This report presents properties of the Discrete Pulse Transform on multi-dimensional arrays introduced by the authors two or so years ago. The main result given here in Lemma 2.1 is also formulated in a paper to appear in IEEE Transactions on Image P
rocessing. However, the proof, being too technical, was omitted there and hence it appears in full in this publication.
The ring operations and the metric on $C(X)$ are extended to the set $mathbb{H}_{nf}(X)$ of all nearly finite Hausdorff continuous interval valued functions and it is shown that $mathbb{H}_{nf}(X)$ is both rationally and topologically complete. Hence
, the rings of quotients of $C(X)$ as well as their metric completions are represented as rings of Hausdorff continuous functions.
The LULU operators, well known in the nonlinear multiresolution analysis of sequences, are extended to functions defined on a continuous domain, namely, a real interval. We show that the extended operators replicate the essential properties of their
discrete counterparts. More precisely, they form a fully ordered semi-group of four elements, preserve the local trend and the total variation.
