  Tensor product real-valued wavelets have been employed in many applications
such as image processing with impressive performance. Though edge singularities
are ubiquitous and play a fundamental role in two-dimensional problems, tensor
product real-valued wavelets are known to be only sub-optimal since they can
only capture edges well along the coordinate axis directions. The dual tree
complex wavelet transform (DTCWT), proposed by Kingsbury [16] and further
developed by Selesnick et al. [24], is one of the most popular and successful
enhancements of the classical tensor product real-valued wavelets. The
two-dimensional DTCWT is obtained via tensor product and offers improved
directionality with 6 directions. In this paper we shall further enhance the
performance of DTCWT for the problem of image denoising. Using framelet-based
approach and the notion of discrete affine systems, we shall propose a family
of tensor product complex tight framelets TPCTF_n for all integers n>2 with
increasing directionality, where n refers to the number of filters in the
underlying one-dimensional complex tight framelet filter bank. For dimension
two, such tensor product complex tight framelet TPCTF_n offers (n-1)(n-3)/2+4
directions when n is odd, and (n-4)(n+2)/2+6 directions when n is even. In
particular, TPCTF_4, which is different to DTCWT in both nature and design,
provides an alternative to DTCWT. Indeed, TPCTF_4 behaves quite similar to
DTCWT by offering 6 directions in dimension two, employing the tensor product
structure, and enjoying slightly less redundancy than DTCWT. When TPCTF_4 is
applied 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 stage
filter bank in DTCWT.
