Some Results Of Periodic Framelet On L~2(Z_N~d)

Periodic wavelet systems in L~2([0,1)~d) have been employed in both the physical sciences and engineering with impressive performance.The fact that the collected data in these areas mentioned above are often discrete turns our attention to discussing the analog of the periodic wavelet systems on[0,1)d,i.e.,the discrete case on Z_N~d.The most common method to construct d-dimensional periodic wavelet frames usually make use of tensor product,in which case edge singularities are ubiquitous and play a fundamental role in high-dimensional problems such as image processing.Therefore,in order to overcome the shortcomings of tensor product periodic wavelet frame,this thesis constructs periodic discrete wavelet frame by dilation matrix.By taking different entries in the diagonal of the dilation matrix in this thesis,an anisotropic phenomena,i.e.,the direction of different axes have different dilation factors,will be derived.This thesis gives the matrix form of high dimensional fast fourier transform,and studies the d-dimensional wavelet frames and various properties,we construct discrete framelet transform.We also develop a fast decomposition and synthesis algorithm approach to implement such discrete framelet transform,which is based on the diagonal matrix with different scaling scales.In this thesis,first,we study the construction of first order periodic discrete wavelet frame transform on Z_N~d,the nature of the perfect reconstruction are presented.Then,we construct a class of finite multi-scale periodic discrete wavelet frames on Z_N~d by iteration step;At the same time we give a fast decomposition and synthesis algorithm of periodic discrete wavelet frames.In order to show the rationality and practicability of the results,two instances are constructed.Finally,we give another form of discrete framelet on Z_N~d.