In this paper we recall the basic mathematical fundamentals of Formal Concept Analysis (FCA) and analyse the possibilities to define soft sets from a concept lattice. We propose two ways of finding soft sets in a concept lattice of FCA model. We give also an example of soft set in the framework of digital medical image. We prove the usefulness of working with soft sets for the ROI - RONI categorization of a medical image.
1. Models whose categorization objects belong to a space X and for which the categorization criteria are also defined within the space X; 2. Models whose categorization objects are in a space X and for which the categorization criteria are defined outside of the space X;
The soft set theory is a theory of categorization giving a categorization of a space X, following a set of parameters outside X.
Both soft set and FCA models are of the second type. It is for this reason that we decided to study the two models from the point of view of their basic mathematical concepts. We found that the FCA model was a special case of soft set. It is claimed that in particular classes of applications soft set modeling may be more profitable.
The paper is organized as follows. Section 2 presents the mathematical FCA model. Section 3 gives the basic concepts of soft sets theory. In section 4, two soft sets models of FCA are proposed. Section 5 is dedicated to an application of soft sets to digital medical image analysis. Finally, some conclusions are given in section 6. 2
The FCA model is presented following [
Definition 1. (Formal context) A formal context, K, is a triple K = (O, A, I) where O is a set of objects, A is a set of attributes and I is a binary relation from O to A defined by: ∀o ∈ O, a ∈ A, I(o, a) = 1 when the object o has the attribute a I(o, a) = 0 otherwise.
Starting from binary relation I, one defines two derivation operators I↑ and I↓.
Definition 2. (Derivation operators) Let P(O) and P(A) be respectively the set of all subsets of O and A.
The operator I↑ is defined as follows: I↑ : P(O) → P(A). For X ⊂ O, The operator I↓ is defined as follows: I↓ : P(A) → P(O). For Y ⊂ A, I↑(X) = {a ∈ A/I(o, a) = 1, ∀o ∈ X} I↓(Y ) = {o ∈ O/I(o, a) = 1, ∀a ∈ Y } (1) (2)
Three properties are established in [
Definition 3. (Formal concept)[
Definition 4. (A formal concepts order)[
Taking into account properties 2 and 3, it is proved [
Mathematical categorization theory contains several categorization models each based on a mathematical theory, be it algebraic, geometric, probabilistic or other. In classical models of categorization, the idea is to split a space X in categories taking into account one or several criteria defined also based on the space X. So these criteria are in some way internal to the space X.
A soft set is a model giving a categorization on a space X taking into account a cognitive element external to X. This feature of externality represents another point of view of categorization.
We give some basic elements from soft set theory [
We explore in this section two possibilities to interpret the FCA model as a soft set. The ”conceptual metaphor ” generating this parallel is that relations between objects and attributes in the FCA model can be viewed as parameters. We have two options : either classify objects or classify formal concepts from the FCA model.
In the first case, one classifies FCA objects and the set A of parameters is the set A of FCA attributes. That is the categorization is constructed following attributes.
In the second case, one classifies FCA formal concepts and the set A of parameters is a subset of cartesian product O x A of FCA model.
Remark 2. The second SS model (Soft set 2) is possible because of the duality between extension and intention in the FCA model and the property of Galois connection. 4.1
Definition 6. (Soft set 1) Let be the formal context K = (O, A, I) and its associated concept lattice G(K).
We define a soft set SS1 as follows: SS1 = {F1, A} where A⊂ A, F1 : A → P(O) defined by:
F1(a) = {o ∈ O / I(o,a) = 1 }, if a ∈ A
F1(a) = ∅, otherwise. 4.2
Definition 7. (Soft set 2) Let be the formal context K = (O, A, I) and its associated concept lattice G(K).
We define a soft set SS2 as follows: SS2 = {F2, A} where A⊂ O x A, F2 : A→ P(G(K)).
Remark 3. In this case, the categorization space is all the concept lattice. 1. One categorizes formal concepts, (not objects as in Soft set 1) and the clusters are related to a set of parameters chosen in the set O x A of FCA model. 2. F2 can be defined in several ways depending on the purpose of the categorization. The advantage is that one can choose as parameters, object-attribute pairs, therefore, take as the set of parameters A a subset of the space (O x A) of the FCA model. 5
From mathematical point of view, a grey-level digital image is a function I(x,y) defined on Z2 with values in [0, 255]. The space X is a discrete space. In medical image case, we need to distinguish the following elements: – the header (HD) containing patient informations, – the anatomical object (AO) – the background (BG) and, – the disease area (DA) inside the anatomical object.
In the following example, we process a medical image with FCA model but taking into account the point of view of soft sets. The image (see Figure 1) is a grayscale image of size 512*512.
The image is divided into 16 non overlapping blocks B1, B1,......B16. The size of each block is 64*64 pixels. Their position is given in Table 1. Locations are dependent on pixels.
The FCA attributes are the following: Entropy, Header(H), Background (BG) and Anatomical Object(AO). The FCA objects are the blocks. One applies Soft set 2 model.
In the digital medical image applications, image characteristics must be splited in two categories: the category of characteristics expressing the position versus the standard partition (background, header, anatomical object, disease area) with values 0,1 and the category of characteristics expressing the values of features related to the intensity function I(x, y).
We split entropy’s values into 4 intervals: Entropy 1 = [0, 0.25[; Entropy 2= [0.25, 0.50[; Entropy 3 = [0.50, 0.75[; Entropy 4 = [0.75, 1]. Table 2 shows the blocks and corresponding entropy values. The FCA context is described in figure 2 and the concept lattice in figure 3.
The set of parameters is defined on O x A of FCA model with O = {B2, B6, B9, B12} and A = {Entropy 4, AO}, so A = {B2, B6, B9, B12} x {Entropy 4, AO}. Function F2, F2 : A → P(G(K)) is defined by: for a pair p in A , F2(p) = the subset of formal concepts in the sub-lattice corresponding to a characteristic considered as the most important in the analysis. In our case, Entropy 4 is chosen as the most important characteristic. This corresponds to concepts in the sub-lattice in blue in figure 4. 6
In this paper we did a bridge between FCA model and Soft Set Theory. We have analyzed the the FCA model from the point of view of Soft Sets. Because of the fact that the attributes in FCA model can be considered as parameters in the categorization of objects, we propose two types of models of FCA as soft set: the first one that categorizes objects of the FCA model, the second one that categorizes formal concepts of FCA model. We can find out that the main theorem in FCA proved more than 20 years ago, notably that the concepts lattice is a Galois lattice make possible to view some sub lattice or path of concepts lattice as soft sets. All the examples of FCA are processed with Conexp 1.5. An application of FCA – Soft Set in the domain of medical image was presented. The point of view Soft Set may be useful in a definition of the texture for the medical image. Developing a tool dedicated in particular to process some soft sets F-type functions can be useful in the library of digital image analysis.