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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.

• 3 publications
• 1 publication
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

 f=N∑n=1Dn(f),

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.

•  (id−Pn)f=γ−(n)∑i=1ϕni+γ+(n)∑j=1φnj, (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

 ∙ Vni∩Vnj=∅ and adj(Vni)∩Vnj=∅, i,j=1,...,γ−(n), i≠j, (2) ∙ Wni∩Wnj=∅ and adj(Wni)∩Wnj=∅, i,j=1,...,γ+(n),i≠j, (3) ∙ Vni∩Wnj=∅ i=1,...,γ−(n) ,j=1,...,γ+(n). (4)
• For every fully trend preserving operator

 Un(id−AUn)=Un−AUn, Ln(id−ALn)=Ln−ALn.

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

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

Then we have

Therefore

 (id−Un)f=γ−(n)∑i=1ϕni (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

 (Vni∪{yni})∩Wnj=∅, i=1,...,γ−(n), j=1,...,γ+(n), (8)

which implies (4).

Further we have

where , , are such that . Hence

 (id−Ln)Unf=γ+(n)∑j=1φnj (9)

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

 (id−Pn)f=(id−Un)f+(id−Ln)Unf=γ−(n)∑i=1ϕni+γ+(n)∑j=1φnj.

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

 g=(id−AUn)(f)=(id−Un)(f)+((id−A)Un)(f). (10)

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

 (id−Un)(f)=γ−(n)∑i=1ϕni, (11)

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

 g=γ−(n)∑i=1ϕni+((id−A)Un)(f). (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

 (id−Un)(g)=γ−(n)∑i=1ϕni (13)

Using (11) and (13) we obtain

 (id−Un)(g)=(id−Un)(f)

Therefore

 (Un(id−AUn))(f) = Un(g) = g−(id−Un)(f) = (id−AUn)(f)−(id−Un)(f) = (Un−AUn)(f).

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