In this paper, we propose a new theory of invariants to Gaussian blur. We introduce a notion of a primordial image as a canonical form of all Gaussian blur-equivalent images. The primordial image is defined in spectral domain by means of projection operators. We prove that the moments of the primordial image are invariant to Gaussian blur and we derive recursive formulas for their direct computation without actually constructing the primordial image itself. We show how to extend their invariance also to image rotation. The application of these invariants is in blur-invariant image comparison and recognition. In the experimental part, we perform an exhaustive comparison with two main competitors: 1) the Zhang distance and 2) the local phase quantization.

We present a content-based image retrieval method which is particularly designed for noisy images. The images are retrieved according to histogram similarity. To reach high robustness to noise, the histograms are described by novel features which are insensitive to convolution with a Gaussian kernel, i.e. insensitive to a Gaussian additive noise in original images. The advantage of the new method is demonstrated experimentally on real data.