^{1}

^{1}

^{2}

In order to let machine better imitate thinking method of people to perform recognition and classification for fuzzy and uncertain thing, this paper puts forward a fuzzy and rough association method to deal with the problem. However, the application of fuzzy rough sets (FRS) will be introduced mainly on pattern recognition. Some related theories on FRS would be discussed, and some fuzzy rough mathematical methods on pattern-recognition will be given. Then, concrete applications of FRS on image processing and recognition will be introduced. Simulation result signifies that this fuzzy and rough association method is not only fast but also closer to nature attribute of thing for processing and recognizing image by comparing with the single neural network and other recognition device. The recognition rate is about 95.78%.

Because the features of many objective things have the uncertainties and ambiguities, the fuzzy set (FS) and the rough set (RS) had been proposed. Some relevant theories and applications of FS had been studied in [

Like single FS and RS approaches, its membership degrees are very artificial and not more accurate than FRS. When fusing old FS and new FS data of sequential process is required, single FS and RS that identify the target are not as good as target recognition of FRS with complete knowledge of FS and RS. As a flexibility of single FS and RS, single FS and RS methods are more thought to misuse than FRS method. The inferior performance of this single FS and RS relative to the FRS is shown by the simulation experiment. A main short of the single FS and RS methods stems from their lack of a systematic degree of membership update, but the FRS method has a systematic solid mechanism for degree of membership update of the weights thanks to the united union of FS and RS.

In previous works [

As an important aspect in target recognition is the image recognition, but the images obtained by the gather equipment not only include the recognized component of target, but also include other parts of non-target and some noises. Because of bright, illumination, hue and other reasons, the information of images is probably incomplete and fuzzy. These conditions will bring some difficulties to implement the feature extraction and exact matching for target image in next step, so it is necessary to perform the eliminating for some influences of side effect that are bring by the above factors. Therefore, in order to complete complex pattern recognition tasks in real-time, the FRS approach based on FRS theories proposed by the literatures [

The definition of FR proximity will be introduced according to the next to degree definition of [

All FRS A = 〈 A L , A U 〉 on the universal set Γ is denoted as F R ( Γ ) .

According to the proximity in [

Definition 1. Assume { Γ , R } to be a given approximate universal set. Let A , B , C ∈ F R ( Γ ) and λ = [ 0 , 1 ] . P ( A , B ) is called a FR proximity of FRS A and B if the mapping N : F R ( Γ ) × F R ( Γ ) → λ satisfies the following several things:

1) P ( A , B ) = P ( B , A ) ⇔ P ( A L , B L ) = P ( B L , A L ) and P ( A U , B U ) = P ( B U , A U ) ;

2) 0 ≤ P ( A , B ) ≤ 1 ⇔ 0 ≤ P ( A L , B L ) ≤ 1 and 0 ≤ P ( A U , B U ) ≤ 1 , P ( Γ , ϕ ) = 0 ⇔ P ( Γ L , ϕ L ) = 0 and P ( Γ U , ϕ U ) = 0 , where Γ is a universal set, ϕ is an empty set;

3) If A ⊆ B ⊆ C , then P ( A , C ) ≤ P ( A , B ) ∧ P ( B , C ) ⇔ P ( A L , C L ) ≤ P ( A L , B L ) ∧ P ( B L , C L ) and P ( A U , C U ) ≤ P ( A U , B U ) ∧ P ( B U , C U ) .

Where the P is called a FR proximity function on F R ( Γ ) . The approximate universal set { Γ , R } is called a FR proximity space.

According to the above definition, here will give a type of FR proximity as follows:

Theorem 1. If Γ = { u 1 , u 2 , ⋯ , u n } , then

P ( A , B ) = Δ 1 − 1 n ∑ i = 1 n | A ( u i ) − B ( u i ) | (1)

is a FR proximity of FRS A and B , where A = 〈 A L , A U 〉 , B = 〈 B L , B U 〉 ∈ F R ( Γ ) , u i ∈ Γ , A ( u i ) = 〈 A L ( u i ) , A U ( u i ) 〉 , B ( u i ) = 〈 B L ( u i ) , B U ( u i ) 〉 .

Define A ( u i ) − B ( u i ) = 〈 A L ( u i ) , A U ( u i ) 〉 − 〈 B L ( u i ) , B U ( u i ) 〉 = 〈 A L ( u i ) − B L ( u i ) , A U ( u i ) − B U ( u i ) 〉

In the real number region, when U is a closed domain Γ , i.e., U = Γ , then

P ( A , B ) = Δ 1 − 1 | Γ | ∫ Γ | A ( u ) − B ( u ) | d u (2)

is a FR proximity of FRS A and B , where | Γ | is a measurement of Γ , which is a length, area or volume.

Proof: the Equality (1) is proved as follows:

1) P ( A L , B L ) = 1 − 1 n ∑ i = 1 n | A L ( u i ) − B L ( u i ) | = 1 − 1 n ∑ i = 1 n | B L ( u i ) − A L ( u i ) | = P ( B L , A L )

and P ( A U , B U ) = P ( B U , A U ) , so P ( A , B ) = P ( B , A ) .

2) Since 0 ≤ A L ( u i ) , A U ( u i ) , B L ( u i ) , B U ( u i ) ≤ 1 , there are 0 ≤ P ( A L , B L ) ≤ 1 and 0 ≤ P ( A U , B U ) ≤ 1 , so 0 ≤ P ( A , B ) ≤ 1 .

Especially, P ( A L , A L ) = 1 − 1 n ∑ i = 1 n | A L ( u i ) − A L ( u i ) | = 1 , P ( A U , A U ) = 1 , so P ( A , A ) = 1 .

The same, P ( U L , ϕ L ) = 1 − 1 n ∑ i = 1 n | U L ( u i ) − ϕ L ( u i ) | = 1 − 1 = 0 , P ( U U , ϕ U ) = 0 , so P ( U , ϕ ) = 0 .

3) If A ⊆ B ⊆ C , then | A ( u i ) − C ( u i ) | ≥ | A ( u i ) − B ( u i ) | and | A ( u i ) − C ( u i ) | ≥ | B ( u i ) − C ( u i ) | ,

so 1 − 1 n ∑ i = 1 n | A ( u i ) − C ( u i ) | ≤ 1 − 1 n ∑ i = 1 n | A ( u i ) − B ( u i ) | and

1 − 1 n ∑ i = 1 n | A ( u i ) − C ( u i ) | ≤ 1 − 1 n ∑ i = 1 n | B ( u i ) − C ( u i ) |

Therefore, there is P ( A , C ) ≤ P ( A , B ) ∧ P ( B , C ) .

So, the Equality (1) is a FR proximity.

Similarly, the Equality (2) can be proved. At the same time, the proximity that is defined by the theorem 1 is called a FR 1-proximity.Q.E.D.

Here, two recognition methods of FRS are given. An immediate method is a max-principle of membership that applies mainly recognition of individuality. A mediate method is based on a principle of proximity that applies recognition of group model generally.

1) Maximum principle of membership

Definition 2. Assume A i ∈ F R ( Γ ) , ( i = 1 , 2 , ⋯ , n ) . For u 0 ∈ Γ , if there are an i 0 and an j 0 in order to make A L i 0 ( u 0 ) = max { A L 1 ( u 0 ) , A L 2 ( u 0 ) , ⋯ , A L n ( u 0 ) } , and A U j 0 ( u 0 ) = max { A U 1 ( u 0 ) , A U 2 ( u 0 ) , ⋯ , A U n ( u 0 ) } , then u 0 is believed to subordinate A L i 0 and A U j 0 relatively.

Moreover, according to the test need and the trial and error method, A i 0 and A j 0 can be determined.

2) Principle of proximity

Definition 3. Let A i , B ∈ F R ( Γ ) , ( i = 1 , 2 , ⋯ , n ) . If there is an i 0 in order to let P ( A L i 0 , B L ) = max { P ( A L 1 , B L ) , P ( A L 2 , B L ) , ⋯ , P ( A L n , B L ) } be true, then B L is believed to most near A L i 0 , i.e., B L and A L i 0 are believed to be congeneric. Similarly, if there is an j 0 in order to let P ( A U j 0 , B U ) = max { P ( A U 1 , B U ) , P ( A U 2 , B U ) , ⋯ , P ( A U n , B U ) } be true, then B U is thought to be the most close to A U j 0 , i.e., B U and A U j 0 are believed to be congeneric. The principle is called a principle of proximity.

The same, according to the test need, A i 0 and A i 0 can be also determined.

According to FRS theories proposed by the literatures [

First, let an image insert in a frame of box. At the same time, the frame is divided into many small grids. According to the degree of clarity of the image point in each small grid, an appropriate degree of membership μ i j of the image point is given. Moreover, according to the size of the positive region that the image point appears in the small grid, the approximate accuracy of the image

point is computed by α i j = | ( Δ i j ) L | | ( Δ i j ) U | . Thus, the important parameter pair

( μ i j , α i j ) is obtained. Where, i and j are the number of rows and columns of the grid, respectively; Δ i j is the image point within the grid which is located in the i t h row and the j t h column; ( Δ i j ) L and ( Δ i j ) U is the lower approximate and upper approximate, respectively; | Δ i j | denotes the size of the measure of Δ i j , that is, the area size of the small box. In this way, a fuzzy rough

relation matrix ( ( μ i j , α i j ) ) n × m can be created. Define μ i j = 1 to denote that

the image point appears clearly in the grid, and let it fill in the black. Similarly, define μ i j = 0 to denote that the image point does not appear in the grid, and let it be white; define 0 < μ i j < 1 to denote that this image point appear intangibly in the grid, and let this grid be a shadow.

By

S = [ ( 0 , 0 ) ( 0 , 0 ) ( 1 , 0.5 ) ( 1 , 0.9 ) ( 1 , 0.4 ) ( 0 , 0 ) ( 1 , 0.3 ) ( 1 , 0.6 ) ( 1 , 0.1 ) ( 1 , 0.2 ) ( 0 , 0 ) ( 1 , 0.2 ) ( 1 , 0.5 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 1 , 0.4 ) ( 1 , 0.4 ) ( 0 , 0 ) ( 1 , 0.1 ) ( 1 , 0.2 ) ( 0 , 0 ) ( 1 , 0.8 ) ( 1 , 0.3 ) ( 0 , 0 ) ( 1 , 0.5 ) ( 1 , 0.6 ) ( 1 , 0.7 ) ( 1 , 0.1 ) ]

This relation matrix is called a fuzzy rough standard matrix. In order to identify the image, the outline of the image is first analyzed to see which standard image i ( i = 1 , ⋯ , n ) it may be belong to. The standard matrices S i ( i = 1 , ⋯ , n ) that the i images are corresponding to are usually put in sample database. Moreover, the image (b) to be identified that is shown in

In simulation, assume 70 known target categories have been trained, and two characteristic parameters that are μ i j and α i j have been selected. Choose randomly 100 characteristic parameters according to the uniformity distribution, and distribute equiprobably and randomly to 70 target categories. Assume the selected error of the unknown target obeys the normal distribution, and the standard variance of selected error is 3 percent of corresponding known characteristic parameter. The calculation of Formula (1) is chosen as the discriminant function for target recognition in simulation, and then after the simulation is carried out to be 160 times, the correct recognition rate that can be obtained is about 95.78% by using the FRS recognition method.

In here, in order to show the effect of FRS is better than that of single FS and RS methods on target recognition, the simulation is given. In the union of fuzzy and rough methods, the target recognition is performed. The sampling is 160 times in simulation and sampling rate T is 1 second. The recognition curve of FRS method comparing with FS and RS methods is shown in

From

To error recognition curve of the difference of recognition value and true value, the recognition error of FRS method reduces gradually and trends towards stability, no matter what it is at the x position direction or at the y position direction. The mean-square error curve of FRS method at two-position direction is shown in

Based on the theory of FRS, this paper gives the theoretical knowledge of FR target recognition, puts forward FR proximity, and then introduces a kind of

thinking method for recognition, and gives a kind of target recognition method. Finally, the application of FRS in image target recognition is discussed. To compare with FS and RS identification devices alone, the simulation results show that FRS method has faster processing speed and the processing result is closer to the natural attributes of target itself. The potentiality of FRS method in application will open up a development space based on the practical application of FRS in many areas.

This work is supported by National 973 Program (No. 613237), Henan Province Outstanding Youth on Science and Technology Innovation (No. 164100510017), respectively.

Gu, D.H., Han, Z.Y. and Wu, Q.E. (2017) Application of FRS on Target Recognition. Open Journal of Applied Sciences, 7, 503-510. https://doi.org/10.4236/ojapps.2017.710036