2-D adaptive smoothing by 3-D distance transformation

Abstract

Sternberg's rolling ball method for image smoothing demands specification of filter size. We want to make filter size adaptive. A criterion for smoothing heaviness stays necessary, but, unlike size, should be scale invariant. Hence we have pursued adaptation by first smoothing at all sizes and then selecting the best result. The usual implementation is: first dilation, then erosion of the binary 3-D grey landscape. We have found an alternative implementation: first the 3-D medial axis of the landscape complement, then erosion of the landscape from the medial axis points. Both implementations are slow, but both can be speeded up through 3-D distance transformation, as needed for multisize operation. Although distance transformation yields virtually parallel multisize erosion, dilation and medial axis calculation on a single landscape, it cannot maintain this efficiency in eroding different dilation results or eroding from the medial axis. We therefore have switched our interest from global to local adaptation. Unlike the usual implementation which can only be extended to global adaptation, the medial axis approach allows extension to local adaptation as well. As a scale invariant criterion for smoothing heaviness we propose the number of objects or features left over after smoothing. The medial axis can easily be broken up into countable fragments, related to image segments. The importance of segments can be estimated from the values on the medial axis fragment. A given number of most important segments can thus be selected for local small size (detail conserving) closing. As both light objects on dark background and dark objects on light background can be important, the method simultaneously processes the grey landscape and its complement.