Radon-Nikodym approximation in application to image analysis

Abstract: 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.