=Paper=
{{Paper
|id=None
|storemode=property
|title=Obtaining of a Minimal Polygonal Representation of a Curve by Means of a Fuzzy Clustering
|pdfUrl=https://ceur-ws.org/Vol-758/paper_8.pdf
|volume=Vol-758
}}
==Obtaining of a Minimal Polygonal Representation of a Curve by Means of a Fuzzy Clustering==
https://ceur-ws.org/Vol-758/paper_8.pdf
Obtaining the Minimal Polygonal
Representation of a Curve by Means of a Fuzzy
Clustering
Alexander Lepskiy
Higher School of Economics, Moscow, Russia
Abstract. The problem of obtaining of a minimal polygonal represen-
tation of a plane digital curve is treated. Means of a fuzzy clustering
method are used. The fuzzy clustering is realized by relations of similar-
ity and dissimilarity that are defined on a planar digital curve.
1 Introduction
As a rule the some set of features is extracted in image to analyse and recognize
an object in the image. We will distinguish between low- and high-level features
in the image. The low-level features are features that may be extracted without
information about a special location of certain parts of the image object [14].
In the contrary the information about special location of certain parts of the
image object is used to detect high-level features. The edges of image object, the
curvature of a curve on the image are the main low-level features in the image.
The low-level features are aggregated for receive a compact representation of an
image object. As a result we will receive a high-level representation of image
object, for example, a curve. The compact curve representation is necessary for
image compression, image vectorization etc. In general digital curve Γ depend
on a set of parameters so the number of parameters may be equal to the number
of points of digital curve. In this case a problem of representation is to find the
curve Γ ′ that depends from a smaller set of parameters and saves a main infor-
mation about the curve Γ . There are many methods of solving of this problem,
which may be divided into two groups – the group of approximate methods and
the group of interpolative methods. The methods of first group are based on the
replacement of digital curve Γ by a such curve of some fixed class that satisfies to
some conditions “nearness”. The methods based on the Bezier curves and on the
B-spline are the most popular approximate methods of finding the curve repre-
sentation [15], [13]. The application of those methods requires a prior detection
of knots of spline and this task is equal to a general task of a curve represen-
tation. The methods of second group assumes the choice of some set of points
on Γ and replacement of every part of curve between the two neighbor points
by the other curve from some fixed class (for example, class of segments, circles,
algebraic curves of some order etc.) in agreement with the defined optimal con-
ditions. The straight-line interpolation is called by polynomial representation of
curve. There are two main approaches of solving the task of polynomial represen-
tation of curve: heuristic and optimization. The algorithms based on extraction
of dominant points, on the using of split-junction procedure for a side of polygon
(for example, Douglas-Peucker algorithm), genetic algorithms [6], algorithms of
multiple smoothing [24], algorithms on the fuzzy logic [11] etc. are referred to
the algorithms of the first approach. These algorithms are rapid but not optimal
as a rule. Algorithms of the second approach assume to find such approximate
polygonal line which satisfied to a defined optimal condition. The conditions
which are considered as an optimal criterion are following: 1) the polygon with
fixed number of vertex must have minimal perimeter [23]; 2) the maximal dis-
tance between the points of the curve and segments of the polygon must be a
minimal [18]; 3) the number of segment of polygon with approximation error
must be a minimal [4]; 4) the area of a symmetric difference between the set
bounded by a closed curve and set bounded by the polygon must be a minimal
[28]; 5) the approximation error of polygon with a fixed length of a segment
must be a minimal [21]. Thus the algorithms of second approach solve tasks of
nonlinear optimization with boundary conditions. The majority of algorithms
mentioned above are suboptimal. Almost all algorithms of finding of compact
polygonal representation assumes the preliminary finding of basis set of curve
points which must be optimized with a respect to the chosen criteria. The set
of points with a high curvature is chosen as a basis set. At this paper we will
consider the approach to find polygonal representations of curve with a help of
fuzzy clustering methods. The main idea of this approach bases on a fact that
the quantitative low-level local features of a curve at the given point may be
considered as a degree of membership of this point to polygonal representation.
The curve itself is considered as a fuzzy set. Then a problem of finding of a
minimal representation of a fuzzy set may be formulated as a solution of a task
of a fuzzy clustering.
2 Statement of Problem
n−1
We will considered the plane discrete curve Γ = (gk )k=0 , gk = xk i + yk j. The
points gk , k = 0, ..., n − 1 , belongs to ZZ 2 and they satisfy to a condition of a
connectivity in the initial contour which will be used in an image processing. We
will consider the set of points of curve Γ as an ordering set. We want to extract
some subset B = {gi1 , ..., gil } of points of a curve Γ that will be a “good”
representation of Γ .
The minimal polygonal representations of curve must consist of such points
g of curve Γ which have a great information capacity relatively to a given set
of features {ωi }i∈I . We will consider only local features: low-level features of
curve. We may be consider those features as some functions of points of curve:
ωi (g), g ∈ Γ , i ∈ I. It will be assumed that ωi (g) ∈ [0, 1] for all g ∈ Γ ,
i ∈ I and ωi (g) ≤ ωi (h) , if the point h ∈ Γ is more informative than point
g ∈ Γ relatively of feature ωi . The normalized estimation of curvature and the
normalized variation of contour length after deletion of point g are by examples
of such features functions [2]. The function ωi (g) characterizes the degree of
membership of point g to set informative points of curve Γ relatively i-th feature.
Therefore the set of informative points of curve Γ relatively i-th feature may be
considered as a fuzzy set {(g, ωi (g)), g ∈ Γ } with membership function ωi . If
we consider the information capacity of points of curve Γ relatively to the set of
features {ωi }i∈I set Γ can be considered in terms of a fuzzy set with membership
function ω(g) = T (ωi (g)),Qwhere T (·) – t-norm on [0, 1]I [9]. For example,
T (ωi ) = min ωi or T (ωi ) = i∈I ωi . In general case we can use some nonnegative
i
function from feature µΓ (g) = f (ω (g)) as a membership function. Then we may
formulated the task of finding of such minimal fuzzy subset B of set Γ for which
the set {ω(g)}g∈B will be an optimal representation of {ω(g)}g∈Γ . Let’s specify
a statement of problem. Let’s consider α-cut Bα = {g ∈ Γ : ω(g) ≥ α{ of fuzzy
set Γ for some fixed value α ∈ [0, 1]. The set Bα is a some representation of a
contour Γ . It is necessary to find such value of parameter α ∈ [0, 1] that the
representation Bα will be minimal on the one hand and will be optimal on other
hand. The finding of minimal representation of a fuzzy set is a task of fuzzy
clustering. The main ways to solve a fuzzy clustering task were considered in the
works [19],[20], [5], [1] etc. The review of fuzzy clustering methods may be found
in [29]. The modern state of problem may is reviewed in [12]. One approach
to fuzzy clustering consists to definition of some functionals on the set of all
representations, which then are optimized to receive a desired clustering.
3 The Using of Similarity Relation
Let us consider representation
Bα of contour Γ , α ∈ [0, 1] with membership
ω ω (g)|Bα |, g ∈ Bα ,
function µα (g) = We are introduced into considera-
0, g∈ / Bα .
tion so called fuzzy similarity relation r(g, h) on Γ that is reflexive, symmetric
fuzzy relation satisfying to inequality |r(g, h) − r(g, e)| ≤ 1 − r(h, e) for all
e, g, h ∈ Γ for construction of identifying functional. The last inequality is an
equivalent to condition for strongly ∆ -transitive relation (respect to t-norm
a∆b = max{a + b − 1, 0}) [7]. The equivalence of strongly ∆-similarity (that is
reflexive, symmetric, strongly ∆-transitive relation) and ∆-similarity was proved
in [7]. The coherent nearness relation [3] is weak. By analogy with E.H.Ruspini
we called set Bα by fuzzy r-representation of set Γ if the inequality
X
r(g, h)µω
α (h) ≥ µΓ (g) (1)
h∈Γ
is holds for all g ∈ Γ . The efficiency of such clustering depends on a fuzzy
similarity relation r(g, h). The choice of this relation is P
defined by classification
features. In particular, the function r(g, h) = 1− n−1 ni=1 ρi (ωi (g), ωi (h)) is
the similarity relation, where ωi (g) is an informativity function of the i-th feature
of point g, ρi is such metric in R1 that ρi (a, b) ≤ 1 for all a, b ∈ [0, 1]. We will
consider the similarity relation r(g, h) = 1− |ω(g) − ω(h)| below. Then (1) take
the form X
|Bα | (1 − |ω(g) − ω(h)|) ω(h) ≥ ω(g)|Γ | (2)
h∈Bα
for all g ∈ Γ . It is obvious to see that Bα = Γ povided (2) is valid. Thus the task
consists of a maximal reduction of a cardinality of Bα (with increased α) until (2)
remains valid. The set Bα of minimum cardinality for which (2) is valid we will
call by minimal r-representation of set Γ and will denote by Bα . The following
inequality may be get from (2) if we considered that ω(g) < α if g ∈ Γ \Bα :
!
X |Γ | X
(1 − ω(h))ω(h) ≥ − ω(h) max ω(g). (3)
|Bα | g∈Γ \Bα
h∈Bα h∈Bα
Thus we proved the validity of the following proposition.
Proposition 1. If set Bα is a fuzzy r-representation of set Γ then (3) is correct.
The contrary of this statement may be is not true in general. The algorithm
of finding of minimal representation Bα consists of two steps: 1) to find the set
(1)
Bα of minimum cardinality for which (3) is valid; 2) to add (if it is necessary)
⌢
(1) (1) |Γ |
the set Bα by such points h ∈ Γ \Bα that (2) is correct. Let Γ = {hi }i=1
be a set of points of contour Γ ordered by decreasing of weights ω(h), h ∈ Γ .
Calculate the function
Xp
Q(p) := (1 − ω(hi ))ω(hi )
i=1
and let
|Γ | Xp
R(p) := − ω(hi ) max ω(gj )
p i=1 p+1≤j≤|Γ |
for p = 1, 2, ..., |Γ |. The minimum p for which Q(p) ≥
n R(p) will be define o
a
⌢ ⌢
(1)
boundary of partition of set Γ on two classes Bα := hi ∈ Γ : i = 1, 2, ..., p
(1)
and Γ \Bα as a consequence from (3). On the second step are to find such point
(1) (1)
h ∈ Γ \Bα for which (1 − |ω(g) − ω(h)|) ω(h) → max for all g ∈ Γ \ Bα ∪ {h} .
(2) (1)
We will check the validity of condition (2) for set Bα = Bα ∪ {h}. If (2) is
(2) (2)
not correct then we add the new point from Γ \Bα to the set Bα etc. We have
a question. Will we get a minimal fuzzy r-representation of curve Γ with help
of suggested algorithm indeed? The following proposition gives us the condition
when we will get the minimal fuzzy r-representation after the first step.
Proposition 2. If we get after the first step of algorithm such a representation
(1)
B = Bα that
X 2 |Γ |
(1 − ω(h)) ≤ 1 + |B| − (4)
B+1
h∈B
2 2 (1)
and |Γ | max ω(g) ≤ α |B| then Bα will be a minimal fuzzy r-representation of
g∈Γ
a curve Γ .
(1)
Proof. Firstly we show that the representation Bα formed on the first step of
algorithm is a minimal representation for which (3) is satisfied. To show this we
consider the set function
, !
X |Γ | X
φ(B) = (1 − ω(h))ω(h) − ω(h) ,
|B|
h∈B h∈B
where B ⊆ Γ such that h∈B ω(h) < |Γ |
P
|B| . Let φ(∅) = 0. Let us show that φ
|Γ |
is monotone set function. Let Si = h∈B ω i (h), δi = |B|+i−1
P
, i = 1, 2. Then
−S2
φ(B) = Sδ11−S 1
. We have
S1 − S2 + ω(g) − ω 2 (g)
ψ(ω(g)) = φ(B ∪ {g}) =
δ2 − S1 − ω(g)
P |Γ |
for such every g ∈ Γ \B that ω(g) + h∈B ω(h) < |B|+1 . Then φ(B ∪ {g}) −
2
(g) 1 −S2 )(δ1 −δ2 +ω(g))
φ(B) = δω(g)−ω
2 −S1 −ω(g)
+ ψ(S
(δ2 −S1 −ω(g))(δ2 −S1 ) ≥ 0 such as S1 ≥ S2 , δ1 ≥ δ2 . There-
fore φ is a monotone set function. In addition ψ (x) is a monotone function on
2
[0, 1]. Indeed, we have ψ ′ (x) = x −2x(δ 2 −S1 )+δ2 −S2
(δ2 −S1 −x)2
. Two cases are possible. At
the first case the minimal value of numerator of derivative ψ ′ (x) up to x = 1
if δ2 − S1 > 1. In this case we have ψ ′ (x) ≤ 0 if δ2 ≥ 1 + 2S1 − S2 ⇔ (4). At
the other case 0 ≤ δ2 − S1 ≤ 1 and the minimal value of numerator of deriva-
2
tive ψ ′ (x) up to x = δ2 − S1 and ψ ′ (x) ≥ 0 if δ2 − S2 − (δ2 − S1 ) ≥ 0. The
last inequality is true because δ2 − S2 − (δ2 − S1 ) ≥ δ2 − S1 − (δ2 − S1 )2 ≥
2
0 = (δ2 − S1 ) (1 − (δ2 − S1 )) ≥ 0. Therefore φ is a monotone set function and
φ(B ∪ {g′ }) ≥ φ(B ∪ {g′′ }) if ω(g′ ) ≥ ω(g′′ ). We may get two cases when we
(1) (1)
form the set Bα : 1) h∈B(1) ω(h) < |Γ(1)| ; 2) there exist such point g ∈ Bα
P
α |Bα |
(1)
that |Γ(1)| − ω(g) < h∈B(1) \{g} ω(h) < |Γ(1)| . The set Bα will be by set with
P
|Bα | α |Bα |
(1)
minimal cardinality which satisfied to (3) since the set Bα in the first case and
(1)
the set Bα \{g} in the second case formed from the points h ∈ Γ with maximal
(1)
value of feature ω(h). We will show the validity of inequality (2) for set Bα
if the condition of proposition obeys to completing of proof of proposition. If
(1)
g ∈ Γ \Bα , an inequality (2) will lead to inequality (3). Consequently it will be
(1) (1)
correct. Let g ∈ Bα . Then |ω(g) − ω(h)| ≤ 1 − α for any h ∈ Bα . Then we
have X X
(1 − |ω(g) − ω(h)|) ω(h) ≥ α ω(h) ≥
(1) (1)
h∈Bα h∈Bα
|Γ | |Γ |
α2 B(1)
α ≥ (1)
max ω(g) ≥ (1)
ω(g).
|Bα | g∈Γ |Bα |
The proposition is thus proved.
(1)
If we get on the first step of algorithm the set Bα for which conditions of
proposition aren’t satisfied we must carry out the second step of algorithm. In
this case we may to get the set Bα near to minimal in general.
Remark 1. The function rs (g, h) = 1 − |ω(g) − ω(h)|s , s ∈ (0, 1], may be used
in the capacity of similarity relation too. This function satisfies to all conditions
of similarity relation because the inequality (a + b)s ≤ as + bs is correct for
a, b ≥ 0, 0 ≤ s ≤ 1. It is obvious that the inclusion Bα (s1 ) ⊇ Bα (s2 ) is correct
for minimal rs -representation Bα (s) if 0 < s1 ≤ s2 ≤ 1 because rs1 ≤ rs2 in this
case.
4 The Using of Dissimilarity Relation
Other relation may be used in task of fuzzy clustering without similarity relation.
For example, the points of minimal polygonal representation must be located far
from each other on a curve Γ . We may introduce the fuzzy dissimilarity relation
regarding these conditions. This relation must be antireflexive, symmetric fuzzy
relation τ (g, h) and obeying to an inequality |τ (g, h) − τ (g, e)| ≤ τ (h, e) for
all e, g, h ∈ Γ . Note that definition of fuzzy dissimilarity relation is coordinated
with a fuzzy similarity relation that is introduced above. Let f (g) be membership
function ofnpoint g ∈ Γ to theoset of informative points. Again we will call the
set Bβ = g ∈ Γ : µfβ (g) ≥ β with membership function µfβ (g) by fuzzy τ -
representation of set Γ if the inequality
X
(1 − τ (g, h)) 1 − µfβ (h) ≥ 1 − f (g) (5)
h∈Γ
is correct for all g ∈ Γ . We will considered that condition (5) is valid if Bβ = ∅
and isn’t valid if Bβ = Γ . Thus the task is to increase maximally the cardinality
of Bβ (with decreased β) until (5) remains is valid. The set Bβ of maximum
cardinality for which (5) is valid we will call a maximal τ -representation of set
Γ and will denote by B̄β .
We will use the function τ (g, h) = l(g, h) as a dissimilarity relation. Here
l(g, h) is a minimal length of arc of the curve Γ located between the points
g, h ∈ Γ , that normed by length of all curve Γ . We will use also the functions
f 1 , g ∈ Bβ ,
f (g) = ω(g)|Γ | and µβ (g) = |Γ \Bβ | as a membership function
f (g), g ∈/ Bβ
of curve Γ and set Bβ correspondingly. Then inequality (5) can be rewritten:
X
|Γ \ Bβ | (1 − l(g, h)) (1 − f (h)) ≥ |Γ | (1 − f (g)) (6)
h∈Γ \Bβ
for all g ∈ Γ . Then the new formulation of task about seach of (r, τ )-representation
of curve Γ follows from (2) and (6). It is necessary to find such set B for which
the system of inequalities
X |Γ |
(1 − |ω(g) − ω(h)|) ω(h) ≥ ω(g),
|B|
h∈B
X |Γ |
(1 − l(g, h)) (1 − ω(h)) ≥ (1 − ω(g))
|Γ \ B|
h∈Γ \B
are holds for all g ∈ Γ . We have a question: in what case does the algorithm give
us the minimal (r, τ )-representation of curve Γ ? The next statement follows
from proposition 2.
Proposition 3. If after the first step of algorithm we get such representation
2
(1) (1)
Bα that (4) is true and |Γ | max ωI (g) ≤ α2 Bα , |Γ | min ωI (g) ≥ |Γ | −
g∈Γ g∈Γ
2
(1) (1)
0.5(1 − α) Γ \Bα then Bα will be a minimal fuzzy (r, τ )-representation of
closed digital curve Γ .
(1)
Proof. If the conditions of proposition are satisfied, then the set Bα will be a
minimal fuzzy r-representation as was shown in proposition 2. Now we should
(1)
proof that thePset Bα will be fuzzy τ -representation too. To show this, it’s
noticed that h∈A l (g, h) ≤ 0.5 |A| for closed curve and any point g ∈ Γ ,
A ∈ 2Γ . Then we have
X X
(1 − l (g, h)) (1 − ω(h)) ≥ (1 − α) (1 − l (g, h)) ≥
(1) (1)
h∈Γ \Bα h∈Γ \Bα
|Γ |
0.5(1 − α) Γ \B(1)
α ≥ 1 − min ω(g) .
(1) g∈Γ
Γ \Bα
The proposition is thus proved.
Fig. 1. The initial contour and the minimal polygonal representation of contour found
by fuzzy clustering method
The results of the algorithm of a research of minimal polygonal representa-
tion of contour are shown in Fig.1. On the Fig.1.a the representation was found
by fuzzy clustering method with help of similarity and dissimilarity relations sep-
arately. On the Fig.1.b the representation was found by fuzzy clustering method
with help of combined using of similarity and dissimilarity relations. We used
normalized estimation of curvature in the capacity of feature function ω(g) (see
[10]). Note that the quality of algorithm work may be improved if we will use
the fuzzy clustering for the few features.
5 Summary and Conclusion
In this paper we have considered two clusters in a polygonal representation
of a curve. The first cluster consists of points that belong to the polygonal
representation. The second cluster consists of points that not belong to the
polygonal representation. In case of the crisp clustering distance within one
cluster is small, whereas clusters are sparse, so two objects from different clusters
are distant. The notion of distance at this paper was replaced by similarity
and dissimilarity fuzzy relation. We have received the fuzzy clustering method
for polygonal representation. The quality of this representation depends on a
similarity and dissimilarity fuzzy relation.
6 Acknowledgement
This work was supported by the grant 11-07-00591 of RFBR (Russian Foun-
dation for Basic Research). The study was undertaken in the framework of the
Programme of Fundamental Studies of the Higher School of Economics in 2011.
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