Adaptive Signal Compression Based Vision Entropy

Following the development of digital age,the information which need to save,transport and transace is increasing exponential and the great deal pressure on it make it nessary to compress the infor-matin befor transport, which require to compression.Compression can be classed into lossless compression and loss compression,the virtue of lossless compression is to get a higher compression ratio,but it can only limit to some situations in whicn the effect after decoding is can be sustained.In the view of mathematic, it concerns the approximation theory and harmonic analysis.Wavelets lead us to study signal compression,noisefilter and other problems from a new point of view.On the one hand because the compaty of wavelets basis and the multiresolution of the decompression of wavelets,nonlinear approximation is equated to an adaptive approximation;and on the other hand because the uncon-ditions of the wavelets basis,it make it the best basises for a class signal in nonlinear approximation,so many signal can get a sparse represention with the help of wavelets basises.In this paper we consider the compression of signal .Inspired by the wavelets,we introduce the concept of geometric wavelets. With no effection on human vision,that is in the meaning of vision entropy, we transform the compression into a best uniform approximation which is familiar to us in mathematic, we approximate the signalwith polynomials.We bring forward a complete arithmetic and make theoretical analysis and error estimate.Comparing with the result in wavelets,we get a satisfied result.In prefce we introduce the compression and make a review on the recent developments.In the second chapter we introduce the basical stepes in compression with the example of wavelets.Compression contain two con-tents,encode and decode.In the course of encode we make wavelets thansform in signal,then quantify the wavelets coefficents and code ,thus we get the compressed data and the inverse course is just the decoding.In the third chapter we introduce the concept of VE:vision entropy,and give it a represention in mathemantic,it correspond best uniform approximation which is konwn to us.We review the theory aspect in best uniform approximation and Remes arithematic in discrete situation, the content here is the foundation of the following chapter.The main work of our owner is in the fourth chapter. According to the explanation in the third chapter,we consider of the error with the distance definition of || ? j^. First we give it a explanation in mathematic.In interval [a, fe],for a given n times piecesmooth function f(x),we hope to find a piecewise polynomials p*n{x) with total degree n, fora given so > O,satisfytha,t is to sayBefore offering the arithmetic we give the concept of geometric wavelets in interval [a, b}Suppose (fii is the best uniform approximation polynomial of / in fi,thenis the geometric wavelets of / on Q.For a function /,we can get its best uniform approximation polynomial in O and complete the first layer approximation for /,and in the first Iyer we have i =f1-(piwheren := r},/i := f\ = /, I with the resolution relies on the Lipsechtz regularity of the character of f(x) in interval (a, 6) .The...