1 Introduction
The LULU operators and the associated Discrete Pulse Transform developed during the last two decades or so are an important contribution to the theory of the nonlinear multiresolution analysis of sequences. The basics of the theory as well as the most significant results until 2005 are published in the monograph [13]. For more recent developments and applications see [1], [4], [7], [8], [14]. Central to the theory is the concept of separator. This concept is defined in [13] only for operators on sequences due to the context of the book. However, it is meaningful in more general settings. In fact, some of the axioms have been used earlier, e.g. see [18], for functions on arbitrary domains. We will give the definition of separator for operators on real functions defined on a domain with a group structure.
Let a be an abelian group. Denote by
the vector lattice of all real functions defined on
with respect to the usual pointwise defined addition, scalar multiplication and partial order. For every the operator given byis called a shift operator.
Definition 1
An operator is called a separator if
Here denotes the identity operator and the operator is defined in terms of the pointwise linear operations for the operators on , that is, . The first two axioms in Definition 1 and partially the third one were first introduced as required properties of nonlinear smoothers by Mallows, [9]. Rohwer further made the concept of a smoother more precise by using the properties (i)–(iii) as a definition of this concept. The axiom (iv) is an essential requirement for what is called a morphological filter, [18], [19], [21]. In fact, a morphological filter is exactly a syntone operator which satisfies (iv). Let us recall that an operator is called syntone if
The coidempotence axiom (v) in Definition 1 was introduced by Rohwer in [13], where it is also shown that it is an essential requirement for operators extracting signal from a sequence.
The LULU theory was developed for sequences, that is, the case . Given a biinfinite sequence and the basic LULU operators and are defined as follows
(1)  
(2) 
It is shown in [13] that for every the operators and as well as their compositions are syntone separators. Hence they are an appropriate tool for signal extraction. Furthermore, these operators form the so called strong LULU semigroup. This a four element semigroup with respect to composition, see Table 1, which is fully ordered with respect to the usual pointwise defined order
(3) 
We have
(4) 
Let us recall that, according to the well known theorem of Matheron [10], in general, two ordered morphological operators generate a six element semigroup which is only partially ordered.
The power of the LULU operators as separators is further demonstrated by their Total Variation Preservation property. Let be the set of sequences with bounded variation, that is,
Total Variation of a sequence is given by .
Definition 2
An operator is called total variation preserving if
(5) 
We should note that since is a seminorm on we always have
Hence, the significance of the equality (5) is that the decomposition does not create additional total variation. In particular, this property is very important for the application of the LULU operators to discrete pulse decompositions of sequences.
The aim of this paper is to generalize the LULU operators to functions on in such a way that their essential properties are preserved. In Section 2 the definitions of the basic operators and on are derived by using a strengthened form of the morphological concept of connection. Then we show that indeed these operators replicate the properties of the LULU operators for sequence. More precisely, we prove that: (i) they are separators (Section 2); (ii) their smoothing effect can be described in a similar way to the monotonicity of sequences (Section 3); (iii) they generate a four element fully ordered semigroup (Section 4). The developed theory can be applied to many problems of Image Analysis and it is the intention of the authors to research such applications in the future. However, as an illustration and demonstration of the power of this approach we apply the newly defined operators to deriving a total variation preserving discrete pulse decomposition of images. Noise removal and partial reconstructions are discussed in Section 6.
2 The basic operators and .
The definition of the operators and for sequences involves maxima and minima over sets of consecutive terms, thus, making an essential use of the fact that is totally ordered. Since , , is only partially ordered the concept of ‘consecutive’ does not make sense in this setting. Instead, we use the morphological concept of set connection, [19].
Definition 3
Let be an arbitrary nonempty set. A family of
subsets of is called a connected class or a connection
on
if
(i)
(ii) for all
(iii) for any family
This definition generalizes the topological concept of connectivity to arbitrary sets including discrete sets like . If a set belongs to a connection then is called connected.
It is clear from Definition 3 that a connection on does not necessarily contain sets of every size. For example, and are connections on but neither of them contain sets of finite size other than 0 and 1. In the definition of the operators and we need sets of every size. We assume that the set is equipped with a connection which satisfies the following three conditions
(6)  
(8)  
The aim of the conditions (6)–(8) is to define a connection which is sufficiently rich in connected sets. This is demonstrated by the following property, which is obtained via iterative application of the property (8):
As usual, is the number of the elements in the set , that is, the size of . For we have . Given a point and we denote by the set of all connected sets of size , which contain point , that is,
(10) 
Now the operators and are defined on as follows.
Definition 4
Let and . Then
(11)  
(12) 
Let us first see that Definition 4 generalizes the definition of and for sequences. Suppose and let be the connection on generated by the pairs of consecutive numbers. Then all connected sets on are sequences of consecutive integers and for any we have
Hence for an arbitrary sequence considered as a function on the formulas (11) and (12) are reduced to (1) and (2), respectively.
Theorem 5
(Order Properties)
Proof. We will only prove the inequalities involving since those involving are proved similarly.
a) Let . For every and we have
Hence
Therefore, , , which implies .
b) Let . For any and , we have . Therefore
Theorem 6
For any the operators and are separators.
Proof. We will only verify the conditions (i)–(v) in Definition 1 for since is dealt with in a similar manner.
(ii) Let , where is a constant function with a value of . Then for every we have
(iii) Let and , . For every we have
(iv) The inequality
is an immediate consequence of Theorem 5. Then it is sufficient to prove the inverse inequality. Let and . We have
(13) 
But implies . Therefore for every and we have
Using that the right hand side is independent of we further obtain
Then it follows from the representation (13) that
(v) The coidempotence of the operator is equivalent to . The inequality is an easy consequence of Theorem 5. Hence, for the coidempotence of it remains to show that . Assume the opposite. Namely, there exists a function and such that . Using the definition of this inequality implies that there exists such that for every we have , or equivalently
(14) 
Let be such that . Then for every we have
(15) 
Taking in (14) and (15) we obtain a contradiction which completes the proof.
3 The operators and as smoothers
Similar to their counterparts for sequences the operators and defined in Section 2 smooth the input function by removing sharp peaks (the application of ) and deep pits (the application of ). The smoothing effect of these operations is made more precise by using the concepts of a local maximum set and a local minimum set given below.
Definition 7
Let . A point is called adjacent to if . The set of all points adjacent to is denoted by , that is,
An equivalent formulation of the property (8) of the connection is as follows:
(16) 
Definition 8
A connected subset of is called a local maximum set of if
Similarly is a local minimum set if
The next four theorems deal with different aspects of the application of and to functions in . Their cumulative effect will be discussed at the end of the section. All theorems contain statements a) and b). Due to the similarity we present only the proofs of a).
Theorem 9
Let
and . Then we have
a) if and only if there exists a local maximum
set such that and ;
b) if and only if there exists local minimum
set such that and .
Proof. a) Implication to the left. Suppose that there exists a local maximum set , . Consider an arbitrary and let . Then, since the size of is larger than the size of we have . Furthermore, by (16) we have . Let . Since , we have that but Then using also that is a local maximum set we obtain
Since the set is arbitrary, this inequality implies that .
Implication to the right. Suppose . Let be the largest (in terms of ) connected set containing such that
(17) 
The set is obviously unique and can be constructed as , where is the morphological point connected opening generated by , see [19] or [20], and . We have , , because otherwise (17) is satisfied on the larger connected set . Therefore
Hence is a local maximum set.
Assume that . It follows from (2) that there exists such that . Then
This contradicts the assumption . Therefore, .
Theorem 10
Let . Then
a) the size of any local maximum set of the function is
larger than ;
b) the size of any local minimum set of the function is
larger than .
Proof. a) Assume the opposite, that is, there exists a local maximum set of such that . By Theorem 9 we have that
Since is idempotent, see Theorem 6, this implies the impossible inequality , which completes the proof.
Theorem 11
Let and let .
a) If then ;
b) If then .
Proof. a) For any the set is connected and of size larger than . Therefore, by (2), for every there exists such that . Then, using also the given inequality, for every and we have
Hence
Theorem 12
Let and .
a) If is a local minimum set of then there exists a
local minimum set of such that .
b) If is a local maximum set of then there exists a
local maximum set of such that .
Proof. a) Let be a local minimum set of . Then
Let be such that and let
An easy application of Theorem 11 shows that . Let and let be the largest (with respect to inclusion) connected subset of which contains . As in the proof of Theorem 9, the set can be obtained through . For every we have . Therefore is a local minimum set of .
Theorems 9–12 provide the following characterization of the effect of the operators and of a function :

The application of () removes local maximum (minimum) sets of size smaller or equal to .

The operator () does not affect the local minimum (maximum) sets in the sense that such sets may be affected only as a result of the removal of local maximum (minimum) sets. However, no new local minimum sets are created where there were none. This does not exclude the possibility that the action of () may enlarge existing local maximum (minimum) sets or join two or more local maximum (minimum) sets of into one local maximum (minimum) set of ().

() if and only if does not have local maximum (minimum) sets of size or less;
Furthermore, as an immediate consequence of Theorem 10 and Theorem 12 we obtain the following corollary.
Corollary 13
For every the functions and have neither local maximum sets nor local minimum sets of size or less. Furthermore,
if and only if does not have local maximum sets or local minimum sets of size less than or equal to .
4 The LULU semigroup
In this section we consider the operators , and their compositions. The main result is that , , and form a semigroup with respect to composition with a composition table as given in Table 1. Furthermore, the semigroup is totaly ordered as in (4) with respect to the pointwise defined partial order (3).
Theorem 14
The operators and are idempotent, that is,
(18)  
(19) 
Theorem 15
For any we have
(20) 
Proof. It follows from Theorem 5 that
(21) 
Assume that (20) is violated. In view of (21), this means that there exists and such that
It follows from Theorem 9 that there exists and such that is a local maximum set for . Then, by Theorem 12, there exists such that is a local maximum set of the function . We have . However, does not have any local maximum sets of size less than or equal to , see Theorem 10. This contradiction completes the proof.
As in the case of sequences, the key result for the set
(22) 
to be closed under composition is the equality in Theorem 15. Now one can easily derive the rest of the formulas for the compositions of the operators in this set. The composition table is indeed as given in Table 1. Furthermore, Theorem 15 implies the total order on the set (22) as in (4). Indeed, we have
Therefore, the operators and for functions on generate via composition a semigroup with exactly the same algebraic and order structure as the semigroup generated by the operators and for sequences.
5 Discrete pulse transform of images
In this section we apply the LULU operators defined and investigated in the preceding sections to derive a discrete pulse decomposition of images. A grayscale image is given through a function on a rectangular domain , the value of being the luminosity at the respective pixel. For the theoretical study it is more convenient to assume that the functions are defined on the whole space . To this end one can, for example, define on the set as a constant, e.g. 0. Hence we consider the set .
Appropriate connections for images are defined through a relation on reflecting what we consider neighbors of a pixel in the given context. Figure 1 gives some examples of the the neighbors of the pixel .
We call a set connected if for any two pixels there exists a set of pixels such that each pixel is neighbor to the next one, is neighbor to and is neighbor to . We assume that the neighbor relation on is such that
(23)  
(24) 
The conditions (23)–(24) ensure that the set of connected set defined through this relation is a connection in terms of Definition 3 and satisfies the conditions (6)–(8). Hence we can apply the operators and discussed in the preceding sections to functions on . Similar to the case of sequences we obtain a decomposition of a function by applying iteratively the operators with increasing from 1 to . This can be done in different ways depending on sequencing of the and . Since this section is intended as a demonstration rather than presenting a comprehensive discrete pulse transform theory, we will take one particular case when follows . Define the operators , , by and . Then for any and we have
(25) 
Definition 16
A function is called a pulse if there exist a connected set and a real number such that
The set is called support of the pulse and is denoted by .
Figure 2 gives an example of a pulse. It should be remarked that the support of a pulse may generally have any shape, the only restriction being that it is connected.
The usefulness of the representation (25) of a function is in the fact that all terms are sums of pulses as stated in the next theorem.
Theorem 17
Let .

For every the function is a sum of discrete pulses with disjoint support, that is, there exist and discrete pulses , , such that
(26) and
(27) 
Let be such that , and . Then
(28)
Proof. a) Denote . We have
(29) 
where the first term in the sum on the right hand side is nonnegative while the second one is nonpositive. Let