New results on the theory of morphological filters by reconstruction

This paper treats the problem of establishing bounds for the morphological filter by reconstruction class. Morphological filters by reconstruction, which are composed of openings and closings by reconstruction, are useful filters for image processing because they do not introduce discontinuities. The main contributions of this paper are: (a) To establish when the combination of openings by reconstruction (Or, respectively, of closings by reconstruction) is an opening by reconstruction (respectively a closing by reconstruction). (b) To establish, for any filter by reconstruction, upper and lower bounds that are, respectively, a closing by reconstruction and an opening by reconstruction. In addition, the paper investigates certain aspects of filters by reconstruction that possess a robustness property called strong property. Some dual and equivalent forms are introduced for a family of multi-level filters recently introduced. A significant side-result is to determine some instances of connected openings composed by openings and closings by reconstruction that are not openings by reconstruction (similarly for closings). (C) 1998 Pattern Recognition Society. Published by Elsevier Science Ltd. All rights reserved.