Projection Operators and Moment Invariants to Image Blurring

In this paper we introduce a new theory of blur invariants. Blur invariants are image features which preserve their values if the image is convolved by a point-spread function (PSF) of a certain class. We present the invariants to convolution with an arbitrary N-fold symmetric PSF, both in Fourier and image domain. We introduce a notion of a primordial image as a canonical form of all blur-equivalent images. It is defined in spectral domain by means of projection operators. We prove that the moments of the primordial image are invariant to blur and we derive recursive formulae for their direct computation without actually constructing the primordial image. We further prove they form a complete set of invariants and show how to extent their invariance also to translation, rotation and scaling. We illustrate by simulated and real-data experiments their invariance and recognition power. Potential applications of this method are wherever one wants to recognize objects on blurred images.