This paper deals with the implementation of Steiner point of fuzzy set. Some definitions and properties of Steiner point are investigated and extended to fuzzy set. This paper focuses on establishing efficient methods to compute Steiner point of fuzzy set. Two strategies of computing Steiner point of fuzzy set are proposed. One is called linear combination of Steiner points computed by a series of crisp

Associated with every closed bounded convex set in

The utility of the Steiner point extends beyond its definition as a robust center of a set of static points. Locating the Steiner point of an object is helpful for many tasks, because Steiner point is an invariant point of an object, while a transform is used on it in certain ways, such as growing uniformly in all directions, moving in a line, and rotating around an axis [

In the early years, much work has been done on some algebraic and analytic structure and behavior of Steiner points, such as linear translation, continuity, and even affine translation of an object. Three important properties were studied and known as basic properties of Steiner points, which are shortly denoted by commutation, addition, and continuity [

This approach focuses on implementing Steiner point of fuzzy set. The motivation is trying to find an efficient method to calculate the Steiner point of a fuzzy set. This paper is arranged as follows. In the second part, some definitions and properties of Steiner point of fuzzy set are investigated referring to the literatures. The third part discusses strategies to compute the Steiner point of a fuzzy set. Two main methods are proposed for calculating the Steiner point of a fuzzy set. In the fourth part, stability analysis of Steiner point of fuzzy set is proposed. In the fifth part, some experiments on image processing are presented. The last part of the paper contains the conclusions.

In the following let us suppose that

The Steiner point of

We denote by

Let

Based on Definition

Let

for any

for

Then

The following theorem is due to [

A function

for any

for any

for

As mentioned in [

Let

Let

Let

Let

For the general case, the Steiner point of a fuzzy set can be calculated as follows.

Let

From Lemma

Let

Now there are two strategies to compute Steiner point of a fuzzy set. One is to find a series of

In the case of step fuzzy set, we have fixed the number of

From Lemma

For the sake of convenience in computing Steiner point of a fuzzy set, we introduce the second strategy (which is similarly approximate). Firstly, consider the following definition.

Let

Let

We prove that

So (S1) and (S2) are satisfied. Rather, more

Another motivation is from [

If

It follows from [

For any Steiner point of fuzzy set

A Steiner point of

If it is reasonable, we can introduce the following definitions.

For

For

Now what we need is computing Steiner point based on crisp set, for a 2-dimensional case, based on polygon or convex polygon. For a 2D set, there are two steps to compute its Steiner point in numerical sense. The first step is to find all the convex points of the set and to form a convex polygon

From [

Supposing

Let us consider an example in the following. Suppose that

In this section, we investigate four fuzzy images as an example, which were proposed in [

Test images used for evaluating two strategies of computing Steiner point of fuzzy set. The linear combination of Steiner point is marked by “o” and the approximate Steiner point is marked by “+.” (A) Synthetic fuzzy image. (a) The Steiner point of image (A). (B) A slice of a 3D MRA fuzzy segmented image of a human aorta. (b) The Steiner point of image (B). (C) Microscopy images of a born implant. (c) The Steiner point of image (C). (D) Synthetic fuzzy image. (d) The Steiner point of image (D).

There are some technical issues that should be interpreted here. First, commonly not always all grays appear in an image, so we can find the minimum gray (denoted by

In this paper, every degree of

In Figure

Comparison of the Steiner points obtained by two methods.

Figure | Approximate S.P. | D-step S.P. | Distance | var1 | var2 |
---|---|---|---|---|---|

Figure |
(83.49, 86.67) | (82.95, 86.42) | 0.17 | 0.04 | 0.06 |

Figure |
(136.84, 123.30) | (136.27, 120.38) | 0.03 | 0.08 | 0.05 |

Figure |
(64.61, 120.46) | (63.39, 120.06) | 0.06 | 0.02 | 0.05 |

Figure |
(243.19, 165.84) | (242.26, 165.89) | 0.01 | 0.04 | 0.04 |

The Steiner point calculated by

This approach focuses on implementing Steiner point of fuzzy set and some properties of Steiner point on fuzzy set. We try to find some efficient methods to compute Steiner point of fuzzy set. Two strategies of computing Steiner point of fuzzy set are proposed, namely, the linear combination of Steiner point, which calculates the Steiner point based on the approximate

See Algorithm

if

define

define

for

compute the convex hull of the object in image

calculate the Steiner point of the object:

//

computing image distance between two neighbor gray:

record the Steiner point corresponding to the minimum varying image gray:

if

end if

end for

Function CalculateSteinerpoint(

denote

for

compute the external angle

end for

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by CMP Laboratory, Department of Cybernetics, Faculty of Electrical Engineering, Czech Technical University. One of the authors was supported by the Agreement between Czech Ministry of Education and Chinese Ministry of Education. This work was also supported by Blue Project of Universities in Jiangsu Province Training Young Academic Leaders Object and the National Natural Science Foundation of China (no. 61170121).

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