Study On Approximation Theory And Application Of Gaussian Filter

Gaussian filter is a kind of ideal filter, which has the minimum product of time duration and frequency bandwidth and perfect time-frequency characteristic. The Gaussian filtering theory is one of the basic theories in image processing,computer vision,communications,metrology and measurement,time-frequency analysis,wavelet and so on, and the Gaussian filter has been widely applied in those fields.Different from those traditional filters like the Butterworth filter and the Chebyshev filter whose design theories are almost perfect, Gaussian filter is not perfect in its design theory. In this dissertation, firstly, some important properties of the Gaussian filter are discussed; secondly, the comprehensive design and implementation of the Gaussian filter are developed; thirdly, some Gaussian filtering algorithms with high accuracy and high efficiency are applied to surface roughness measurement,roundness measurement and image edge detection,etc. Of course, the new design theory and implementation of the Gaussian filter are useful not only in these fields above but also in many other fields. To develop the design theory of the Gaussian filter, the main research subjects are explored as follows:(1) The main characteristics of Gaussian filter are analyzed including unit impulse response,unit step response,amplitude-frequency characteristic,phase- frequency characteristic and the minimum product of time duration and frequency bandwidth and so on. Analyzing these characteristics is helpful for us to study the Gaussian filter and its design theory as well as its applications more thoroughly.(2) Some approximation design methods of Gaussian filter are explored including the rational approximation functions, the cascaded butterworth filters and the cascaded moving average filters. The design methods of analog Gaussian filter,IIR type of digital Gaussian filter and FIR type of digital Gaussian filter are discussed. These new design methods make the difficult design issue of Gaussian filter ago easy now and the complicated design issue ago simple now.(3) The maximum and minimum amplitude characteristic deviations of the cascaded butterworth filters and the cascaded moving average filters from the ideal Gaussian filter are very close in position and opposite in direction. A better Gaussian approximation filter is the linear combination of the two kinds of cascaded filters above, its amplitude transmission characteristic deviation is less than 0.11%, whereas the amplitude characteristic deviations of the two kinds of cascaded approximation filters above are 2.2% and 1% respectively. The linear combination Gaussian filtering is also efficient and more accurate.(4) The B-spline function is studied for implementing the Gaussian filtering. Based on the variational principle with the tension condition, the Gaussian approximation filter from the spline function is presented. The two-cascaded Gaussian approximation filter, with the amplitude characteristic deviation less than 1% and the zero-phase-shift characteristic, is a novel approximation method. The filtering algorithm for this method is both simply in its algorithm structure and high in its filtering efficiency.From different respects, we can conclude many different design methods for the Gaussian filter. In other words, the design method of the Gaussian filter is not unique. The advanced design method yields high accuracy,high efficiency and easy realization for Gaussian filtering. The good design theory must result in many excellent application achievements as follows.(1) The novel digital filtering algorithms of Gaussian filter have been successfully applied in surface roughness measurement. The amplitude transmission characteristic in the ISO11562 standard can be realized by these Gaussian approximation filtering algorithms with high accuracy and high efficiency.The 2D Gaussian filter from extending 1D Gaussian filter is used to determine the reference surface for 3D surface topography measurement. The calculational quantity of the 2D filtering algorithm decreases more than one hundred times and the calculational efficiency increases very highly, compared with the traditional filtering method.(2) The novel digital filtering algorithms of Gaussian filter have been successfully applied in roundness measurement. A method for removing the end effects of Gaussian filtering is presented according to the periodicity of roundness signal. (3) The novel digital filtering algorithm of Gaussian filter has been successfully applied in the implementation of LOG filtering for image edge detection. Applying high accuracy and high efficiency Gaussian approximation filter in LOG filtering is helpful for multi-scale LOG filtering. In the case of multi-scale LOG filtering, only one parameter in the algorithm needs to be changed. This method is more convenient than the mask method in which the coefficients of the whole mask of LOG filter with different scales need to be refeshed. This is a new implementation for LOG filter.