We extend the theory of polynomial moments by proving their spectral behavior with respect to Gaussian noise. This opens the door to doing computations on the signal-to-noise ratios of polynomial filters and with this, the comparability to classical filters is made possible. The compactness of the information in the polynomial and Fourier spectra can be compared to determine which solution will give the best performance and numerical efficiency. A general formalism for filtering with orthogonal basis functions is proposed. The frequency response of the polynomials is determined by analyzing the projection onto the basis functions. This reveals the tendency of polynomials to oscillate at the boundaries of the support; the resonant frequency of this oscillation can be determined. The new theory is applied to the extraction of 3-D embossed digits from cluttered surfaces. A three-component surface model is used consisting of a global component, corresponding to the surface; a Gaussian noise component, and local anomalies corresponding to the digits. The extraction of the geometric information associated with the digits is a preprocessing step for digit recognition. It is shown that the discrete polynomial basis functions are better suited than Fourier basis functions to fulfill this task.