Tensor product real-valued wavelets have been employed in many applicationssuch as image processing with impressive performance. Though edge singularitiesare ubiquitous and play a fundamental role in two-dimensional problems, tensorproduct real-valued wavelets are known to be only sub-optimal since they canonly capture edges well along the coordinate axis directions. The dual treecomplex wavelet transform (DTCWT), proposed by Kingsbury [16] and furtherdeveloped by Selesnick et al. [24], is one of the most popular and successfulenhancements of the classical tensor product real-valued wavelets. Thetwo-dimensional DTCWT is obtained via tensor product and offers improveddirectionality with 6 directions. In this paper we shall further enhance theperformance of DTCWT for the problem of image denoising. Using framelet-basedapproach and the notion of discrete affine systems, we shall propose a familyof tensor product complex tight framelets TPCTF_n for all integers n>2 withincreasing directionality, where n refers to the number of filters in theunderlying one-dimensional complex tight framelet filter bank. For dimensiontwo, such tensor product complex tight framelet TPCTF_n offers (n-1)(n-3)/2+4directions when n is odd, and (n-4)(n+2)/2+6 directions when n is even. Inparticular, TPCTF_4, which is different to DTCWT in both nature and design,provides an alternative to DTCWT. Indeed, TPCTF_4 behaves quite similar toDTCWT by offering 6 directions in dimension two, employing the tensor productstructure, and enjoying slightly less redundancy than DTCWT. When TPCTF_4 isapplied to image denoising, its performance is comparable to DTCWT. Moreover,better results on image denoising can be obtained by using TPCTF_6. Moreover,TPCTF_n allows us to further improve DTCWT by using TPCTF_n as the first stagefilter bank in DTCWT.
展开▼
