Conversion based fuzzy fractal dimension integrating self-similarity and porosity, via DFS and FIS (Mamdani and Sugeno systems)

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Highlights

  • In this study, a novel approach in geometry is introduced and developed.

  • An essential advantage of this study would be non-Euclidean geometry via integrating Fuzzy sets and Fractal geometry.

  • Other advantages of proposed FFDs would be that it brings into account every dimension category including the corresponding memberships.

  • The best feature of the cbFFDs can be served as their expertise definition which makes it closer to each other all engaging materials.

Abstract

Fuzzy logic via Fractal geometry is a novel mathematical combination. Using Conversion Based Fuzzy Fractal Geometry (cbFFG), researchers can approach to fractal geometry with the utilization of Direct Fuzzy Sets DFS and Fuzzy Inference Systems FIS1 in their immense fields of studies. It is proposed a new fuzzy fractal dimension FFD using membership functions instead of conventional log-log linear regression processes which needed more time and calculations. The results are derived from IP2 by developing the defined FIS via Fuzzy Analyses and/or for the defined direct membership function. As that of FD, the cbFFD range is [0, 3] for solid geometry and is [0, 2] for plane geometry with discrimination that the output results are upon fuzzy inference and so identifiable and interpretable. Here, cbFFDs are computed by two main inputs: 1) Porosity Conversion Based Factor and 2) Self-Similarity Conversion Based Factor. The outputs of the cbFFD process can be both interpreted by linguistic terms and graduated by the fuzzy numbers. The cbFFD makes the geometric dimensioning more understandable. The final values of cbFFDs can accurately be ranged between geometric and arithmetic means of the proposed cbFFDs or logically computed by prepared FISs.

Introduction

Fuzzy sets theory which was constructed by Zadeh in the 1960s has proven that many natural phenomena can be interpreted via fuzzy language and fuzzy inference systems [1].

A fuzzy language can be explained as a fuzzy branch of the set of strings over a finite alphabet [2]. The operations: union, intersection, etc. can be defined in AND, OR, NAND, NOR and other algebraic operations in fuzzy sets [3].

Fuzzy sets are classes of objects and/or members with a continuum of grades of membership [4]. These theoretical sets can be extended to real-life and there they are characterized by membership (characteristic) functions which devote to each object grades of memberships ranging between 0 (as never) and 1 (as always) [2].

In real life, many diverse sets of things can be classified as fuzzy sets. After all, their modification adverbs often fluctuate between never and always or better saying because they are probabilistic. As an essential part of real physics is the problem of dimensioning of objects –a branch of geometry– which differs from Euclidean to non-Euclidean. For example, no ideal triangle or ideal circle can be found in real objects simply as Pi number stretches to infinity, and the right angle is divided into too many infinitesimal degrees.

The definitions in daily spoken languages may deviate from the identity of scientific geometrical evaluations [5]. For bringing the languages closer to scientific geometry, it is proper to define probability into the existence of languages, thus it has been followed as the perception of stochastic languages [6].

Fractal, as a new branch of non-regular geometry, was introduced first by Mandelbrot in the 1960s decade. Afterward, in 1977 Mandelbrot introduced Fractal sets to prepare a calculation of degrees of regularity and similarity of natural structures in the physical systems. It is used in different ways with different results.

For years, fractal and multi-fractal geometries including dimensions were used vastly in various fields of study like medicine, (1D, 2D or 3D) analysis applications, pattern recognition, texture analysis, signal analysis and segmentation [7]. To utilize fractals, it is needed to recognize the fractalization and its linguistic terms [8]. Numerous tools were introduced to recognize the fractal dimension or multi-fractal dimension of shapes and images [9]. Describing natural phenomena by studying statistical scaling methods is not new [10,11]. Even in old ages, people were informed of these methods, but they were unable to calculate them mathematically [12]. The following studies were performed on this field to impact public views about geometry [9], [10], [11], [12], [13], [14], [15]. A large number of physical systems tend to present similar behaviors on different scales of observation [16].

The main problem with fractal geometry is finding the fractal dimensions FDs via an optimized methodology. There are many different methods for calculating FD, yet the real problem is the quickness of response and the number of stages of calculations. On the other hand, the exactness of calculations is an important factor which encounters with different errors in binarization, counting, rescaling, and regression steps. However, day to day needs for novel methodologies to decrease the naming errors are felt.

A new mathematical method of approaching the fractal geometry and also finding fractal dimensions can be known as fuzzy-based fractal analysis which can engage statistical scaling and probabilistic views to irregular geometry. In this case, essential factors in characterizing fractalization are known, and their parts in fractal dimensions are statistically calculated. The fuzzified fractal analysis prepares well-inferred dimensions for applications such as image processing, sound physics, physical space-time, medical image analysis, electrochemical patterns, digital images, signal analysis, etc.

The fuzzy fractal dimension of complex networks was published by [17]. Features of their article were: 1) A new fractal dimensional model based on fuzzy sets theory was proposed, 2) The complexity in their model reduced significantly from NP-hard problems and 3) This model could obtain a deterministic fractal dimension for a specific network [18].

It is published in an article entitled "A study on fractal dimensions and convergence in fuzzy control systems" by [19]. The above paper showed a problem in the realm of intelligence, knowledge-based systems, i.e., knowledge generation by preparing the automation of this duty [19].

As the introduction of our project in this paper, it is proposed two types of fuzzy fractal dimensions via conversion factors using percentages of two Inputs: 1) POROSITY and 2) SELF-SIMILARITY.

  • Type 1: DFS3 for finding cbFFD.4

  • Type 2: FIS5 for finding cbFFD

The idea of the present study is actually to systematically integrate naming theories: fuzzy sets and fractal geometry via a conversion basis. The main purpose is to derive more comprehensive and accurate results on the fractal dimension from preparing better resolutions for engineering, medical, environmental, etc. problems.

Novel PCBs6 are defined and utilized for the first time, they were not presented even nor pointed out in the literature. This is one of the outstanding novelties of this valuable project.

New word classes: Porosity and Self-Similarity are defined to cover the fractals more efficiently. The integration of the naming word classes in the premise of the fuzzy theory is and advantage of this paper for the problem of fractal dimensioning. The proposed cbFFDs are calculated via two methodologies: direct fuzzy set and fuzzy inference system. The result shows that naming methodologies outputs are close to those of traditional ones.

Extra parts such as rescaling, reboxing, log-log regression and repetitions are dropped; hence, this paper is known as the best saving time and energy methodology among fractal dimensioning methods.

It has relied on experimental experiences in calculations of cbFFDs (Nonlinear Calibration). The uncertainty gaps in various defined MFs7 are covered via Gaussian distribution. Gaussian MFs are utilized for more generality and more applicability in various case studies. The cbFFD is calculated only for DSSV8 of image or body, not for rescaled and repeated views, So many steps and calculations are dropped and the process is simplified.

Both Mamdani and Sugeno FIS types are invited for estimations of cbFFDs. The final output cbFFDs are mathematically calculated via arithmetic and geometric means and are logically estimated via prepared FIS. Then, cbFFDs are classified as linguistic terms and exhibited by numbers, words, and percent of each class. Conventional Fractal Dimensioning is the statistical and relative subject which is a time consuming and complex calculating method for estimation of fractalization of shapes in the multifaceted geometries. The following is noted as motivators for the present study:

  • 1.

    As the subject is statistical and relative so it can get help from the Fuzzy Sets Theory to cover the statistical population, minimize the relative errors and enhance the integration of fractalization with its components i.e. Porosity, Self-Similarity and so on.

  • 2.

    Fractalization in fuzzy mode can be degreed, leveled and so interpreted by cross points, guidelines and highlighted word classes.

  • 3.

    cbFFDs consume much less cost, time and calculations which reduces or eliminates numerous steps of conventional Fractal Dimensioning.

By cbFFDs researchers can approach FD by their insight and manipulated (Underestimations or overestimations) to overcome Field to Field variations, while conventional FDs are constant and in some cases not justified by researchers of a definite field.

Section snippets

Fuzzy sets

The theory of fuzzy set has been promoted for modeling of nonlinear, uncertain, and complex systems [20]. The important terms of a fuzzy inference system comprise membership function MF, fuzzy sets operations, fuzzifier, defuzzifier, inference unit, and knowledge base [21]. The membership function is defined as a curve or a set formula representing the degree of presence of elements in the scope of the set [21]. The input and output domains range between 0 and 1 or between 0 and 100 in the

Methods

The very experienced from the following projects helped much in the development and advancement of the present study: [7,23,24,[27], [28], [29], [30], [31], [32], [33], [34], [35]].

The idea of this method is rather different from the conventional fractal analyses. This methodology is the Conversion Based attitude toward fractalization and then fractal dimension. The concept arises from the notion that a body/image dimension is dependent upon a function of fractalization in definite scales:

Results and discussion

The methodology for analyzing conversion based fuzzy fractal geometry proposed in the study is a novel idea integrating two theories of fuzzy sets and fractal geometry through the determination of PCB via DFS or FIS. Yet, the concept of dimension introduced in the present scheme is a fuzzified approach toward the geometric dimension. Any figure/object can have an FFD in the range [0, 3] which is based upon deviation from the ideal three-dimensional background defined by the researcher. Of great

Conclusion

A novel approach toward the fuzziness of fractal geometry was studied to integrate diverse fields’ parameters with geometric physical dimensions. The computed FFDs introduced via the study are so friendly and case knowledge dependent. They can be utilized as helpful tools in practices such as image analysis, image processing, pattern recognition, defect detection, medical research, art psychology, astronomy, and so on.

An essential advantage of the study would be that the fuzzy fractal dimension

Author declaration template

We wish to draw the attention of the Editor to the following facts which may be considered as potential conflicts of interest and to significant financial contributions to this work. [OR]

We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.

We confirm that the manuscript has been read and approved by all named authors and that there are no other

CRediT authorship contribution statement

Hamid Sarkheil: Conceptualization, Project administration, Writing - review & editing. Shahrokh Rahbari: Conceptualization, Data curation, Formal analysis, Software. Behzad Rayegani: Validation, Writing - original draft, Writing - review & editing.

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