Median and morphological scale space filtering and zero-crossings

Abstract

Until recently, attention has been focused on linear methods for achieving multiscale decomposition, indeed certain filters have been considered uniquely appropriate in part because they, Gaussian filters for example, localise features well and partly because they do not create new inflexions with increasing scale. Unfortunately even filters, such as Gaussians, produce decompositions in which information, associated with edges and impulses, is spread over many, or all, scale space channels and this both compromises edge location and potentially pattern recognition.An alternative is to use nonlinear filter sequences (filters in series, known as sieves) or banks (in parallel). Recently multiscale decomposition using both erosion (dilation) and closing (opening) operations with sets of increasing scale flat structuring elements have been used to analyse edges over multiple scales and the granularity of images. These do not introduce new edges as scale increases. However, they are not at all statistically robust in the face of, for example, salt and pepper noise. This paper shows that sieves also do not introduce new edges, are very robust and perform at least as well as discreet Gaussian filters when applied to sampled data. Analytical support for these observations is provided by the morphology decomposition theorem discussed elsewhere in this volume.