Compactly Supported Tensor Product Complex Tight Framelets with Directionality

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

The Growth and Structure of Dark Matter Haloes

In this paper, we analyse in detail the mass-accretion histories and structural properties of dark haloes in high-resolution N-body simulations. Modeling the density distribution in individual haloes with the NFW profile, we find, for all main progenitors of a given halo, there is a tight correlation between its inner scale radius $r_s$ and the mass within it, $M_s$, which is the basic reason why halo structural properties are closely related to their mass-accretion histories. This correlation can be used to predict accurately the structural properties of a dark halo at any time from its mass-accretion history. We also test our model with a large sample of GIF haloes. The build-up of dark haloes in CDM models generally consists of an early phase of fast accretion and a late phase of slow accretion [where $M_h$ increases with time approximately as the expansion rate]. These two phases are separated at a time when the halo concentration parameter $c\sim 4$. Haloes in the two accretion phases show systematically different properties, for example, the circular velocity $v_h$ increases rapidly with time in the fast accretion phase but remain almost constant in the slow accretion phase,the inner properties of a halo, such as $r_s$ and $M_s$ increase rapidly with time in the fast accretion phase but change only slowly in the slow accretion phase. The potential well associated with a halo is built up mainly in the fast accretion phase, even though a large amount of mass (over 10 times) can be accreted in the slow accretion phase. We discuss our results in connection to the formation of dark haloes and galaxies in hierarchical models.Comment: 26 pages, including 10 figures. v2: some conceptual changes. Accepted for publication in MNRA