signal represented by the dimension pair (n1,ni) and superscript (Li) means the levels of decomposition for the ith level filter bank. Note that, starting from the second level, we attach an IRC filter bank to each output channel from the previous level, and hence the entire filter has a total of 2(L1+...+LN) output channels. Multidimensional oversampled filter banks Oversampled filter banks are multirate filter banks where the number of output samples at the analysis stage is larger than the number of input samples. It is proposed for robust applications. One particular class of oversampled filter banks is nonsubsampled filter banks without downsampling or upsampling. The perfect reconstruction condition for an oversampled filter bank can be stated as a matrix inverse problem in the polyphase domain. For IIR oversampled filter bank, perfect reconstruction have been studied in Wolovich and Kailath. in the context of control theory. While for FIR oversampled filter bank we have to use different strategy for 1-D and M-D. FIR filter are more popular since it is easier to implement. For 1-D oversampled FIR filter banks, the Euclidean algorithm plays a key role in the matrix inverse problem. However, the Euclidean algorithm fails for multidimensional (MD) filters. For MD filter, we can convert the FIR representation into a polynomial representation. And then use Algebraic geometry and Gröbner bases to get the framework and the reconstruction condition of the multidimensional oversampled filter banks. Multidimensional nonsubsampled FIR filter banks Nonsubsampled filter banks are particular oversampled filter banks without downsampling or upsampling. The perfect reconstruction condition for nonsubsampled FIR filter banks leads to a vector inverse problem: the analysis filters are given and FIR, and the goal is to find a set of FIR synthesis filters satisfying. Using Gröbner bases As multidimensional filter banks can be represented by multivariate rational matrices, this method is a very effective tool that can be used to deal with the multidimensional filter banks. In Charo, a multivariate polynomial matrix-factorization algorithm is introduced and discussed. The most common problem is the multidimensional filter banks for perfect reconstruction. This paper talks about the method to achieve this goal that satisfies the constrained condition of linear phase. According to the description of the paper, some new results in factorization are discussed and being applied to issues of multidimensional linear phase perfect reconstruction finite-impulse response filter banks. The basic concept of Gröbner bases is given in Adams. This approach