Spatial derivatives and the progagation of noise in Gaussian scale-space

Image structure analysis requires the computation of local spatial derivatives of the intensity distribution. These are determined in a natural way by convolution with Gaussian derivative operators. These "fuzzy derivatives" are calculated at any scale, and remove the problem of the ill-posed nature of numerical differentiation. However, noise is enhanced by the high-pass nature of differentiation, particularly at high order. On the other hand, the Gaussian-weighted averaging gives rise to noise reduction. This paper gives an analysis of the propagation of spatially uncorrelated as well as spatially correlated additive noise in scale space, when the noise is subjected to fuzzy derivative operators of any order. The propagation of noise variance is always substantially reduced when scale is increased, the effect being greater for higher order derivatives. The spatial blurring is always predominant, or, the representation of the noise and its derivatives is substantial only at the original (inner) scale. Expressions are derived for the propagation of noise in functions of derivatives, like the Laplacian and isophote curvature. All expressions are evaluated for a D-dimensional (image) data structure. Determining derivatives, even up to high order, combined with scale space, is a very robust and stable operation. The important conclusion is that the use of differential geometrical methods in scale space, particularly in noisy images, is justified.