Elimination of Gibbs and Nyquist-Shannon Phenomena in 3D Image Reconstruction

Fracture surfaces are often modelled by Fourier two-dimensional (2D) series that can be converted into digital 3D reliefs mapping the morphology of solid surfaces. Such digital replicas may suffer from various artifacts when processed inappropriately. The Gibbs phenomenon and spatial aliasing are two of those artifacts that may devalue Fourier replicas. The Gibbs phenomenon involves the fact that Fourier sums overshoot at a jump discontinuity, and that this overshoot does not die out as the frequency increases. According to the Nyquist-Shannon sampling theorem spatial aliasing occurs when Fourier frequencies exceed the Nyquist critical frequency. In this paper it is shown how we can fully eliminate these unpleasant effects. Graphical Abstract A new method for an approximation of fracture surfaces by a partial sum of the Fourier series which does not contain any Gibbs effects (a), any artefacts caused by convergence in the mean (b) and any alias (c). If number N of members of Fourier replica is equal to a resolution of a scanned original, the replica goes through the same points as the original. If it is higher, the original and replica differs in subpixel level only. Illustration (d) represents critical abrupt increases of aliasing artifacts in original 120 9 120 data matrix for sum of 480 members in case of standard Fourier series, case (e) is the sum of 480 members in the new series. [GRAPHICS]