# Recognition Algorithm Based on Improved FCM and Rough Sets for Meibomian Gland Morphology
## 1 Introduction
#### 2.1.3 Core and Attribute Reduction  
The concepts of core and attribute reduction are two fundamental concepts of the rough sets theory. The attribute reduction is the essential part of an information system, which can discern all dispensable objects from the original information system. The core is the basis of attribute reduction. The information system may not have only one reduction, the intersection of all reductions is called the core of the information system.  
Let \(P\) be a set of equivalence relations, and \(P\subseteq R\) and \(P\
eq\varnothing\). If \(ind(R)=ind(R-P)\), then the set \(P\) can be dispensed in the set \(R\), otherwise it cannot be dispensed. If each \(P\) in the set \(R\) is not dispensable, \(P\) is independent, otherwise it is dependent. If the set of condition attributes is independent, one may be interested in finding all possible minimal subset of attributes and the set of all indispensable attributes (core).  
Given an information system \(S=(U,A)\), in which \(U\) is a non-empty finite set and \(A=C\cup D\) and \(C\cap D\
eq\varnothing\), \(C\) indicates the set of the condition attributes, and \(D\) indicates the set of decision attributes. If \(B\subseteq C\) and \(d\subseteq D\), \(pos(d)=\cup\{\underline{B}(X)|X\in U/ind(d)\}\) is the relative positive region of the decision attribute \(d\) with respect to \(B\).  
Figure 1: Schematic diagram of the upper approximation and lower approximation.  
Let \(P\) and \(Q\) be equivalence relationship sets. If \(pos_{ind(P)}(ind(Q))=pos_{ind(P-\{R\})}(ind(Q))\), then \(R\) can be reduced by \(P\). The set of all irreducible equivalence relationships of \(Q\) in \(P\) is called the core of \(P\), and is denoted as \(core_{Q}(P)\). Core is the set containing the most important attributes for classification in the condition attributes, and without them, the quality of the classification will drop.  
The relation between the reduction of attributes' set and the core is as follows:  
\[core(P)=\cap red(P)\]  
The expression \(red(P)\) represents all the reductions of \(P\). The expression \(core(P)\) contains all the equivalence relations in the reduction of \(P\), which is the important and indispensable attributes' set in \(P\).  
The concept of \(core(P)\) has two meanings:  
1. the \(core(P)\) is used as the basis for the calculation of attribute reduction.
2. the \(core(P)\) is a feature set that cannot be eliminated in attribute reduction.  
The concept of \(core(P)\) provides a powerful mathematical tool for extracting important attributes and their values from the condition attributes by attribute reduction. The attributes in the set of condition attributes are not equally important, even some of them are redundant. The processing of attribute reduction aims to reduce the unnecessary condition attributes or remove redundant attributes in the information system, and obtain the smallest set of condition attributes that can ensure correct classification. In other words, the classification quality of the reduced attributes' set is the same as that of the original attributes' set. Under the condition of guaranteeing the classification ability of the information system, attribute reduction can get a simpler and more effective decision rule. Lastly, attribute reduction is not only the approach and method of obtaining classified knowledge from an information system, but also the focus and essence of the rough sets theory research.