Properties of a Natural Ordering Relation for Octagonal Neighborhood Sequences

Transcription

1 Properties of a Natural Ordering Relation for Octagonal Neighborhood Sequences Attila Fazekas Image Processing Group of Debrecen Faculty of Informatics, University of Debrecen P.O.Box 12, H-4010 Debrecen, Hungary Attila.Fazekas@inf.unideb.hu András Hajdu Image Processing Group of Debrecen Faculty of Informatics, Univerity of Debrecen P.O.Box 12, H-4010 Debrecen, Hungary hajdua@inf.unideb.hu Lajos Hajdu Number Theory Research Group of the Hungarian Academy of Sciences, and Institute of Mathematics, University of Debrecen P.O.Box 12, H-4010 Debrecen, Hungary hajdul@math.klte.hu Abstract Neighborhood sequences play an important role in several branches of discrete geometry and image processing. The literature of such sequences is very wide. In this paper we give a survey on results on a natural partial ordering relation for generalized nd octagonal neighborhood sequences. As this ordering does not have nice properties for each subset of such neighborhood sequences, we also investigate another relation and provide several properties for it. We put special emphasize on neighborhood sequences which generate metrics on Z n. In certain applications it can be useful to compare any two neighborhood sequences - however, none of these partial orderings is a total ordering. For this purpose, we investigate a norm-like concept, called velocity, which fits very well to the natural ordering relation. We also define a metric for neighborhood sequences, and investigate its properties. 1. Introduction In their classical paper, Rosenfeld and Pfaltz [15] investigated two types of motions in the two-dimensional digital space. The cityblock motion allows horizontal and vertical movements only, while in the case of chessboard motion one can diagonal movements, as well. The octagonal distances can be obtained by the mixed use of these motions. This concept has been investigated and extended by many authors, in several directions. Here for the general history and the basic properties of these and related concepts of digital topology, we only refer to the survey paper Research supported in part by the János Bolyai Research Fellowship of the Hungarian Academy of Sciences [13] and the book [16]. In this paper we give a brief survey of the literature on so-called octagonal neighborhood sequences which are derived from the above notions in a direct way. In particular, we focus on results about the structural behavior of such sequences under a nice natural partial ordering relation, which can also be important in several applications. First we briefly summarize the most important results and directions of investigations of this area. Das, Chakrabarti and Chatterji (see [5]) considered arbitrary periodic sequences of cityblock and chessboard motions, called periodic neighborhood sequences, and also their corresponding generalizations in Z n. Moreover, they established a formula for calculating the distance d(p, q; A) of any two points p, q Z n, determined by such a neighborhood sequence A. They introduced a natural partial ordering relation for periodic neighborhood sequences: if for two periodic neighborhood sequences A, B we have d(p, q; A) d(p, q; B) for all p, q Z n then A is faster than B. Das [1] investigated the lattice properties of the set of periodic neighborhood sequences and some of its subsets under this relation, in 2D. He obtained some positive, but also some negative results. Later, Fazekas [6] made similar investigations and obtained related results in 3D. Later, the present authors [7] extended the investigations to arbitrary, not necessarily periodic neighborhood sequences. Such sequences have important applications. For example, it turns out that neighborhood sequences which provide the best approximations to the Euclidean distance in Z 2 in some sense, are not periodic (see [10]). In [7] it was shown that the results of Das [1] and Fazekas [5] about the structure of periodic neighborhood sequences can be extended to arbitrary dimension, even in this more general case. Furthermore, the authors investigated the structure of the set and some subsets of the generalized octagonal neighborhood sequences in nd, under this ordering.

2 They involved into their investigations all those sets which were studied by Das [1] and Fazekas [5] in the periodic case. We note that similar investigations have been made in [11], for ultimately periodic and Lyndon sequences, and also for sequences having density. Unfortunately, in several cases negative results were obtained: some of the structures considered do not have nice properties. Hence instead of this natural partial ordering the authors proposed another relation, which is in close connection with the original one. More precisely, the natural ordering is a refinement of the new relation. As it turns out, under this new ordering, the examined sets usually form lattices with some nice properties (see [7]). Those neighborhood sequences which generate metrics on the digital space Z n naturally play a special role in several problems. Hence it is important to analyze the structural properties of these sequences. Such an investigation was performed in [8]. It turns out that in 2D the set of such sequences has a nice algebraic structure under the above mentioned natural partial ordering relation. For the sake of completeness, the new relation has also been considered in this case, too. Summarizing the above investigations, it turned out that the natural ordering relation has some unpleasant properties in several cases. In particular, it fails to be a complete ordering, practically on any subset of the general octagonal neighborhood sequences. However, in certain applications it can be useful to compare any two neighborhood sequences, i.e. to decide which one spreads faster in Z n. For this purpose, A. Hajdu and L. Hajdu [9] introduced a norm-like concept, called velocity, on the set of such sequences, and investigated its properties. This concept was introduced in a way to fit the natural ordering relation. Further, a related metric was also defined for neighborhood sequences, and the resulted metric space turned out to be a complete, separable one. The authors in [8] extended these investigations to neighborhood sequences generating metrics on Z n. The structure of this paper is the following. In the next section we introduce our notation. Then we summarize the most important results about the above mentioned natural ordering relation for periodic and general octagonal sequences, from [1] and [7]. In the fourth section we investigate the structural properties of metrical neighborhood sequences, see [8]. Finally, summarizing some results from [9] and [8], we introduce velocity for such sequences, and show that this notion is in a good accordance with the natural ordering. 2. Basic definitions In this section we introduce some standard notation concerning neighborhood sequences. Let n, m N with m n. The points p =(p 1,...,p n ) and q =(q 1,...,q n ) in Z n are m-neighbors, if the following two conditions hold: p i q i 1 (1 i n), n p i q i m. The sequence A =(A(i)), where A(i) {1,...,n} for all i N, is called an n-dimensional (shortly nd) neighborhood sequence. If for some l N we have A(i + l) =A(i) for i N then A is called periodic with period l. The set of the nd-neighborhood sequences will be denoted by S n, while the set of periodic ones by P n. Further, let P n (l ) and P n (l) be the sets at most l-periodic and l-periodic (l N) nd-neighborhood sequences, respectively. Let p, q Z n and A S n. The point sequence p = p 0,p 1,...,p t = q, where p i 1 and p i are A(i)-neighbors in Z n (1 i t), is called an A-path from p to q of length t. The A-distance d(p, q; A) of p and q is defined as the length of the shortest A-path(s) between them. As a brief notation, we also use d(a) for the A-distance. We introduce two partial orderings, and on S n. These orderings are defined in the following way. For A, B S n write A B d(p, q; A) d(p, q; B) for every p, q Z n, and set A B A(i) B(i) for every i N. The ordering was introduced by Das et al. [5] for P n and was investigated in [1], also in the periodic case. The authors [7] extended this ordering to S n and introduced, as well. Now we recall a few basic concepts and facts from lattice theory. They will be used throughout the paper without any further reference. Let H be a partially ordered set. We say that H is a lattice, if for any A, B H the greatest lower bound A B and the least upper bound A B of these elements exist. If for any L H the greatest lower bound L and the least upper bound L of L also exist, then the lattice H is called complete. It is well-known that if L exists for all subset L of H, then L also exists for any subset, and vice versa. The lattice H is distributive, if for any A, B, C H we have (A B) C =(A C) (B C) and (A B) C =(A C) (B C). The notation H and H will refer to the greatest lower bound and least upper bound in H, with respect to the actual ordering. 3. The lattice properties of S n For any i N and j {1,...,n} put A (j) (i) = min{j, A(i)}. From [7] we know that for A, B S n we have A B A (j) (i) B (j) (i) for any k N.

3 Moreover, the ordering is a proper refinement of, that is A B implies A B. Further, one can easily check that neither nor is a total ordering on S n. We summarize the results about the structure of all those sets which were studied by Das [1], Fazekas [6], and by the present authors [7]. That is, we look at the sets S n, P n, P n (l ) and P n (l). Unfortunately, in most cases these sets do not form nice structures with respect to. For the proofs of these results see [1], [6], and [7], respectively. The most important positive result in this direction is the following. Theorem 1 [7]. (S 2, ) is a complete distributive lattice. However, this theorem does not hold in higher dimensions, that is Theorem 2 [7]. (S n, ) is not a lattice for n 3. Concerning some special sets of periodic sequences, similar unkind properties of may also occur. The only positive statement holds only in two dimensions, and is due to Das [1]. Theorem 3 [1]. (P 2 (l), ) for any l N and (P 2 (l ), ) for l 4 are complete distributive lattices. Now we give some negative results about such subsets. Theorem 4 [7]. (P n, ) for n 2, (P 2 (l ), ) for l 5, and (P n (l ), ), (P n (l), ) for l 2, n 3 are not lattices. The above results show that under the relation we cannot obtain a nice structure neither in S n (atleastin higher dimension), nor in its various subsets. Now we describe the structure of S n, P n, P n (l ) and P n (l) with respect to. As we will see, the structures obtained are much nicer than in case of. Theorem 5 [7]. (S n, ) is complete distributive lattice with greatest lower bound S n = {1} and least upper bound S n = {n}. The ordering relation has similar, but somewhat worse properties in P n. Theorem 6 [7]. (P n, ) is a non-complete distributive lattice with greatest lower bound P n = {1} and least upper bound P n = {n}. The forthcoming theorem shows that P n (l ) is not a good subset of S n, also with respect to. Theorem 7 [7]. (P n (l ), ) is not a lattice for n 2 and l 6. We note that (P n (l ), ) is a distributive lattice if l 5, for every n N. For details see [7]. However, P n (l ) has a nice structure under. Theorem 8 [7]. (P n (l), ) is a distributive lattice for every n, l N. 4. Lattices of metrical neighborhood sequences It is obviously not true that d(a) is a metric on Z n for every A S n. With the following result of Nagy [14] we can decide whether the distance function related to A is a metric on the n-dimensional digital space, or not. Theorem 9 [14]. Let A S n. Then d(a) is a metric if and only if k+t 1 A (j) (i) A (j) (i) for any k, t N and j {1,...,n}. We note that a similar assertion is proved by Das et al. [5], for periodic neighborhood sequences. For each n N let M n denote the set of those ndneighborhood sequences which generate metrics on Z n.if A M n then A is called metrical. Because of their importance, there are several results in the literature about metrical octagonal neighborhood sequences, see e.g. the papers [4, 5, 1, 3, 2, 8, 9, 14] and the references given there. In this section we overview the structural results on the set of metrical neighborhood sequences with respect to both and. The next result shows that it is not true that any two metrical neighborhood sequences can be compared using these orderings. Theorem 10 [8]. The partial orderings and are not total orders on M n. Now we look at the structural properties of M n under and in separate sections. Again, we omit the proofs of the results mentioned, they can be found in [8]. We start with, as the situation is much simpler in this case. i=t 4.1. The structure of M n with respect to In [7] the authors introduced to obtain better structural results for S n and P n than with. The following result shows the slightly surprising fact that M n does not form a nice structure under. Theorem 11 [8]. (M n, ) is not a lattice for n The structure of M n with respect to The situation for (M n, ) is similar to (M n, ) at least when n 3. However, this is not that surprising, since (S n, ) is also not a lattice in this case. Theorem 12 [8]. (M n, ) is not a lattice for n 3. The following theorem shows that contrary to the higher dimensional case, metrical 2D-neighborhood sequences form a nice structure with respect to. Theorem 13 [8]. (M 2, ) is a non-distributive complete lattice. Moreover, for any subset M of M 2 we have M 2 M = S 2 M. It is an interesting property of M 2 that while for any A, B M 2 we have A S2 B M 2, the same statement does not hold for A S2 B. For example, if we choose A = and B = then by Theorem 9 it is easy to verify that A, B M 2. However A S2 B = , which sequence does not belong to M 2. On the other hand, the least upper bound of A and B also exists in M 2, since M 2 is a complete lattice. Hence it is easy to determine A M2 B for any A, B M 2 (see [8]), however it is not clear how to determine A M2 B?

4 The following theorem gives an answer to this problem in a more general form. Theorem 14 [8]. For any A S 2 there exists a B M 2 with B A, such that for any C M 2 with C A, C B holds. Moreover, B(1) = A(1) and if the first k elements of B are already given, then 1, B(k +1)= if k+1 A(i) k B(i)+1and l B(i) i=k l+2 for every l =1,...,k, 2, otherwise. B(i)+1 We define the metrical closure MC(A) of the neighborhood sequence A S 2 as the sequence B given by the above theorem. Then in case of B 1,B 2 M 2, MC(B 1 S2 B 2 )=B 1 M2 B 2. Now we give an infinite procedure from [8] which produces the metrical closure of A S 2.Tosimplify the description, we define switching and switching back as changing a sequence element from 1 to 2 and vice versa, respectively. Moreover, we call a finite word l (C(i)) k metrical if C(i) C(i) holds for every l {1,...,k}. i=k l+1 c:=0 % Invoking the counting variable % for switching. k:=1 % Invoking the slice length for % checking metricity. ITER: B[k]:=A[k] % Setting the next element of the % metrical closure B of A. IF SUB(B,1,k) is not metrical THEN BEGIN B[k]:=2 % Making SUB(B,1,k)=(B(1),...,B(k)) % metrical by switching. c:=c+1 % Updating the number of switchings. END ELSE IF (B[k]=2 AND c>0) THEN IF SUB(B,1,k-1)+"1" is metrical BEGIN B[k]:=1 % Switching back B[k]. c:=c-1 % Updating number of switchings. END END k:=k+1 % Increasing the slice length for the % next metricity check. GO TO ITER: % Finding the next element of B. Note that in the above algorithm the counter of switchings for the k-th step can be calculated as c =#{l B(l) = 2, 1 l k} #{l A(l) =2, 1 l k}. As one can see, this procedure is a kind of greedy algorithm: it keeps c as small as possible, beside keeping the metricity. 5. Velocity of neighborhood sequences As we have seen in Section 3, the natural ordering has some unpleasant properties in certain cases. It fails to be a complete ordering on S n, moreover, the structure obtained is not even a lattice in higher dimension. However, in certain situations it can be useful to compare any two neighborhood sequences, i.e. to decide which one spreads faster on the digital plane Z n. For this purpose, in this section we present a norm-like concept from [9], called velocity, and investigate its properties. This concept has to be introduced in a way to fit the natural ordering relation faster, so we need some preliminaries before defining velocity. Hence we start with listing some natural conditions, which should be met by this concept. Velocity must be sensitive for the behavior of the sequences in different dimensions. It may happen that a sequence is faster than another one in higher dimensions, but they have the same speed in lower dimensional subspaces. For example, in 3D the sequences (3,3,3,3,...) and (2,2,2,2,...) have the same velocity on the planes {x, y}, {x, z}, {y, z} defined by the coordinate axes; or the sequences (1,3,1,3,...) and (2,2,2,2,...) behave differently in certain subspaces of Z 3. Velocity should be sensitive for these properties. The elements of the sequences must be weighted with a suitable weight function. This condition comes from two reasons. First, it is natural to consider the early elements of the sequences more important than the elements occurring later. The second reason has a theoretical background. Namely, if we intend to take into consideration all elements of the sequences, then we have to guarantee the convergence of certain sums or series of the (weighted) elements of the sequences. Velocity must be defined in accordance with the natural ordering. This condition is rather natural: velocity should preserve the ordering. If a neighborhood sequence is faster than another one, its velocity should certainly be larger. As is only a partial ordering, we cannot demand the opposite statement to hold. However, the velocity concept introduced has the nice property that in a certain sense this opposite statement is also valid. We start with the concept of a weight system. Definition 1. Let n N. The set of functions δ j : N R (1 j n) is called a weight system, if the following three conditions hold: δ j (i) > 0(1 j n, i N), δ j (i) < (1 j n),

5 δ j is monotone decreasing (1 j n). We introduce the concept of velocity in two steps. First, we assign an n-tuple to every neighborhood sequence. The elements of this n-tuple belong to the velocity of the given neighborhood sequence in the subspaces of Z n of dimensions from 1 to n. Then we define one descriptive velocity value. Definition 2. Let A S n, and δ j (1 j n) be a weight system. The j-dimensional velocity of A is defined as v A j = A (j) (i)δ j (i). Let T be the linear space of bounded real sequences over R, and let δ j (1 j n) be a weight system. It is wellknown (see e.g. [12]) that for every j, with the norm (x i ) = x i δ j (i), ((x i ) T ), T becomes a Banach space. Thus, for any A =(A(i)), v A j could be defined as va j = (A(j) (i)). Obviously, for every A S n we have δ j (i) vj A n δ j (i). We define the velocity of A by the help of the j- dimensional velocities. Definition 3. Let A S n. The velocity of A is given by v A = 1 n n vj A. j=1 This shows that regardless of the system δ j, the j- dimensional velocities and the velocity of A is determined by the first few terms of A. By Definitions 2 and 3 the first two preliminary conditions are met by this velocity concept. The next two theorems show that this concept is in close connection with the natural ordering. Theorem 15 [9]. Let A, B S n with A B, and let δ j (j =1,...,n) be a weight system. Then vj A vj B for every j =1,...,n. By the definition of the velocity, the above theorem implies that if A B, then v A v B. As we mentioned earlier, is only a partial ordering. Hence, the above statement cannot be reversed. However, the next statement shows that in a certain sense the opposite statement is still valid. Theorem 16 [9]. Let A, B S n. If for any weight system δ j (1 j n), vj A vj B holds for all j =1,...,n, then A B. It can be easily verified that the condition vj A vj B for all j =1,...,ncannot be replaced by v A v B, for details see [9] Metric space of the neighborhood sequences We introduce a metric on the set of neighborhood sequences in a similar fashion as we did it for velocity. Definition 4. Let ={δ j j =1,...,n} be a weight system and A, B S n. The distance ϱ of these sequences is defined by ϱ (A, B) = 1 n n j=1 A (j) (i) B (j) (i) δ j (i). One can easily verify that in case of any weight system, the function ϱ isametricons n. Further, the metric space (S n,ϱ ) is bounded. Its diameter is diam(s n,ϱ )=ϱ ((1, 1,...), (n, n,...) )= n 1 n n j=1 δ j (i). In what follows, we present some useful and interesting properties of these metric spaces. Theorem 17 [9]. For any weight system, (S n,ϱ ) is a complete metric space. The last result of this subsection shows that the set of periodic neighborhood sequences is a dense subset of (S n,ϱ ). As the set of periodic neighborhood sequences is clearly countable, this also yields that (S n,ϱ ) is a separable metric space. Theorem 18 [9]. For any weight system, the set of periodic neighborhood sequences is dense in (S n,ϱ ) Topological Properties of M n In this subsection we investigate the topological propertiesofthesetm n, as the subset of S n with respect to above defined metrics. Note that as clearly (S n,ϱ ) is the product of compact spaces, it is also compact. The following statement shows that M n is an isolated subset of S n. For the proof, see [8]. Theorem 19 [8]. The set M n \{n} is a perfect subset of the metric space (S n,ϱ ). M n is a compact subset of (S n,ϱ ). References [1] P.P. Das. Lattice of octagonal distances in digital geometry. Pattern Recognition Lett., 11: , [2] P.P. Das. Best simple octagonal distances in digital geometry. Journal Approx. Theory, 68: , [3] P.P. Das, B.N. Chatterji. Octagonal distances for digital pictures. Inform Sci., 50: , [4] P.P. Das, P.P. Chakrabarti, B.N. Chatterji. Generalised distances in digital geometry. Inform Sci., 42:51-67, 1987.

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