计算机识别系统综述：第十届国际会议论文集

计算机识别

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《第十届计算机识别系统国际会议论文集》是"智能系统与计算进展"系列丛书中的一本，由马雷克·库尔济尼斯基、米哈伊尔·沃日尼亚克和罗伯特·布尔杜克共同编辑。该书聚焦于人工智能领域中最具有前景的方向之一——计算机识别系统。它收录了52篇经过精心挑选的文章，由模式识别领域的专家撰写，涵盖了该领域的核心方法论和当前研究的应用。书中内容分为几个关键部分： 1. 特征、学习和分类器：这部分讨论了在设计计算机识别系统时，如何选择和提取特征，以及如何运用机器学习算法训练分类器，如支持向量机、决策树或神经网络等。 2. 生物识别：这部分关注个人身份验证的生物特征分析，如指纹识别、面部识别、虹膜扫描等，这些技术在安全验证、移动设备解锁等领域有着广泛应用。 3. 数据流分类与大数据分析：随着大数据时代的到来，如何处理实时数据流并进行高效分类成为热点问题，这包括流式算法、分布式计算和实时数据分析的方法。 4. 图像处理与计算机视觉：涵盖了图像处理技术的发展，如图像增强、目标检测、图像分割，以及计算机视觉在自动驾驶、无人机监控等场景中的应用。 5. 医疗应用：展示了计算机识别技术在医疗领域的重要角色，如医学图像分析、疾病诊断辅助、基因组学数据分析等，推动了精准医疗的发展。 6. 应用篇：总结了计算机识别系统在各个行业的实际应用案例，如工业自动化、智能家居、金融服务、电子商务等，展示了其广泛的影响力。这本书对于计算机科学、人工智能和机器人技术的研究人员以及学生来说，是一本宝贵的参考工具。它不仅提供了深入的理论知识，还展示了最新的研究成果和技术趋势，对于理解和开发更先进、高效的计算机识别系统具有重要意义。通过阅读这些论文，读者可以了解到该领域的最新进展，并启发创新思维。

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Recognition of Fuzzy or Incompletely

Described Objects

Juliusz L. Kulikowski

(

)

Nalecz Institute of Biocybernetics and Biomedical Engineering,

Polish Academy of Sciences, 4 Ks. Trojdena Street, 02-109 Warsaw, Poland

jkulikowski@ibib.waw.pl

Abstract. Typical pattern recognition problem consists in assigning of

a given object (result of observation) to one of previously deﬁned similar-

ity classes of objects. The problem has an unique solution if the classes

are disjoint; otherwise it may happen that the considered object can be

assigned to a class only on a limited certainty level. A more general prob-

lem arises if the object being to be recognized has not been described

with a full accuracy. The situations of uncertainty consisting in miss-

ing some components of objects description and in inaccuracy of some

objects’ features or parameters description are considered. An approach

to the solution of the ill-described objects recognition based on the con-

cepts of relative logic is proposed. This makes the proposed approach

closer to a natural human decision making supported by intuition and,

as such, useful in the case of uncertainty concerning the input data of

the recognition problem.

Keywords: Pattern recognition

· Limited input data · Input data

uncertainty

· Information variables · Relative logic · Semi-ordering

1 Introduction

Classical pattern recognition problem is an idealized theoretical model of deci-

sion making consisting in assigning to an observed object the name of a class of

objects (a pattern) satisfying a predeﬁned criterion of similarity [1,2]. The notion

of an object may denote in this case a physical or graphical object, an acoustic sig-

nal or linguistic expression, a scene, a result of observation etc. Such formulation

of the problem admits not only some uncertainty about the membership of the

object to a similarity class but also a primary ignorance about the exact borders

of the similarity classes. The last can be described analytically or geometrically

(this being an exceptional situation), by probability distributions, membership

functions or as some environments of enumerable sets of exemplary representa-

tives of the classes. On the other hand, the objects to be recognized are repre-

sented by strongly deﬁned attributes, formally represented by points in obser-

vation space: vectors of measured parameters or strings of qualitative features.

 Springer International Publishing AG 2018

M. Kurzynski et al. (eds.), Proceedings of the 10th International Conference

on Computer Recognition Systems CORES 2017, Advances in Intelligent

Systems and Computing 578, DOI 10.1007/978-3-319-59162-9

Recognition of Fuzzy or Incompletely Described Objects 3

of partially visible simple geometrical image details [10]. Decision making in

the case of missing data have been analyzed in a wider context of data mining

methods [11]. However, the I-DOR problems have some speciﬁcity that should

be taken into consideration when attempts to solve them by methods adopted

from other areas are made. The consecutive sections of this paper are devoted

to introduction of basic notions (Sect. 2) and to the methods of solution of the

above-mentioned I-DOR problems (Sects. 3, 4 and 5). In the last section some

conclusions are given.

2 Basic Notions

2.1 General Notions

Along this paper the following notions will be used. A set of all possible objects ω

under consideration constitutes an object space denoted by Ω. The objects may

be of any ﬁxed physical or abstract nature, however, they are assumed to be

mutually distinguishable (for this purpose the symbols ω will be indexed). The

objects are characterized by their observed attributes which may be of numerical

(real, integer etc.), Boolean or any qualitative nature. The attribute space U will

thus be deﬁned as a Cartesian product of the sets U

(i)

of the values of simple

attributes:

U =

[i]

(i)

. (1)

It is assumed that in the attribute space U a similarity (reﬂexive, symmetrical

and transitive) relation σ holds and it generates a partition of U into a family

C =



(κ)



,κ∈ K, of subsets called similarity classes, satisfying the conditions:

(a)

∀κ, λ, κ = λ, C

(κ)

∩ C

(λ)

= ∅,

(b)



κ∈K

(κ)

= U .

The set K of tags assigned to the similarity classes is assumed to be ﬁnite (at

least two-element) and its elements are called patterns. Moreover, it is taken into

consideration a family

F =2

(2)

of subsets of the attribute space, called an observation space.Weusethesymbol

v (indexed, when necessary) for denoting the elements of F (the observations).

Remark that the observations satisfy the set algebra rules including the opera-

tions of sum (∪), intersection (∩) and asymmetrical diﬀerence (\)ofsets[12].

2.2 Information Variables

Let us take into consideration an observation space F . First, let us remind two

basic deﬁnitions:

Deﬁnition 1. A binary: (a) reﬂexive, (b) symmetrical and (c) transitive rela-

tion ≈ described in a non-empty set A is called equivalence of the elements of A.

4 J.L. Kulikowski

Equality of the elements (observations) of F is a simple example of their equiv-

alence.

Deﬁnition 2. A binary relation  satisfying the conditions of: (a) reﬂexivity,

(b) weak asymmetry (α



 α



∧ α



 α



holds if and only if α



≈ α



)and

The expression α



 α



should be read as: α



precedes α



or α



is followed by



. The well-known notion of a subset ⊆ (in the set theory sense [12]) introduces,

in particular, a semi-ordering of the elements of any family B of sets. Really, for

any sets β



,β



∈ B if β



⊆ β



then it also can be put β



 β



. We call a natural

semi-ordering of the elements of B the semi-ordering following from the notion of

subset. This, in particular, can be applied to the observations in F . Let us remark

that Deﬁnition 2 admits existence in A some pairs of incomparable elements,

for which is neither α



 α



nor α



 α



. Natural semi-ordering satisﬁes the

following mini-maximization properties: for any β



,β



∈ B if β



 β



then

(β



∪ β



) ≈ β



, (3)

(β



∩ β



) ≈ β



. (4)

This property, when necessary, can be extended on other types of semiordering.

Deﬁnition 3. If F is an observation space described on the basis of attribute

space U then we call information variable a pair:

=[F, ] (5)

where  is a semi-ordering relation described in F satisfying the following con-

ditions:

(a) it contains a natural semi-ordering of F as its sub-relation,

(b) it satisﬁes the mini-maximization properties.

The symbol  canbereadas“is not more certain that” while ≈ in this context

can be interpreted as “is equally certain as”.

Deﬁnition 4. Let S =



(1)

(2)

,...,v

(k)



be a sub-family of observations

of an information variable Z

. Any observation v

(i)

∈ S such that no other

observation v

(n)

∈ S, if not equivalent to v

(i)

, satisﬁes the condition:

(a) v

(i)

 v

(n)

, is called to be of maximal certainty in S,

(b) v

(n)

 v

(i)

, is called to be of minimal certainty in S.

It follows from Deﬁnition 4 that: (a) U is of maximal certainty in F , (b) empty

observation θ is of minimal certainty in F . It also follows from Deﬁnition 3 that

if F is a family of all measurable subsets of U and μ is a probability measure

described on U then μ generates a semi-ordering in F according to the principle:

(i)

 v

(j)

⇐⇒ μ



(i)



≤ μ



(j)



. (6)

Recognition of Fuzzy or Incompletely Described Objects 5

In similar way, it can be shown that observations described by fuzzy sets (with

a membership function generating the semi-ordering relation [13]) or by rough

sets (where only three classes of uncertainty are taken into consideration [14])

can also be considered as particular cases of information variables [9].

2.3 Relative Logic

The relation of semi-ordering used above to deﬁne the information variables is

also a basis of logical systems called topological [15]orrelative [16], which quan-

titative logical values of statements replace by their relative logical assessment.

The correspondence between topological logic and other types of logical systems

as well as a concept of logical space were described in [9]. Below, for our pur-

poses, the concept of relative logic will be presented in a slightly simpliﬁed form.

Deﬁnition 5. An ordered string of elements:

Λ =[Σ, S, Σ

∗

, ≈, ] (7)

where:

– Σ denotes a set of simple, linguistically correct, assertive sentences

(statements),

– S is a set of logical operators (∨, ∧, ¬, ⇒),

– Σ

∗

is a set of composite sentences based on Σ and S,

– ≈ is a relation of logical equivalence and

–  is a relation of weak logical semi-ordering described in Σ

∗

, and satisfying

the mini-maximization properties: ∀q





∈ Σ

∗

such that q



 q



it is



∨ q



) ≈ q



, (8)



∧ q



) ≈ q



(9)

is called a relative logical system (RLS).

The symbol ≈ in this context can be read as “is (logically) equivalent to”, while

the symbol  as “is (logically) not more true than”. Let us remark that in RLS

the concepts of “absolute false”or“absolute true” do not exist.

Example 1. Let X be an information variable described on a parameter space

U =[35÷ 42]. We take into consideration the following statements concerning

observations of X:

(1)

:“xisnear36.6

◦

C”,q

(2)

:“x is higher than 36

◦

C”,

(3)

:“x is less than 35.5

◦

C”,

Then the following logical assessment of simple and composite statements in

RLS are possible:

(1)

 q

(2)

(1)

≈ q

(2)



(1)

∨ q

(2)



≈ q

(2)



(1)

∧ q

(2)



≈ q

(1)

but also it may happen: q

(3)

 q

(2)



(2)

∧ q

(3)



≈ q

(2)

, etc. On the other hand,

it may happen that q

(2)

and q

(3)

occur mutually incomparable; such situation

can be denoted by q

(2)

? q

(3)

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