Software systems and computational methodsReference:
Fuzzification of convex bodies in Rn
Abstract.The paper is dedicated to the generalization of Matheron’s theorem about covariogram to the case where possible estimation error occurs, modelled by fuzzification of convex bodies. The classic case of identification of convex bodies does not consider the cases when input information and measurement contain error term. This is general issue when applying line segment distributions to recover covariogram and later the body itself. The authors define a body and ask what result will be for the length distribution for the given fuzzy body. Through this procedure the authors generalize Matheron’s theorem for this case. The authors extensively use fuzzy statistics and fuzzy random variables to extend convex bodies and length distribution functions to the fuzzy case. The authors use several properties of fuzzy numbers and fuzzy calculus techniques (mainly Aumann integration).The authors introduce generalized fuzzy distribution to apply them in a general setting of fuzzy convex bodies. Fuzzy convex bodies are defined by adding to the convex body and subtracting (in Hukuhara sense) from its fuzzy numbers in Rn. Then the generalization of Matheron’s theorem for a fuzzy case is derived, based on fuzzy function calculus techniques. Fuzzy convex bodies can be seen as a collection of convex bodies. The authors introduced fuzzy covariogram based on fuzzy convex bodies.
Keywords: Åstimation error, V-lines, Gaussian field, Aumann integration, Integral geometry, Fuzzy covariogram, Fuzzy distribution, Ñonvex body, Matheron’s theorem, Hukuhara Differentiability
Article was received:30-05-2019
The problem of identifying of convex bodies is widely known problem in integral geometry and several tools have been introduced to deal with it (starting from support function determination, and integral representation ending with random line length distribution and more complex structures like V-lines) [1-4].
There is also a general way of identifying the convex body through its covariogram, determination of which requires other sort of measurements as well. One of the techniques suggests taking line segments’ length distributions (for all directions) and recover from that covariogram of the body (for several types of convex bodies exact formula of reversion are known [5-6]).
Matheron’s theorem provides differential tool connecting covariogram and length distribution of a parallel line of given direction. However, in most cases, measurements will contain errors. This makes precise identification of body flawed. Instead of dealing with error bounds (in covariogram) and finding upper values for it, we deal with other form of error incorporation, fuzzy sets. By doing so, we try to describe all possible convex bodies which we could be recovered for given.
So we deal with the inverse problem in some sense. We define a body and ask what result we will get for length distribution for the given fuzzy body.
Through this procedure, we generalize Matheron’s theorem for this case. We extensively use fuzzy statistics and fuzzy random variables to extend convex bodies and length distribution functions to a fuzzy case. Use several properties of fuzzy numbers and fuzzy calculus techniques (mainly Aumann integration).
First, we generalize a fuzzy distribution function for our purposes. Then define fuzzy convex bodies as a natural generalization of the convex body to the fuzzy case.
Next, we derive some useful properties of fuzzy convex bodies and fuzzy covariogram. And derive Matheron’s theorem generalization for fuzzy case.
1. General fuzzy distribution
Suppose we have fuzzy random variable, i.e. random variable which takes on fuzzy numbers as values.
More formally we have , with
α-levels. As fuzzy numbers these α-levels must obey inclusion principle, i.e. for all ω-s for .
Generally, fuzzy densities are given by precise definition (for example in Viertl’s book [7, p. 55]; ). However, no case of discrete distribution is discussed. To define fuzzy distribution, we should analyze the idea of taking values smaller than or equal to given x.
Suppose we have two possible outcomes i.e. , and we have their probabilities .
Suppose we take and . Now we consider some x.
It is obvious that if then . It is also obvious that if then .
Now let’s consider the case where x is in between two fuzzy number.
So if we have then
However, if x belongs to the interval of α-levels we get the following:
For simplicity let’s suppose that , so that they have no intersection (or strictly ).
And suppose . Here we either can have that outcome corresponding either took value less than or equal to x, or value greater than x. So in this case
So in general will not be fuzzy number for each x.
The general case for arbitrary Ω would be.
This is general formula, for fuzzy distributions.
When we are speaking about fuzzy densities we require some interval for values of cdf-s.
The general framework is as follows.
with finite integral of both and . And such that there is some true density function in the interval with highest belief.
And fuzzy distributions for continuous case will be defined
So this obviously become fuzzy function in sense that for each x it will be fuzzy number.
This structure is somewhat intuitive, while the first structure is purely constructive.
For our case we can unintendedly encounter non continuous probabilities. So we should use general formulation.
2. Fuzzy convex bodies
Now let’s skip to our main problem. First of all, let’s define a fuzzy case of convex body . We will do it in the following way.
where is some fuzzy number with support in and with α-cuts. Here D is understood as fuzzy number with support in , whose membership function is D for any level α. Next
where minus is understood as Hukuhara difference. So is some fuzzy number for which it is possible. must be satisfied.
Now the final part is
and is defined as a fuzzy concept having as its α-levels .
Proposition 1. One can note that each level is a convex set, from the fact that Minkowski sum of two convex bodies is again convex. Putting it another way, if , then , for .
Proof. For cases a=0, and a=1, the proposition is obvious. Next, we give two simple arguments.
and are convex bodies, and so is convex. Moreover
Summing up, we get
Proposition 2. For , .
Proof: We must show, that each element of is also contained in .
Let’s first note that and . From which we have
is a convex set that must be summed with to get D. is a convex set that must be summed with to get D. As , . So we have
From which obviously if , then . So
and as G is convex. So .
Remark 1. So defined satisfy all criteria of fuzzy number , except that of elements of are convex bodies, not numbers.
3. Fuzzy covariogram and its properties.
Now we can define covariogram for (for covariogram in classical case ; ). Though as a function of this fuzzy argument it has not developed approaches for further investigation, we try to show that it is a fuzzy number, and will try to recover its membership function.
Let be the (n-1)-dimensional unit sphere with center at the origin.
Definition1. We will name fuzzy covariogram of the fuzzy concept, , with following α-levels
where and , with being n-dimensional Lebesgue measure.
Remark2. So fuzzy covariogram has α-levels, and this are collection of numbers.
Proposition 3. For , we have .
Proof. From Proposition 2 we have , so if , than , and
Proposition 4. For each , is a fuzzy number.
Proof. By the inclusion property for (proposition 2), we have, if
Next for completeness we should proof the continuity of membership function, which can be done in the following way. First, we will show, that the membership function satisfies the intermediate value property. And from boundedness we can imply that it is continuous.
To do it take the two layers. and , with . Let’s look at the smallest values and . From the inclusion property we have. . Whenever the equality is hold the intermediate value property is immediate. Suppose now .
Let’s take and , with and . This is guaranteed by the properties of . Now look at the convex combination of this two: . For each there is .
Obviously, there is such that for each , and
This proves that intermediate value property is satisfied. And from bounded of membership function, and monotonicity of its left and right bounds of each α-levels we can say that it is continuous. The same can be done with maximal values of covariogram. And for the remaining part, the function is constant.
Remark 3. In the above proof we extensively used the property of convex bodies, for dealing with Lebesgue measure of Minkowski sum of convex bodies, and to interchange the sign of intersection and addition. In general this is forbidden, as the sum may be even non-measurable .
Next, if we define covariogram as a function of the magnitude of translation (i.e. ), then we have the following. Here the translation is a vector (t,u), where defines direction, and t determines magnitude.
Proposition 5. defined by α-levels
Is the fuzzy function of t, i.e. the function of , which take fuzzy values.
Remark 4. The later proposition is crucial, as it gives the opportunity to use Hukuhara differentiation and Aumann integral concepts.
Proposition 6. is not t-Hukuhara-differentiable a.s. in .
The idea behind is that is getting “thinner”. Or in other word is 0 starting from some t. Suppose, then is not definable in Hukuhara sense.
So we should use Aumann integration concept instead.
First, we should define distribution functions for our case. The difference from the crisp(classical) case is that the projection space is different for different elements of . So first we should define .
We know that this is homeomorphic to 
And we assign a measure to each element of , .
Though we can combine them into a fuzzy concept, it is rather easier to leave it like this.
To use the definition of (1) for the fuzzy random variable, we should have the same universe of discourse, or in other words we should have the same Ω. However, Ω is different for different shapes. To bypass this problem, we first note that Ω has not to be the same for all α-levels. So we can deal with different Ω–s on different levels, but we should exactly fix Ω for any α-level. So we will consider the following distribution functions.
Where is length of intersection of a line parallel to u and intersecting or at some point ω.
The crucial theorem here is the following
Proof: To show this, we should literally show that any value of has corresponding with for which .
Not to embark our paper with much text, let’s just give the idea of proof. For any value of and , the maximal value is fixed by which corresponds to
Note that this set is always non-empty. This is guaranteed by the convexity of all sets considered. So we can use the convex hull of the union of sets. Moreover, this set contains only one element. The next bound is also fixed. So any value in-between can be obtained by considering a convex combination of these two bounds. No matter which convex body is considered, while in a given direction and for given magnitude it has the desired value of distribution function.
So, when we fix the Ω we can easily speak about .
And next theorem which comes along as generalization of already known result
Theorem 2. . (compare with the classical case in ). Here the integral used is Aumann integral.
Proof. This is immediate consequence of the previous theorem and Matheron theorem about covariogram. We have
where in the last line integration in Aumann sense is used, and on the previous line in Riemann-Lebesgue sense.
Measurement error could be incorporated into model trough different error addition. An example can be error described random variable from given distribution (or distribution family). However, this approach can have several issues. First of all, this could be misleading, as the reconstruction of covariogram will require reversion. More simply suppose that you have some error field added to the given convex body. From that point, you should get rid of the error part to get the body itself. However, this will again lead to many possible alternatives of convex bodies, which you could get, as you don’t know the error exactly. So the covariogram will itself represent the random field. In that situation, other approaches like support function representation can be more fruitful [13 – 15].
Next problem is concerned with possible unboundedness of error term. If we have Gaussian field the possible infinite of (very big error can occur), which is not the case of measurement error. So measurement error when dealing with convex bodies, are more likely to be bounded .
For further we are going to try to get a generalization of other formulas, connecting direction dependent distribution of length, covariogram and distribution of length of line segments .
Fuzzy convex bodies are the natural generalization of convex bodies when we want to incorporate measurement error into our model and derive all possible convex bodies that could give rise of the given direction dependent length distribution. From this point of view, fuzzy convex bodies can be seen as a collection of convex bodies (though at each level it contains not only convex bodies).
We introduced the natural generalization of fuzzy distribution function based on the ordinary distribution function. Introduced the concept of a fuzzy convex body. Discussed some of its properties making it similar up to domain with fuzzy numbers.
We introduced fuzzy covariogram based on fuzzy convex bodies.
Next, we construct a generalized fuzzy length distribution based on fuzzy distributions introduced previously. And lastly, we give equation connecting it with fuzzy covariogram, thus generalizing Matheron’s theorem to the fuzzy case.
The present research was partially supported by the RA MES State Committee of Science and Russian Federation Foundation of Innovation Support in the frame of the joint research project 18RF-019 and 18-51-05010 Àðì-à accordingly.
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