Fuzzy Logic in the Wavelet Framework

The translation of knowledge contained in databank into linguistically interpretable fuzzy rules has proven in real applications to be difficult. A solution to this problem is furnished by multiresolution techniques. A dictionary of functions forming a multiresolution is used as candidate membership functions. The membership functions are chosen among the family of scaling functions that have the property to be symmetric, everywhere positive and with a single maxima. This family includes among others splines and some radial functions. The main advantage of using a dictionary of membership functions is that each term, such as „small“, „large“ is well defined beforehand and is not modified during learning. After reviewing the connection between a Takagi-Sugeno fuzzy model and spline modeling, we show how a multiresolution fuzzy system can be developed from data by using wavelet techniques. For regularly spaced datapoints, a matching pursuit algorithm may be used to determine appropriate fuzzy rules and membership functions. For on-line problems, biorthogonal splines wavenets are taken to determine the fuzzy rules and the resolution of the membership functions. An alternative technique, based on wavelet estimators is also presented. Multiresolution fuzzy techniques, also known as „fuzzy-wavelet“, have found applications in fire detection. For instance, wavelet analysis has been combined with fuzzy logic in flame detectors for on-line signal processing. The resulting algorithms have greatly contributed to translate a new understanding of flames‘ dynamics into algorithms that are capable of discriminating between a real fire and possible interferences, such as those caused by the sun‘ s radiation.