Properties of the Discrete Pulse Transform for Multi-Dimensional Arrays

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 $f \in A (Z^{d})$ (where $A (Z^{d})$

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

f

, given by

f = N \sum n = 1 D_{n} (f),

can be written as the sum of individual pulses of each resolution layer $D_{n} (f)$ . The second result in the lemma below indicates a form of linearity for the nonlinear LULU operators.

Lemma 2.1
Let $f \in A (Z^{d})$ , $s u p p (f) < \infty$ , be such that $f$ does not have local minimum sets or local maximum sets of size smaller than $n$ , for some $n \in N$ . Then we have the following two results.

(i d - P_{n}) f = γ^{-} (n) \sum i = 1 ϕ_{n i} + γ^{+} (n) \sum j = 1 φ_{n j},

(1)

where $V_{n i} = s u p p (ϕ_{n i}), i = 1, 2, . . ., γ^{-} (n)$ , are local minimum sets of $f$ of size $n$ , $W_{n j} = s u p p (φ_{n j}), j = 1, 2, . . ., γ^{+} (n)$ , are local maximum sets of $f$ of size $n$ , $ϕ_{n i}$ and $φ_{n j}$ are negative and positive discrete pulses respectively, and we also have that

$∙$	$V_{n i} \cap V_{n j} = \emptyset and a d j (V_{n i}) \cap V_{n j} = \emptyset, i, j = 1, . . ., γ^{-} (n), i \neq j,$	(2)
$∙$	$W_{n i} \cap W_{n j} = \emptyset and a d j (W_{n i}) \cap W_{n j} = \emptyset, i, j = 1, . . ., γ^{+} (n), i \neq j,$	(3)
$∙$	$V_{n i} \cap W_{n j} = \emptyset i = 1, . . ., γ^{-} (n), j = 1, . . ., γ^{+} (n) .$	(4)

For every fully trend preserving operator $A$

$U_{n} (i d - A U_{n}) = U_{n} - A U_{n},$

$L_{n} (i d - A L_{n}) = L_{n} - A L_{n} .$

Proof.
a) Let $V_{n 1}, V_{n 2}, . . ., V_{n γ^{-} (n)}$ be all local minimum sets of size $n$ of the function $f$ . Since $f$ does not have local minimum sets of size smaller than $n$ , then $f$ is a constant on each of these sets, by [4, Theorem 14]. Hence, the sets are disjoint, that is $V_{n i} \cap V_{n j} = \emptyset$ , $i \neq j$ . Moreover, we also have

a d j (V_{n i}) \cap V_{n j} = \emptyset, i, j = 1, . . ., γ^{-} (n) .

(5)

Indeed, let $x \in a d j (V_{n i}) \cap V_{n j}$ . Then there exists $y \in V_{n i}$ such that $(x, y) \in r$ . Hence $y \in V_{n i} \cap a d j (V_{n j})$ . From the local minimality of the sets $V_{n i}$ and $V_{n j}$ we obtain respectively $f (y) < f (x)$ and $f (x) < f (y)$ , which is clearly a contradiction. For every $i = 1, . . ., γ^{-} (n)$ denote by $y_{n i}$ the point in $a d j (V_{n i})$ such that

f (y_{n i}) = min y \in a d j (V_{n i}) f (y) .

(6)

Then we have

Therefore

(i d - U_{n}) f = γ^{-} (n) \sum i = 1 ϕ_{n i}

(7)

where $ϕ_{n i}$ is a discrete pulse with support $V_{n i}$ and negative value (down pulse).

Let $W_{n 1}, W_{n 2}, . . ., W_{n γ^{+} (n)}$ be all local maximum sets of size $n$ of the function $U_{n} f$ . By [4, Theorem 12(b)] every local maximum set of $U_{n} f$ contains a local maximum set of $f$ . Since $f$ does not have local maximum sets of size smaller than $n$ , this means that the sets $W_{n j}$ , $j = 1, . . ., γ^{+} (n)$ , are all local maximum sets of $f$ and $f$ is constant on each of them. Similarly to the local minimum sets of $f$ considered above we have $W_{n i} \cap W_{n j} = \emptyset$ , $i \neq j$ , and $a d j (W_{n i}) \cap W_{n j} = \emptyset$ , $i, j = 1, . . ., γ^{+} (n)$ . Moreover, since $U_{n} (f)$ is constant on any of the sets $V_{n i} \cup {y_{n i}}$ , $i = 1, . . ., γ^{-} (n)$ , see [4, Theorem 14], we also have

(V_{n i} \cup {y_{n i}}) \cap W_{n j} = \emptyset, i = 1, . . ., γ^{-} (n), j = 1, . . ., γ^{+} (n),

(8)

which implies (4).

Further we have

where $z_{n j} \in a d j (W_{n j})$ , $j = 1, . . ., γ^{+} (n)$ , are such that $U_{n} f (z_{n j}) = max z \in a d j (W_{n j}) U_{n} f (z)$ . Hence

(i d - L_{n}) U_{n} f = γ^{+} (n) \sum j = 1 φ_{n j}

(9)

where $φ_{n j}$ is a discrete pulse with support $W_{n j}$ and positive value (up pulse).
Thus we have shown that

(i d - P_{n}) f = (i d - U_{n}) f + (i d - L_{n}) U_{n} f = γ^{-} (n) \sum i = 1 ϕ_{n i} + γ^{+} (n) \sum j = 1 φ_{n j} .

b) Let the function $f \in A (Z^{d})$ be such that it does not have any local minimum or local maximum sets of size less than $n$ . Denote $g = (i d - A U_{n}) (f)$ . We have

g = (i d - A U_{n}) (f) = (i d - U_{n}) (f) + ((i d - A) U_{n}) (f) .

(10)

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

(i d - U_{n}) (f) = γ^{-} (n) \sum i = 1 ϕ_{n i},

(11)

where the sets $V_{n i} = s u p p (ϕ_{n i})$ , $i = 1, . . ., γ^{-} (n)$ , are all the local minimum sets of $f$ of size $n$ and satisfy (2). Therefore

g = γ^{-} (n) \sum i = 1 ϕ_{n i} + ((i d - A) U_{n}) (f) .

(12)

Furthermore,

where $v_{i} = f (y_{n i}) = min y \in a d j (V_{n i}) f (y)$ . Using that $A$ is fully trend preserving, for every $i = 1, . . ., γ^{-} (n)$ there exists $w_{i}$ such that $((i d - A) U_{n}) (f) (x) = w_{i}$ , $x \in V_{n i} \cup {y_{n i}}$ . Moreover, using that every adjacent point has a neighbor in $V_{n i}$ we have that $min y \in a d j (V_{n i}) ((i d - A) U_{n}) (f) (y) = w_{i}$ . Considering that the value of the pulse $ϕ_{n i}$ is negative, we obtain through the representation (12) that $V_{n i}$ , $i = 1, . . ., γ^{-} (n)$ , are local minimum sets of $g$ .

Next we show that $g$ does not have any other local minimum sets of size $n$ or less. Indeed, assume that $V_{0}$ is a local minimum set of $g$ such that $c a r d (V_{0}) \leq n$ . Since $V_{0} \cup a d j (V_{0}) \subset Z^{d} ∖ γ^{-} (n) ⋃ i = 1 V_{n i}$ it follows from (12) that $V_{0}$ is a local minimum set of $((i d - A) U_{n}) (f)$ . Then using that $(i d - A)$ is neighbor trend preserving and using [4, Theorem 17] we obtain that there exists a local minimum set $W_{0}$ of $U_{n} (f)$ such that $W_{0} \subseteq V_{0}$ . Then applying again [4, Theorem 17] or [4, Theorem 12] we obtain that there exists a local minimum set ${~ W}_{0}$ of $f$ such that ${~ W}_{0} \subseteq W_{0} \subseteq V_{0}$ . This inclusion implies that $c a r d ({~ W}_{0}) \leq n$ . Given that $f$ does not have local minimum sets of size less than $n$ we have $c a r d ({~ W}_{0}) = n$ , that is ${~ W}_{0}$ is one of the sets $V_{n i}$ - a contradiction. Therefore, $V_{n i}$ , $i = 1, . . ., γ^{-} (n)$ , are all the local minimum sets of $g$ of size $n$ or less. Then using again (7) we have

(i d - U_{n}) (g) = γ^{-} (n) \sum i = 1 ϕ_{n i}

(13)

Using (11) and (13) we obtain

(i d - U_{n}) (g) = (i d - U_{n}) (f)

Therefore

$(U_{n} (i d - A U_{n})) (f)$	$=$	$U_{n} (g) = g - (i d - U_{n}) (f)$
	$=$	$(i d - A U_{n}) (f) - (i d - U_{n}) (f)$
	$=$	$(U_{n} - A U_{n}) (f) .$

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

References

[1] R Anguelov and I N Plaskitt (now Fabris-Rotelli), A Class of LULU Operators on Multidimensional arrays, University of Pretoria, Tech. Rep. UPWT2007/22, 2008, URL=http://arxiv.org/abs/0712.2923.
[2] R Anguelov and I N Fabris-Rotelli, Discrete Pulse Transform of Images, Proceedings of ICISP 2008: International Conference on Image and Signal Processing, Cherbourg-Octeville, Normandy, France, 1-3 July 2008, Lecture Notes in Computer Science, 5099 (2008) 1 -9.
[3]
I Fabris-Rotelli and S J van der Walt, The Discrete Pulse Transform in Two Dimensions, Proceedings of the Twentieth Annual Symposium of the Pattern Recognition Association of South Africa, 30 November 1 December 2009 Stellenbosch, South Africa.
[4] R Anguelov and I N Fabris-Rotelli, LULU Operators and Discrete Pulse Transform for Multi-dimensional Arrays, IEEE Transactions on Image Processing, to appear, 2010.

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	$U_{n} (i d - A U_{n}) = U_{n} - A U_{n},$
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Properties of the Discrete Pulse Transform for Multi-Dimensional Arrays

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