The Promotion Of V-System And Its Application

To explore the effective mathematical tools(orthogonal function and orthogonal transformation)for signal analysis more deeply,a new orthogonal functions system composed of piecewise polynomials is constructed from the Legendre polynomials,and it is called generalized V-system of degree k.Compared to the V-system,the expression construction of the generalized V-system of degree k is more concise,without solving any equations or orthogonalization.Then the equivalence relation between the generalized V-system of degree k and the V-system of degree k is proved,and the discrete matrix corresponding to the generalized V-system of degree k is built,by which the generalized V-transform of degree k is constructed.It is proved that the generalized V-transform maintains almost all the properties of the V-transform,and can accurately reconstruct the discontinuous information,also has a fast algorithm for degree one,which makes it superior to the V-transform as designing fast algorithms for the V-transform is difficult.The experimental results show that the V-transform and generalized V-transform can reconstruct the original image accurately,however,since the basis function of the Fourier transform is continuous,the Fourier transform cannot accurately reconstruct such separated(intermittent)objects with finite terms.The speed of the generalized V-transform is much higher than that of the V-transform,which is faster than the Fourier transform,and the larger the number of interpolation points,the more obvious the time efficiency advantage.Constructing two other orthogonal fast transforms based on fast generalized V-transform:one is the improved fast generalized V-transform,and the other is the combination of the generalized V-transform and the oblique transform—the fast generalized V-slope transform,reflects the diversity of fast algorithms for generalized V-systems.The time complexity of the six orthogonal transforms of V-transform,fast generalized V-transform,improved fast generalized V-transform,oblique transform,fast generalized V-slant transform and DCT transform are given.When orthogonal transformation is performed on the one-dimensional vector of N points,the fast generalized V-transform and the improved fast generalized V-transformation have the least amount of operations,which greatly reduces the time complexity.The experiment also compares the mean square error of the six orthogonal transformed images,and finds that the fast generalized V-transform can maintain good performance in image compression while the computational complexity is greatly reduced.