A general axiomatic system for image resolution quantification

In this paper, we generalize our previously published axiom system for quantification of image resolution and prove that any resolution measure consistent with the new axiom system must be a homogeneous symmetric function of order 1/2 of the eigenvalues of the covariance matrix of the PSF. We demonstrate that the previous axiom system is not consistent with the affine transformation axiom. We propose a weak combination axiom to replace the previous strong combination axiom and use it to solve this conflict. It is remarkable that the original finding in one-dimension by Wang and Li can be easily rediscovered with aid of the weak combination axiom, instead of using the previous strong combination axiom. If the previous axiom system is modified with the weak combination axiom and augmented with the affine transformation axiom, the resolution measure is shown to be proportional to the squared root of the geometric mean of the eigenvalues of the covariance matrix of the PSF. Relevant discussions and possible extensions are also provided.