Multidimensional quasi-eigenfunction approximations and multicomponent AM-FM models
We develop multicomponent AM-FM models for multidimensional signals. The analysis is cast in a general n-dimensional framework where the component modulating functions are assumed to lie in certain Sobolev spaces. For both continuous and discrete linear shift invariant (LSI) systems with AM-FM inputs, powerful new approximations are introduced that provide closed form expressions for the responses in terms of the input modulations. The approximation errors are bounded by generalized energy variances quantifying the localization of the filter impulse response and by Sobolev norms quantifying the smoothness of the modulations. The approximations are then used to develop novel spatially localized demodulation algorithms that estimate the AM and FM functions for multiple signal components simultaneously from the channel responses of a multiband linear filterbank used to isolate components. Two discrete computational paradigms are presented. Dominant component analysis estimates the locally dominant modulations in a signal, which are useful in a variety of machine vision applications, while channelized components analysis delivers a true multidimensional multicomponent signal representation. We demonstrate the techniques on several images of general interest in practical applications, and obtain reconstructions that establish the validity of characterizing images of this type as sums of locally narrowband modulated components.