Sufficiently Near Sets of Neighbourhoods⋆
James F. Peters
Computational Intelligence Laboratory,
Department of Electrical & Computer Engineering, Univ. of Manitoba,
E1-526, 75A Chancellor’s Circle, Winnipeg, MB R3T 5V6
[email protected]
Abstract. The focus of this paper is on sets of neighbourhoods that
are sufficiently near each other as yet another way to consider near sets.
This study has important implications in M. Katětov’s approach to topologising a set. A pair of neighbourhoods of points are sufficiently near,
provided that the C̆ech distance between the neighbourhoods is less than
some number ε. Sets of neighbourhoods are sufficiently near, provided
the C̆ech distance between the sets of neighbourhoods is less than some
number ε.
Keywords: Approach space, C̆ech distance, collection, ε-near collections, merotopy, near sets, neighbourhood, topology.
1
Introduction
The problem considered in this paper is how to measure the nearness of sets
of neighbourhoods. The solution to this problem stems from the work by M.
Katĕtov [1] and S. Tiwari [2] on merotopic spaces. M. Katĕtov observed that
merotopic spaces are obtained by topologising certain parts of a nonempty set.
The term mero comes from the Greek word meros (part). Historically, a consideration of merotopic distance starts with a study of approach spaces (see,
e.g., [2,3,4]). Usually, an approach space distance function δ ∶ X × P(X) →
[0, ∞] that maps a member of a set and a subset (part) of a set to a number
in [0, ∞]. In this work, one starts with a generalised approach space defined in
terms of a distance function ρ ∶ P(X) × P(X) → [0, ∞] that maps a pair of
nonempty sets to a number in [0, ∞].
In its most basic form, an approach merotopy is a measure of the nearness of members of a collection. For collections A, B ∈ P 2 (X), a function ν ∶
P 2 (X) × P 2 (X) ∶→ [0, ∞] satisfying a number of properties is a called an εapproach merotopy. A pair of collections are near, provided ν(A, B) = 0. For
⋆
Many thanks to S. Tiwari, S. Naimpally, C.J. Henry & S. Ramanna for their insights
concerning topics in this paper. This research has been supported by the Natural
Sciences and Engineering Research Council of Canada (NSERC) grant 185986, Manitoba NCE MCEF grant, Canadian Arthritis Network grant SRI-BIO-05.
J.T. Yao et al. (Eds): RSKT 2011, LNCS 6954, pp. 17–24, 2011.
© Springer-Verlag Berlin Heidelberg 2011
18
J.F. Peters
ε ∈ (0, ∞], the pair A, B are sufficiently near, provided ν(A, B) < ε. In this work,
a determination of nearness and sufficient nearness results from considering the
distance between descriptions of sets of objects to determine the perceived similarity or dissimilarity of the sets. A consideration of descriptive forms of sufficient
nearness has considerable practical significance, since the distance between most
collections of sets in science, engineering, and the arts is usually not zero (the
descriptions of such collections of subsets are seldom identical).
This paper has the following organisation. The Lowen form of an approach
space and the more recent Peters-Tiwari generalised approach space are briefly
explained in Sect. 2. This leads to description-based neighbourhoods, in general,
and visual neighbourhoods, in particular, in Sect. 3. An approach to measuring
the nearness of collections of neighbourhoods is presented in Sect. 4.
2
Preliminaries
Let X be a nonempty ordinary set. The power set of X is denoted by P(X), the
family of all subsets of P(X) is denoted by P 2 (X).
Definition 1. A function δ ∶ X × P(X) → [0, ∞] is called a distance on X, if,
for any A, B ⊆ X and x ∈ X, the following conditions are satisfied:
(D.1) δ(x, {x}) = 0,
(D.2) δ(x, ∅) = ∞,
(D.3) δ(x, A ∪ B) = min{δ(x, A), δ(x, B)},
(D.4) δ(x, A) ≤ δ(x, A(α) ) + α, for α ∈ [0, ∞], where A(α) ≑ {x ∈ X ∶ δ(x, A) ≤ α}.
The pair (X, δ) is called an approach space [3,4].
Definition 2. A generalized approach space (X, ρ) [5,6] is a nonempty set X
equipped with a generalized distance function ρ ∶ P(X) × P(X) → [0, ∞] if and
only if for all nonempty subsets A, B, C ∈ P(X), ρ satisfies properties (A.1)(A.5), i.e.,
(A.1) ρ(A, A) = 0,
(A.2) ρ(A, ∅) = ∞,
(A.3) ρ(A, B ∪ C) = min{ρ(A, B), ρ(A, C)},
(A.4) ρ(A, B) = ρ(B, A),
(A.5) ρ(A, B) ≤ ρ(A, B (α) ) + α, for every α ∈ [0, ∞], where B (α) ≑ {x ∈ X ∶
ρ({x}, B) ≤ α}.
It has been observed that the notion of distance in an approach space is closely
related to the notion of nearness [2,7]. In particular, consider the Čech distance
between sets.
Definition 3. C̆ech Distance [8]. For nonempty subsets A, B ∈ P(X), ρ(a, b)
is the standard distance between a ∈ A, b ∈ B and the C̆ech distance Dρ ∶ P(X) ×
P(X) → [0, ∞] is defined by
⎧
⎪
⎪inf {ρ(a, b) ∶ a ∈ A, b ∈ B},
Dρ (A, B) ≑ ⎨
⎪
⎪
⎩∞,
if A and B are not empty,
if A or B is empty.
Sufficiently Near Sets of Neighbourhoods
19
Remark 1. Observe that (X, Dρ ) is a generalized approach space. The distance
Dρ (A, B) is a variation of the distance function introduced by E. C̆ech in his
1936–1939 seminar on topology [8] (see, also, [9,10,11]).
3
Description-Based Neighbourhoods
For N. Bourbaki, a set is a neighbourhood of each of its points if, and only if,
the set is open [12, §1.2] [13, §1.2, p. 18]. A set A is open if, and only if, for
each x ∈ A, all points sufficiently near 1 x belong to A.
For a Hausdorff neighbourhood (denoted by Nr ), sufficiently near is explained
in terms of the distance between points y and x being less than some radius r [14,
§22]. In other words, a Hausdorff neighbourhood of a point is an open set such
that each of its points is sufficiently close to its centre.
Traditionally, nearness of points is measured in
terms of the location of the points. Let ρ ∶ X × X ∶→
[0, ∞] denote the standard distance2 between points
y
in X. For r ∈ (0, ∞], a neighbourhood of x0 ∈ X is
ρ(x0 , y) < r
the set of all y ∈ X such that ρ(x0 , y) < r (see, e.g.,
x0
Fig. 1, where the distance ρ(x, y) between each pair
x0 , y is less than r in the neighbourhood). In that
case, a neighbourhood is called an open ball [15,
§4.1] or spherical neighbourhood [16, §1-4]. In the
plane, the points in a spherical neighhourhood (nbd)
Fig. 1. Nbd Nr (x0 )
are contained in the interior of a circle.
Next, consider visual neighbourhoods in digital images, where each point is an
image pixel (picture element). A pixel is described in terms of its feature values.
Pixel features include grey level intensity and primary colours red, green, and
blue with wavelengths 700 nm, 546.1 nm and 435.8 nm. respectively)3 , texture,
and shape information. Visual information (feature values) are extracted from
each pixel with a variety of probe functions.
A visual neighbourhood of a point x0 is an open set A such that the
visual information values extracted from all of the points in A are sufficiently
near the corresponding visual information values at x0 . In its simplest form, a
nbdv (denoted by Nrφ ) is defined in terms of a real-valued probe function φ used
to extract visual information from the pixels in a digital image, reference point
x0 (not necessarily the centre of the nbdv ) and ‘radius‘ rφ such that
1
2
3
...tous les points assez voisins d’un point x [12, p. TG I.3].
i.e., for x, y ∈ X ⊂ R, ρ(x, y) = ∣x − y∣.
The amounts of red, green and blue that form a particular colour are called tristimulus values. Let R, G, B denote red, green, blue tristimulus values, respectively. Then
define the following probe functions to extract the colour components of a pixel.
r=
R
,
R+G+B
g=
G
,
R+G+B
b = 1 − r − g.
20
J.F. Peters
X = {drawing visual pixels}, x, y ∈ X,
φ ∶ X → [0, ∞], (probe function, e.g., probe φ(x) = pixel x intensity),
ρφ (x0 , y) = ∣φ(x0 ) − φ(y)∣, (visual distance),
x0 ∈ X, (nbdv reference point),
rφ ∈ (0, ∞], (sufficient nearness bound),
Nrφ (x0 , rφ ) = {y ∈ X ∶ ρφ (x0 , y) < rφ }, (perceptual nbdv ).
φ(x1 )
φ(x2 )
φ(x3 )
φ(x0 , x2 ) < rφ
φ(x0 , x3 ) < rφ
φ(x0 , x1 ) < rφ
φ(x0 )
φ(x4 )
φ(x0 , x4 ) > rφ
Fig. 2. Sample Visual Nbd Nrφ (x0 ) in a Drawing
At this point, observe that the appearance of a visual neighbourhood can be
quite different from the appearance of a spherical neighbourhood. For this reason,
x0 is called a reference point (not a centre) in a nbdv . A visual neighbourhood
results from a consideration of the features of a point in the neighbourhood and
the measurement of the distance between neighbourhood points4 . For example,
φ(x0 ) in Fig. 2 is a description of x0 (probe φ is used to extract a feature
value from x in the form of pixel intensity). Usually, a complete description of
a point x in a nbdv is in the form of a feature vector containing probe function
values extracted from x (see, e.g., [17, §4], for a detailed explanation of the
near set approach to perceptual object description). Observe that the members
y ∈ Nrφ (x0 ) in the visual neighbourhood in Fig. 2 have descriptions that are
sufficiently near the description of the nbd reference point x0 , i.e.,
ρφ (x0 , y) = ∣φ(x0 ) − φ(y)∣ < rφ .
4
It is easy to prove that a visual neighbourhood is an open set.
Sufficiently Near Sets of Neighbourhoods
21
φ(x2 )
φ(x0 )
φ(x1 )
3.2: Nbd Nrφgrey
3.1: Monet meadow
Fig. 3. Sample Monet Visual Neighbourhood, rφgrey =10
Example 1. Visual Neighbourhood of a Point
For example, each of the points in the green shaded regions in Fig. 2 have
intensities that are very close to the intensity of the point x0 . By contrast, many
points in the purple shaded region have higher intensities (i.e., more light) than
the pixel at x0 , For example, consider the intensities of the points in the visual
nbd represented by the green wedge-shaped region and some outlying green
circular regions and the point x4 in the purple region in Fig. 2, where
rφ = 5 low intensity difference,
φ(x0 , x1 ) = ∣φ(x0 ) − φ(x1 )∣ < rφ ,
φ(x0 , x2 ) = ∣φ(x0 ) − φ(x2 )∣ < rφ ,
φ(x0 , x3 ) = ∣φ(x0 ) − φ(x3 )∣ < rφ , but
φ(x0 , x4 ) = ∣φ(x0 ) − φ(x4 )∣ > rφ , where φ(x4 ) = high intensity (white)
In the case of the point x4 in Fig. 2, the intensity is high (close to white),
i.e., φ(x4 ) ∼ 255. By contrast the point x0 has low intensity (less light), e.g.,
φ(x0 ) ∼ 20. Assume rφ = 5. Hence, ∣φ(x0 ) − φ(x4 )∣ > rφ . As in the case of
C. Monet’s paintings5 , the distance between probe function values representing
visual information extracted from image pixels can be sufficiently near a centre
x0 (perceptually) but the pixels themselves can be far apart, i.e., not sufficiently
near or far apart, if one considers the locations of the pixels.
Example 2. Visual Neighbourhood in a Landscape by C. Monet
For example, consider a xixth century, St. Martin, Vetheuil landscape by C.
Monet rendered as a greyscale image in Fig. 3.1. Let φgrey (x) denote a probe
that extracts the greylevel intensity from a pixel x and let rφ = 10 and obtain
the single visual neighbourhood shown in Fig. 3.2. To obtain the visual nbd in
5
A comparison between Z. Pawlak’s and C. Monet’s waterscapes is given in [18].
22
J.F. Peters
I
⊕⊕⊕
⊗⊗
⊕
⊗
Ω
⊗
⊗
⊕⊕ ⊕
⊕
Fig. 4. Sample Sets of Near Neighbourhoods
Fig. 3.2, replace the graylevel intensity of each point sufficiently near the intensity
φgrey (x0 ) with a green colour. The result is green-coloured visual nbd Nrφgrey
in Fig. 3.2. Notice that the pixel intensities for large regions of the sky, hills
and meadow are quite similar. This is the case with the sample pixels (points of
light) x0 , x1 , x2 , where the in ∣φ(x0 ) − φ(x1 )∣ < rφ and ∣φ(x0 ) − φ(x2 )∣ < rφ .
4
Nearness of Sets of Neighbourhoods
This section briefly introduces sets of visual neighbourhoods. Recall that a neighbourhood of a point x is a set of all points that are sufficiently near x. To measure
the nearness of collections of neighbhourhoods, we introduce a norm version of
the Čech distance.
First, a distance function ρ∥⋅∥ is defined in the context of a normed space. Let
X be a linear space over the reals with origin 0. A norm on X is a function ∥ ⋅ ∥∶
X → [0, ∞] satisfying several properties for a normed space [19]. Each norm on X
induces a metric d on X defined by d(x, y) =∥ x−y ∥ for x, y ∈ R [20]. For example,
let a, b denote a pair of n-dimensional vectors of numbers representing object
feature values (e.g., positive real values representing intensities of light reflected
from objects in a visual field), i.e., a = (a1 , . . . , ai , . . . , an ), b = (b1 , . . . , bi , . . . , bn )
such that ai , bi ∈ R0+ . Define a norm version of the Hausdorff lower distance [14,
§22] in the following way.
⎧
⎪
⎪inf {ρ∥⋅∥ (a, b) ∶ b ∈ B}, if B is not empty,
Dρ∥⋅∥ (a, B) = ⎨
⎪
if B is empty,
⎪
⎩∞,
where ρ∥⋅∥ ∶ Rn ×Rn → [0, ∞] is defined, e.g., by the ∥ ⋅ ∥1 norm called the taxicab
distance, i.e., ρ∥⋅∥ (a, b) =∥ a − b ∥1 = ∑ni=1 ∣ai − bi ∣. Then, a norm Čech distance
function Dρ∥⋅∥ ∶ P(X) × P(X) → [0, ∞] is defined by
⎧
⎪
inf{Dρ∥⋅∥ (a, B) ∶ a ∈ A,
⎪
⎪
⎪
⎪
Dρ∥⋅∥ (A, B) = ⎨
Dρ∥⋅∥ (b, A) ∶ b ∈ B}
⎪
⎪
⎪
⎪
⎪
⎩∞,
if A and B are not empty,
if A or B is empty.
Then Dρ∥⋅∥ (A, B) measures the lower distance between the descriptions of objects
in a pair of non-empty sets A, B.
Sufficiently Near Sets of Neighbourhoods
23
Let A, B ∈ P 2 (X) denote collections of sets. A merotopic distance νDρ ∶
P (X) × P 2 (X) ∶→ [0, ∞] defined as
2
νDρ (A, B) ∶=
∥⋅∥
sup
A∈A,B∈B
Dρ (A, B).
This is an example of what is known as a ε-approach merotopy termed an εapproach nearness on X [5].
Example 3. Merotopic Distance Between Collections of Nbds . A pair of
in Fig. 4. Let
greyscale digital images Ω, I is represented by the rectangles
X be the set of pixels of images Ω and I, i.e., X = Ω ∪ I. For simplicity, the
only feature considered is the greylevel intensity of the pixels in each image. Let
A ∈ P 2 (Ω) and B ∈ P 2 (I) denote sets of neighbourhoods containing geometric
shapes in Fig. 4.
The nearness of collections of neighbourhoods A ∈ P 2 (X), B ∈ P 2 (X) is measured with νDρ∥⋅∥ . In this example, it is reasonable to assume that νDρ∥⋅∥ (A, B) < ε
for some small ε, since, e.g., the disc-shaped objects in Fig. 4 have identical
shades of the grey and the remaining members of the collection have identical
greylevel intensities (Ω and I, are ε-approach near, i.e., sufficiently near). The
basic approach in this example has numerous applications in classifying digital
images (see, e.g., [17,21,22,23,5,24,25]). For an implementation of the merotopy
νDρ∥⋅∥ , see [26].
5
Concluding Remarks
Two recent outcomes of the proposed approach to measuring the nearness of collections of sets is the recent discovery of a number of new supercategories [24,25]
as well as a considerable number industrial applications (see, e.g., the use of
merotopies in detecting anomalies in power system equipment [25] and contentbased image retrieval [27]). A recent application of the proposed approach in
measuring nearness of neighbourhoods can be in the visual arts (see, e.g., [18]).
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