In 1992, Bamberger and Smith proposed the directional filter bank (DFB) for an efficient directional decompo- sition of two-dimensional (2-D) signals. Due to the nonseparable nature of the system, extending the DFB to higher dimensions while still retaining its attractive features is a challenging and previously unsolved problem. This paper proposes a new family of filter banks, named 3DDFB, that can achieve the directional decomposition of 3-D signals with a simple and efficient tree-structured construction. The ideal passbands of the proposed 3DDFB are rectangular-based pyramids radiating out from the origin at different orientations and tiling the whole frequency space. The proposed 3DDFB achieves perfect reconstruction. Moreover, the angular resolution of the proposed 3DDFB can be iteratively refined by invoking more levels of decomposition through a simple expansion rule. We also introduce a 3-D directional multiresolution decomposition, named the surfacelet transform, by combining the proposed 3DDFB with the Laplacian pyramid. The 3DDFB has a redundancy factor of 3 and the surfacelet transform has a redundancy factor up to 24/7.

Directional information is an important and unique feature of multidimensional signals. As a result of a separable ex- tension from one-dimensional (1-D) bases, the multidimen- sional wavelet transform has very limited directionality. Furthermore, different directions are mixed in certain wavelet subbands. In this paper, we propose a new transform that fixes this frequency mixing problem by using a simple “add- on” to the wavelet transform. In the 2-D case, it provides one lowpass subband and six directional highpass subbands at each scale. Just like the wavelet transform, the proposed transform is nonredundant, and can be easily extended to higher dimensions. Though nonseparable in essence, the proposed transform has an efficient implementation based on 1-D operations only.