The heuristic method simply uses an appropriate predefined T1 FMF function, such as triangular, trapezoidal, Gaussian, S, or p function, to name a few, to initially represent the distribution of the pattern data. The following are some frequently used heuristic membership functions. Choi and Rhee, Membership functions for fuzzy sets can be constructed by any method exact, heuristic and Meta heuristic, such as triangular, trapezoidal, Gaussian, S, or p function in the domain.
Four possible membership functions are presented in Fig. Where type III and polar are new generation of fuzzy membership function that can be used in several application in the control and classification domains. In the field of pavement management system this new generation of MF play a powerful link between several tools such as multi-resolution methods wavelet and beyond the wavelet methods , image processing, NN and expert system.
A possible membership function can be defined for every category by expert with any tools. For example using image processing techniques and Radon transform, several membership function generated and shows in Fig. More simple and complex functions can be used under the form of discrete and continues. Additionally several more advanced membership function which generate by automatic generator introduced. More applications in image processing frequently used heuristic membership functions that can be generally categorized in Table. By control parameters, one can select a various interval pattern.
Theses parameters usually trained and learned by experts. For feature i. Histogram based method HBM for membership function generation is another method which is more flexible than heuristic methods. Using smoothed histograms which generated by hyper-cube or triangular window and then normalized, the upper and lower membership function flourished and mapped to real data.
Selection a well trained parameters function to model the smoothed histograms has a tangible ramification on performances of MF generator system. To avoid over fitting lowest, the suitable degree of the polynomial function PF is stood out as the knee point of error. In our case, as a real example in control, Type, severity and extents of cracking in pavement surface transform in a transform realm to generate a simple features.
Simple features can use for generation of T1 FMF. Approximate parameter values such as the number, height, and location of peaks which related to cracking used to determine the optimal parameter values of the function. This means that we have a threshold that it considers as a crisp threshold. New again histograms crystallized upper and lower MFs fitted to PF. As dimensional parameters or overall size of problem increase, undesirably become more and more.
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These complexities arise due to the high process in smoothing and fitting. This is a challenging point that set in motion to product a new heuristic to handle computational load. Intersections operation which proposed for this aggregation expressed as. These methods enable them to transfer the knowledge when expert facing with N dimensional features.
These methods are applicable for images realm. We assert that this contribution is valuable. Nevertheless we would like to highlight that high process in discrete smoothing and fitting first 1DUMF and 1DLMF calculation and then aggregation faced us to problem to products an effective MF generator. The fuzzifier m in FCM, can be fired as a membership generator. According to IT2 FCM, two fuzzifier m 1 , m 2 are employed to control the blurring area in fuzzy domain.
Using type-2 fuzzy operations therefore is essential. The crisp center obtained mean of centers of defuzzification as the centroid obtained by the type-reduction according Eq. Their heuristic method summarized in Fig 6.
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These parameters have significant role in designing the FOU for a data set. In general, select unsuitable fuzzifier worth poor clustering. The fitting objective is to minimize the equation. Compactness: measures how close the spline is to the data that reflect to the summation term which weighed by the smoothing factor p,. Smoothness: measures the spline smoothness using its second derivative that reflect to the integral term weighed by 1 - p.
The smoothing spline f minimizes when. The default value for the weight vector w in the error measure is ones size x. Further, D 2 f denotes the second derivative of the function f. The default value for the smoothing parameter, p, is chosen in dependence on the given data sites x and y Pal and Bezdek, The smoothing parameter determines the relative weight to place on the contradictory demands of having f be smooth vs having f be close to the data. As p moves from 0 to 1, the smoothing spline changes from one extreme to the other.
See Fig. Using a simple method, we turned ultrafuzzy to the 3DRT fuzzy set. According a type II membership function, MF must be in [0,1]. Select a bigger h is worth a more enhanced distress for example in pavement distress detection and classification problem and smoother noisy background see Fig.
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In order to define a type II fuzzy set, one can define a type I fuzzy set and assign upper and lower membership degrees to each element to re construct the footprint of uncertainty Fig. For example, when Radon Transform is applied to wavelet modulus, a distress crack is transformed into a peak in radon domain. For example mean of 3DRT have variety range. According to the above Eq.
To extend the fuzzy membership to type II fuzzy sets, ultrafuzziness should be zero, if the MF can be selected without any ambiguous such as type I. The amount of ultrafuzziness will increase by rising uncertainly bound. The extreme case of maximal ultrafuzziness, equal 1, is worth to completely vagueness. The A n e a r and linear or quadratic H k a A cab be determined by q-norms,.
This basic definition relies on the assumption that the singletons sitting on the FOU are all equal in height which is the reason why the interval-based type II is used , Tizhoosh, Similarly, We are evaluated, proposed method, based four conditions Minimum ultrafuzziness, Maximum ultrafuzziness, Equal ultrafuzziness and Reduced ultrafuzziness that every measure of fuzziness should satisfy, which introduced by Kaufmann Kaufmann, In a similar way, we established that the new index is qualified for measure of ultrafuzziness in 3D domain with these conditions.
Image processing is one among interesting applications of 3DMF. Radon, , Miao et al. For the spatial case such as 3DMF, the fuzziness can be calculated as follows Tizhoosh, ;.
Similar 3DMF presented in section 4. The estimation of the MF also exanimate from the fitting of a cubic smoothing Spline, Mora et al. In the polar transform, as p moves from 0 to 1, the smoothing spline changes from one extreme to the other. Miao et al. ISBN All rights reserved. When Lotfi Zadeh invented fuzzy sets in , he never dreamt that the field in which they would be most widely used would arguably be the one that became the most hostile to the concept of fuzziness, namely control.
Introduction To Type-2 Fuzzy Logic Control : Jerry M. Mendel :
Perhaps this was because the word "fuzzy" in Western civilization does not have a positive connotation and suggests an abandonment of mathematical rigor, one of the cornerstones of control. NO YES. Selected type: Hardcover. Added to Your Shopping Cart. View on Wiley Online Library.
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