Approximation Properties And High-dimensional Singularity Analysis Of Shearlet Transform

Because shearlet transform has highly directional sensitivity and the optimal sparse approximation,the shearlet transform has been widely used in application areas,such as image denoising,image segmentation,edge detection and analysis,image enhancement,fusion,texture classification,singularity analysis and inversion problem etc.In this paper,we commit ourselves to derive construction and properties of the cone-adapted shearlet frame and the shearlet frames arising from a group representation,approximation of the inverse shearlet transform and significance analysis of shearlet transform.Specifically,we obtain the main results are as follows:(1)Necessary conditions and sufficient conditions for the irregular shearlet systems to be frames are studied.We show that the irregular shearlet systems to possess upper frame bounds,the space-scale-shear parameters must be relatively separated.We prove that if the irregular shearlet systems possess the lower frame bound and the space-scale-shear parameters satisfy certain condition,then the lower shearlet density is strictly positive.We apply these results to systems consisting only of dilations,obtaining some new results relating density to the frame properties of these systems.We prove that for a feasible class of shearlet generators introduced by P.Kittipoom et al,each relatively separated sequence with sufficiently hight density will generate a frame.Explicit frame bounds are given.We also study the stability of shearlet frames and show that a frame generated by certain shearlet function remains a frame when the space-scale-shear parameters and the generating function undergo small perturbations.Explicit stability bounds are given.Using pseudo-spline functions of type ? and ?,we construct a family of irregular shearlet frames consisting of compactly supported shearlets to illustrate our results.(2)The generalized pseudo-splines and shearlet frames are studied.In this paper,we propose the generalized pseudo-splines of type ? and ?.This family is compact supported and has very good regularity.We use Fourier analysis to derive further important properties such as regularity,stability,convergence and linear independence.Furthermore,we construct a family of shearlet frames consisting of symmetric compactly supported shearlets.Specially,using generalized pseudo-splines,we construct shearlet frames having explicit analytical forms which is important for applications.For this family,we prove that they provide optimally sparse approximation of cartoon-liked images.(3)The approximation of the inverse shearlet transform using discrete series is studied.We first introduce the infinite series and finite series defined by the inverse shearlet transforms.We then investigate the shearlet generator space by introduced by P.Kittipoom et al,some important properties are given.For the shearlet generator space,we show that the infinite series tend to the function to be reconstructed in L2-norm as the sampling density tends to infinity.For the admissible shearlet space,we show that the finite series also tend to the function to be reconstructed in L2-norm as the sampling density tends to infinity.(4)The pointwise convergence of inverse shearlet transform in arbitrary space dimensions is study,which is new research objectives.The pointwise convergence is more important in practical application.For every pair of admissible shearlets,we show that although the integral involved in the inversion formula from continuous shearlet transform is convergent in L2 sense,it is not true in general whenever pointwise convergence is considered.We give some sufficient conditions for the pointwise convergence to hold.Moreover,for any pair of admissible shearlets we show that the Riemannian sums defined by inverse shearlet transform are convergent to the original function as the sampling density tends to infinity.(5)The significance analysis of shearlet transform is studied.In the processing process of high dimensional data,it is of important significance identifying the singularity of high dimensional square integrable functions,which can be used as foundation of high dimensional data processing.Firstly,in the paper,the reconstruction formula of high dimensional square integrable functions is given by using the continuous shearlet transform;Secondly,the decay property of shearlet coefficients of several special functions is studied;Finally,the singular support of square integrable functions is characterized by using the shearlet coefficients of reconstruction formula.Our result improves some known ones given by G.Kutyniok,S.Dahlke etc.