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Properties of the Discrete Pulse Transform for Multi-Dimensional Arrays

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 Processing. However, the proof, being too technical, was omitted there and hence it appears in full in this publication.

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12/18/2007

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1 Introduction

This report presents properties of the Discrete Pulse Transform on multi-dimensional arrays introduced earlier in [1]. The main result given here in Lemma 2.1 is also formulated in [4, Lemma 21]. However, the proof, being too technical, was omitted there and hence it appears in full in this publication.

2 The Lemma

The lemma which follows deals with two technical aspects of the Discrete Pulse Transform of a function (where

denotes a vector lattice). The first is that the Discrete Pulse representation of a function

, given by

can be written as the sum of individual pulses of each resolution layer . The second result in the lemma below indicates a form of linearity for the nonlinear LULU operators.

Lemma 2.1
Let , , be such that does not have local minimum sets or local maximum sets of size smaller than , for some . Then we have the following two results.

  • (1)

    where , are local minimum sets of of size , , are local maximum sets of of size , and are negative and positive discrete pulses respectively, and we also have that

    (2)
    (3)
    (4)
  • For every fully trend preserving operator

Proof.
a) Let be all local minimum sets of size of the function . Since does not have local minimum sets of size smaller than , then is a constant on each of these sets, by [4, Theorem 14]. Hence, the sets are disjoint, that is , . Moreover, we also have

(5)

Indeed, let . Then there exists such that . Hence . From the local minimality of the sets and we obtain respectively and , which is clearly a contradiction. For every denote by the point in such that

(6)

Then we have

Therefore

(7)

where is a discrete pulse with support and negative value (down pulse).

Let be all local maximum sets of size of the function . By [4, Theorem 12(b)] every local maximum set of contains a local maximum set of . Since does not have local maximum sets of size smaller than , this means that the sets , , are all local maximum sets of and is constant on each of them. Similarly to the local minimum sets of considered above we have , , and , . Moreover, since is constant on any of the sets , , see [4, Theorem 14], we also have

(8)

which implies (4).

Further we have

where , , are such that . Hence

(9)

where is a discrete pulse with support and positive value (up pulse).
Thus we have shown that

b) Let the function be such that it does not have any local minimum or local maximum sets of size less than . Denote . We have

(10)

As in a) we have that (7) holds, that is we have

(11)

where the sets , , are all the local minimum sets of of size and satisfy (2). Therefore

(12)

Furthermore,

where . Using that is fully trend preserving, for every there exists such that , . Moreover, using that every adjacent point has a neighbor in we have that . Considering that the value of the pulse is negative, we obtain through the representation (12) that , , are local minimum sets of .

Next we show that does not have any other local minimum sets of size or less. Indeed, assume that is a local minimum set of such that . Since it follows from (12) that is a local minimum set of . Then using that is neighbor trend preserving and using [4, Theorem 17] we obtain that there exists a local minimum set of such that . Then applying again [4, Theorem 17] or [4, Theorem 12] we obtain that there exists a local minimum set of such that . This inclusion implies that . Given that does not have local minimum sets of size less than we have , that is is one of the sets - a contradiction. Therefore, , , are all the local minimum sets of of size or less. Then using again (7) we have

(13)

Using (11) and (13) we obtain

Therefore

This proves the first identity. The second one is proved in a similar manner.   

References