that removes or suppresses the blurring. Both these steps have been extensively studied in the literature, we refer to the recent surveys on registration and on deconvolution/restoration techniques. Generally, image normalization is an ill-posed problem whose computing complexity can be extremely high. In the invariant approach we look for image descriptors (features) that do not depend on h(x, y), τ(x, y) and a. In this way we avoid a difficult inversion of Eq. (1). In many applications, the invariant approach is much more effective than the normalization. Typical examples are the recognition of objects in the scene against a database, template matching, etc.