  For an image pixel information can be converted to the moments of some basis
$Q_k$, e.g. Fourier-Mellin, Zernike, monomials, etc. Given sufficient number of
moments pixel information can be completely recovered, for insufficient number
of moments only partial information can be recovered and the image
reconstruction is, at best, of interpolatory type. Standard approach is to
present interpolated value as a linear combination of basis functions, what is
equivalent to least squares expansion. However, recent progress in numerical
stability of moments estimation allows image information to be recovered from
moments in a completely different manner, applying Radon-Nikodym type of
expansion, what gives the result as a ratio of two quadratic forms of basis
functions. In contrast with least squares the Radon-Nikodym approach has
oscillation near the boundaries very much suppressed and does not diverge
outside of basis support. While least squares theory operate with vectors
$<fQ_k>$, Radon-Nikodym theory operates with matrices $<fQ_jQ_k>$, what make
the approach much more suitable to image transforms and statistical property
estimation.
