Research On Fractional V-System And Its Application In Digital Watermarking

The development of modern information technology needs the support of mathematical theory.Among these,the orthogonal function system has attracted more and more attention in the field of computer image processing,industrial design etc.The V-system of degree k is a non-continuous orthogonal function,which includes both continuous differentiable polynomial and polynomial spline function of degree k,also contains the function of derivative discontinuous.Because of this,V-system of degree k has a strong expression ability for both continuous and discontinuous signal.V-system of degree k not only contains the orthogonality,completeness and regeneration properties,but also has multi-wavelet characteristics that different from other orthogonal functions.With the development of computer network technology,information resources are more rich.The cost of access to these information resources is lower.So the information resources security problems such as easily forged,tampered,copy should not be underestimated.Information security technology is developed based on these problems.The birth of the digital watermarking technology,leading a new trend of information security technology,provides a strong support for the protection of information security.Along with the advance of digital watermarking technology,digital watermarking technology based on frequency domain shows unique advantages.Based on the development of Fourier transform,wavelet transform,non-continuous orthogonal function system--V-system and digital watermarking technology,combined with the advanced achievements of this field at home and abroad,this paper mainly present a new kind of non-continuous orthogonal function system-fractional V-system and application it in digital watermarking.In this paper,the main research work is as follows:This paper constructs the fractional V-system.Through the research on construction of B-spline and fractional B-spline basis functions,we got the idea of starting the order of the basis function.Through researching V-system and Orthogonal Spline Functions Franklin function,we master that linearly independent functions expressed by truncated monomials set and expanded from Franklin functions,can get V-system through Gram-Schmidt orthogonalization.Inspired by this,we take the truncation monomial set that before "Gramm-Schmidt orthogonal as a breakthrough point to construct fractional V-system.Furthermore,it is proved that the fractional V-system has the properties of compact support,completeness,and regeneration and so on.This paper applies the fractional V-system in digital watermarking.Because of the fractional V-system has good characteristics,this paper presents when 0<??1,? = 1/2,the first 16 function of fractional V-system,after the image through fractional V-transform,by changing the frequency coefficients,add watermarking information.In this paper,we compared watermarking results of the fractional V-system and DCT,DWT,V-system the results show that the fractional order V-system is efficient.