  Although tensor product real-valued wavelets have been successfully applied
to many high-dimensional problems, they can only capture well edge
singularities along the coordinate axis directions. As an alternative and
improvement of tensor product real-valued wavelets and dual tree complex
wavelet transform, recently tensor product complex tight framelets with
increasing directionality have been introduced in [8] and applied to image
denoising in [13]. Despite several desirable properties, the directional tensor
product complex tight framelets constructed in [8,13] are bandlimited and do
not have compact support in the space/time domain. Since compactly supported
wavelets and framelets are of great interest and importance in both theory and
application, it remains as an unsolved problem whether there exist compactly
supported tensor product complex tight framelets with directionality. In this
paper, we shall satisfactorily answer this question by proving a theoretical
result on directionality of tight framelets and by introducing an algorithm to
construct compactly supported complex tight framelets with directionality. Our
examples show that compactly supported complex tight framelets with
directionality can be easily derived from any given eligible low-pass filters
and refinable functions. Several examples of compactly supported tensor product
complex tight framelets with directionality have been presented.
