A Kind Of Discrete Shearlets Coefficients Calculation

One of the most useful features of wavelets is their ability to efficiently approximate one-dimensional signal containing pointwise singularities. However, wavelets fail to capture the geometric regularity along the singularies of surfaces because of their isotropic support. For example, we have to use lots of the elements of2-D wavelet basis to represent precisely the images containing the edges, that is, more discontinuities along curves. Limitation of classical wavelets has led to several new representations in order to handle efficiently the geometrical features of multidimensional signals with spatially distributed discontinuities. Such representations contain basis elements with many more shapes and directions than the classical wavelet basis.Recently, a new representation scheme has been introduced by Gitta Kutyniok, Demetrio Labate, Wang-Q Lim and so on. This is so called the shearlets representation which yields nearly optimal approximation properties. A lots of researches has been done on shearlets. The fast algorithm for calculating the shearlets coefficients in discrete domain is proposed in this paper. As a result, the shearlet approach associated to a multiresolution analysis and the fast algorithm can be correlated to the classical mallat algorithm.The paper is structured as follows.(1)We describe some basic properties of shearlets and explain an intui-tive idea of frame. Furthermore, we analyse in detail how shearlets can pr-ovide efficient geometric representation.(2)We introduce the construction of a kind of discrete shearlets and some sufficient conditions of the frame.(3)Firstly, filters associated with shearlets generating functions are con-structed. Secondly, we show the relationshipof the filters at different jth level. Finally, the relationship of the shearlets coefficients at different jth level is given on the basis of the mallat algorithm.(4)We obtain the original coefficients by function sampling value. Furthermore, Application of the fast algorithm in image approximations is discussed.