Shearlet Fast Transform And Application

As a multiscale geometric transform, shearlets has many advantages. Incomparison, as a multiscale geometric analysis, shearlets have many advan-tages. Compare with curvelets, there are three main advantages. First is thatangles are replaced by slopes when parameterizing directions which signifi-cantly supports the treating of the digital setting. Second is that shearletsfit within the general framework of a?ne-like systems, which provides an ex-tensive mathematical machinery. Third is that shearlets provide a unifiedtreatment for the continuous and digital world similar to wavelets.In this essay, we mainly introduced a multiscale analysis structure, pro-posed by B. Han, etc., applied on the tight shearlet framework , which demon-strates the good quality of multiscale structure of wavelet. We also introducedthe corresponding UEP rule and a fast decomposing algorithm. According tothe methods mentioned above, we constructed the tight shearlets frameworksystem, programmed for it, and applied it to the specific image processingproblems.In the theory of multiscale analysis, the decomposition technique will stillrely on subdivision and transition operators. In general a?ne-like systems,we denote as follows: For a d×d invertible integer matrix M and a finitelysupported sequence al, 1≤l≤r, the subdivision operator Sa,M and thetransition operator Ta,M are defined by Shearlet systems can be regarded as a special case of the a?ne-like systems. Inshearlet systems, matrix M are composed by two matrices, where one indexesscale, and the other indexes orientation. For each c > 0 and s∈R, let theparabolic scaling matrix A_c and the shear matrix Ss denote of the formTo illustrate the idea of MRA in the situation of shearlet systems, wewill discuss one step of the decomposition algorithm. For v∈l(Z~2), letSl, l = 1,···,r be a selection of 2×2 shear matrices, let al, l = 1,···,r befinitely supported masks, and let s, 1≤s≤r be the separator between low-and high-frequency part. Similarly with the a?ne-like systems, we computethe next level of low-frequency coe?cients and the high-frequency coe?cientsbyThe next step continues with decomposing vl, l = 1,···,s. The total numberof decomposition steps is J. Ensure that the necessary and su?cient condi-tions for perfect reconstruction is the following theorem, which we could cointhe'Shearlet UEP'.Theorem 0.1 Let Ml ,1≤l≤r be a section of 2×2 shear matrices,and let al ,1≤l≤r be finitely supported masks. Then the followingconditions are equivalent(i) For all v∈l(Z2),(ii) For anyω∈Ω=∪lr=1ΩSlA4, whereΩSlA4 Following is the application of the theorem mentioned above. In thisessay, we only considered one scale, which is M0 = 2I2, i.e., the dilationmatrix is M = M0Ss, where Ss is shear matrix or shear-like matrix. Then,we only considered one kind of lattice point set,i.e., M0Z2. For this kind oflattice point set, and one fixed shear or shear-like matrix Ss, we can get a onedimensional filtering sequencewhere u1 is the low frequency filter, and u2, u3 is the high dimensional fre-quency filter. Then by using tensor product, we got a two dimensional filteringsequence a1,a2,···,a9, in which a1 is the low frequency filter, and the othersis the high dimensional frequency filter.According to the corresponding conclusion of the a?ne-like system, forlattice point set M0Z2, we have su?cient freedom to select the shear orshear-like matrix. To achieve the directionality, there are totally 12 suchmatrices, Sl, l = 1,···,12, which are all related to di?erent rotation angles.These radians of 12 rotation matrices approximately cover the whole plane.So we can achieve su?cient directionality. These 12 matrices are respec-tivelySince for a fixed Ss, we have the filter sequence a1,···,a9, then for signalv∈l(Z~2), we totally have 12 sets of filters, which are completely independent.They construct the tight framework sets together. Notice that we have torenormalize every involved filter in the sets. It can be proved that we canachieve directionality by using the simplest system: dilation matrix M = 2I2and tensor product filters, but using shear matrices in the middle. Here wedid the shearlet or shearlet-like transform for 12 radians, which can cover the whole plane. Therefore, the method in this essay can achieve su?cientdirectionality.So far, we obtained the complete shearlet transform process. Firstly, forthe noisy image, and a fixed matrix from those 12 shear matrices or shear-likematrices, and do the image shear transform. Secondly, use one set of tightframework to do the decomposition on the image transformed in the first step.We need to keep the low-pass coe?cients after the decomposition, and dothreshold de-noising for those high-pass coe?cients in each scale. Thirdly, wemake the reconstruction to those coe?cients after the treatment in the secondstep. Finally, do the corresponding shear or shear-like inverse transform forthe reconstructed image, which obtains the image after de-noising.Through numerical implementation, the de-noising e?ect of the proposedshearlet transform is better than that of the framelet transform. However, formore complex images, such as the texture image, our approach is still to beimproved.