Improved Jacobi-Fourier Moments

Image moments are important tools for image processing and analysis,and Jacobi-Fourier moments(JFMs),as generalized orthogonal moments,have attracted the attention of many scholars.Fractional-order Jacobi-Fourier moments(FJFMs)are further extensions of JFMs.By changing the radius of the radial basis function(RBF)r→ rα,JFMs have the characteristics of local feature extraction.But FJFMs cannot be used to extract the information of the middle area of the image and cannot accurately extract the information of a specific area.On the other hand,Zernike moments(ZMs),as the most popular special case of JFMs,have been applied to various fields such as image analysis and pattern recognition.However,zeros of ZMs’RBF bias toward large radial distance from the origin,which is not effective in extracting small image feature information at low order.Fractional Zernike moments(FrZMs)can adjust the zeros of ZMs’ RBF to make them uniform by changing the fractional parameters,but there is numerical instability at high order.Based on the above problems,this thesis proposes improved JFMs,the specific works are as follows:The JFMs are generalized to transformed Jacobi-Fourier moments(TJFMs),and based on this,adjustable Jacobi-Fourier moments(AJFMs)are proposed.AJFMs can compress the zeros of RBF to the interested interval through adjustable function,and then accurately obtain the local information of the image.Compared with FJFMs,AJFMs not only accurately extract the local feature information inside and outside the image,but also can specifically extract the feature information of the middle area of the image.A large number of experiments show AJFMs have better local reconstruction performance and stronger robustness to local noise than FJFMs.By generalizing ZMs into transformed Zernike moments(TZMs),logarithmic Zernike moments(LoZMs)are proposed.Both ZMs and FrZMs can be regarded as special cases of TZMs.By using the logarithmic function,LoZMs can not only adjust the zeros of ZMs’s RBF to make them relatively uniform in the[0,1]interval,but also avoid the numerical instability of FrZMs near r=0.Experimental results show that LoZMs have better performance in image reconstruction and pattern recognition than ZMs and FrZMs.In addition,the zero-watermarking algorithm based on quaternion also shows that LoZMs have certain advantages.