introduction The theory of shearelets分析和总结分析和总结.docx

The Theory of Shearlets In the following we will give a short introduction into the theory of shearlets. Unlike the traditional wavelet transform does not posses the ability to detect directionality, since it is merely associated with two parameters, the scaling parameter a and the translation parameter t. The idea now is to define a transform, which overcomes this vice, while retaining most aspects of the mathematical framework of wavelets, e.g., the fact that the associated system forms an affine system, the transform can be regarded as matrix coefficients of a unitary representation(正表示) of a special group, there is an MRA-structure associated with the systems. The shearlets satisfy all these properties in addition to showing optimal behavior with respect to the detection of directional information. The Continuous Theory: The basic idea for the definition of continuous shearlets is the usage of a 2-parameter dilation group(扩张组), which consists of products of parabolic scaling matrices and shear matrices. Hence the continuous shearlets depend on three parameters, the scaling parameter a 0, the shear parameter s ∈ R and the translation parameter t ∈ R2, and they are defined by ψa,s,t(x)=a-3/4 ψ ((Da,s-1 (x-t)), where Da,s = [a,-a1/2s;0,a1/2]. The mother shearlet function ψ is defined almost like a tensor product (张量积)by ψ(ξ ,ξ ) =ψ (ξ )ψ (ξ /ξ ), where ψ is a wavelet and ψ is a bump 1 2 1 1 2 2 1 1 2 function(凹凸函数). The figure on the right hand side illustrates the behavior of the continuous shearlets in frequency domain assuming that ψ and ψ are chosen to be compactly supported(紧支撑) in frequency 1 2 domain. The associated continuous shearlet transform again depends on the scaling parameter a, the shear parameter s and the translation parameter t, and is defined by SH (a,s,t) = f,ψ f a,s,t This transform can also be regarded as matrix coefficients of the unitary representation(σ(a,s,t)ψ)(x) = ψ (x)=a-3/4 ψ ((D (x-t)) of the a,s,t a,s-1 shearlet group S=R+