Normalized weighted cross correlation for multi-channel image registration

The normalized cross-correlation ( $NCC$ ) is widely used for image registration due to its simple geometrical interpretation and being feature-agnostic. Here, after reviewing $NCC$ definitions for images with an arbitrary number of dimensions and channels, we propose a generalization in which each pixel value of each channel can be individually weighted using real non-negative numbers. This generalized normalized weighted cross-correlation ( $NWCC$ ) and its zero-mean equivalent ( $ZNWCC$ ) can be used, for example, to prioritize pixels based on signal-to-noise ratio. Like a previously defined $NWCC$ with binary weights, the proposed generalizations enable the registration of uniformly, but not necessarily isotropically, sampled images with irregular boundaries and/or sparse sampling. All $NCC$ definitions discussed here are provided with discrete Fourier transform ( $DFT$ ) formulations for fast computation. Practical aspects of $NCC$ computational implementation are briefly discussed, and a convenient function to calculate the overlap of uniformly, but not necessarily isotropically, sampled images with irregular boundaries and/or sparse sampling is introduced, together with its $DFT$ formulation. Finally, examples illustrate the benefit of the proposed normalized cross-correlation functions.