Transformations between rotational and translational invariants formulated in reciprocal spaces

Correlation functions play an important role in the theoretical underpinnings of many disparate areas of the physical sciences: in particular, scattering theory. More recently, they have become useful in the classification of objects in areas such as computer vision and our area of cryoEM. Our primary classification scheme in the cryoEM image processing system, EMAN2, is now based on third order invariants formulated in Fourier space. This allows a factor of 8 speed up in the two classification procedures inherent in our software pipeline, because it allows for classification without the need for computationally costly alignment procedures. In this work, we address several formal and practical aspects of such multispectral invariants. We show that we can formulate such invariants in the representation in which the original signal is most compact. We explicitly construct transformations between invariants in different orientations for arbitrary order of correlation functions and dimension. We demonstrate that third order invariants distinguish 2D mirrored patterns (unlike the radial power spectrum), which is a fundamental aspects of its classification efficacy. We show the limitations of 3rd order invariants also, by giving an example of a wide family of patterns with identical (vanishing) set of 3rd order invariants. For sufficiently rich patterns, the third order invariants should distinguish typical images, textures and patterns.