the application requirements. The synthesis filters should be designed to reconstruct the input signal back from the subbands when the outputs of these filters are combined together. Processing is typically performed after the analysis stage. These filter banks can be designed as Infinite impulse response (IIR) or Finite impulse response (FIR). In order to reduce the data rate, downsampling and upsampling are performed in the analysis and synthesis stages, respectively. Existing approaches Below are several approaches on the design of multidimensional filter banks. For more details, please check the ORIGINAL references. Multidimensional perfect-reconstruction filter banks When it is necessary to reconstruct the divided signal back to the original one, perfect-reconstruction (PR) filter banks may be used. Let H(z) be the transfer function of a filter. The size of the filter is defined as the order of corresponding polynomial in every dimension. The symmetry or anti-symmetry of a polynomial determines the linear phase property of the corresponding filter and is related to its size. Like the 1D case, the aliasing term A(z) and transfer function T(z) for a 2 channel filter bank are: A(z)=1/2(H0(-z) F0 (z)+H1 (-z) F1 (z)); T(z)=1/2(H0 (z) F0 (z)+H1 (z) F1 (z)), where H0 and H1 are decomposition filters, and F0 and F1 are reconstruction filters. The input signal can be perfectly reconstructed if the alias term is cancelled and T(z) equal to a monomial. So the necessary condition is that T'(z) is generally symmetric and of an odd-by-odd size. Linear phase PR filters are very useful for image processing. This two-channel filter bank is relatively easy to implement. But two channels sometimes are not enough. Two-channel filter banks can be cascaded to generate multi-channel filter banks. Multidimensional directional filter banks and surfacelets M-dimensional directional filter banks (MDFB) are a family of filter banks that can achieve the directional decomposition of arbitrary M-dimensional signals with a simple and efficient tree-structured construction. It has many distinctive properties like: directional decomposition, efficient tree construction, angular resolution and perfect reconstruction. In the general M-dimensional case, the ideal frequency supports of the MDFB are hypercube-based hyperpyramids. The first level of decomposition for MDFB is achieved by an N-channel undecimated filter bank, whose component filters are M-D "hourglass"-shaped filter aligned with the w1,...,wM respectively axes. After that, the input signal is further decomposed by a series of 2-D iteratively resampled checkerboard filter banks IRCli(Li)(i=2,3,...,M), where IRCli(Li)operates on 2-D slices of the input