Reply to "Comments on 'Analysis of the properties of soft morphological filtering using threshold decomposition' "

Constraints and proofs are provided by Shih and Pu (see ibid., vol. 43, no. 2, p. 539, 1995) to justify their previous analysis of the idempotent soft morphological filters. It is shown that for a special case, the idempotency in Theorems 3 and 4 of a previous paper by the authors will not hold in the first stage, but the root signal will be produced in the second stage. The exact constraint is added to ensure the idempotency to be valid for the soft morphological closing and opening.