Morphological scale-spaces

Two scaled morphological operations, the multiscale dilation-erosion and the multiscale closing-opening, have been introduced for the scale-space smoothing of signals. The multiscale operations are translation invariant, nonlinear, increasing, and dependent on a real-scale parameter, which can be negative. The smoothed signals across all scales can be considered as a function on the scale-space. The scale-space image exists for negative and positive scale and, thus, the information in the signal is more expanded than in the linear (Gaussian) scale-space image, which only exists for non-negative scale. The use of flat structuring functions on gray-scale images leads to flat regions in the output signal around the local extrema, and the local extrema are no longer exactly localized in position. Modem scale-space theory concentrate on powerful mathematical results describing the axiomatic bases and the differential structure and invariants of the various scale-spaces.