The Research Of Fractional Fourier Transform In Signal Processing And Image Filtering

Since 1980 the theory of fractional power was build up by Namias, the fractional Fourier transform has become a kind of new mathematics tool, expanding a new space for the research and application of signal processing field. The basic theory of fractional order system includes the defining and sense of fractional calculus, the method and implementation of fractional Fourier transform and fractional Laplace transform, and the description method and imitation implemented method of fractional order equation and fractional order systems that are based of these transforms. The wide application of fractional order system includes the study on fractional control field taking the basic on the theory of fractional order calculus operator and fractional order calculus equation and implementation of fractional Fourier transform in optics system and so on. The application field on FRFT is much more widely: self-adaptive filtering of chirp signal based on FRFT; the FRFT imaging theory on optic; the image signal filtering and compressing based on FRFT. These have been focused in research gradually.In the thesis, the text starts with the definition, property and the numeric compute method of fractional Fourier transform and carries out numeric compute imitation of one and two dimension signal. It describes different points between traditional Fourier transform and fractional Fourier transform in a figure expression manner. Then it discusses the sample process of one and two dimension fractional sample. The text focuses on the application of FRFT used in fractional order filtering system. Firstly it recommends the elements about fractional calculus filter and fractional Kalman filter in theory. It discusses some problems in imitation from the point of view of fractional order state equation. Then it uses the stained image as the study template, aims at the energy distributing of gray image in fractional order field and applies multi-channel and multi-stage filtering method which is basic on one-dimension fractional parameter estimation (optional quasi-Newton algorithm) to wiping the noise from images stained by Gaussian noise. It implements the work of wiping noise in fractional order field,obtains a preferable image and takes on the practical maneuverability as for the cost. The result of experiment indicates: comparing the image filtered by FRFT with the one filtered by FFT, the contour of foreground is much plain and the contour of background is blurry. The determining factor is the parameter estimated area which range should be chosen. As concerning the cost of experiment time, it also has the application value.