based on multivariate matrix factorization can be used in different areas. The algorithmic theory of polynomial ideals and modules can be modified to address problems in processing, compression, transmission, and decoding of multidimensional signals. The general multidimensional filter bank (Figure 7) can be represented by a pair of analysis and synthesis polyphase matrices and of size and , where N is the number of channels and is the absolute value of the determinant of the sampling matrix. Also and are the z-transform of the polyphase components of the analysis and synthesis filters. Therefore, they are multivariate Laurent polynomials, which have the general form: . Laurent polynomial matrix equation need to be solve to design perfect reconstruction filter banks: . In the multidimensional case with multivariate polynomials we need to use the theory and algorithms of Gröbner bases. Gröbner bases can be used to characterizing perfect reconstruction multidimensional filter banks, but it first need to extend from polynomial matrices to Laurent polynomial matrices. The Gröbner-basis computation can be considered equivalently as Gaussian elimination for solving the polynomial matrix equation . If we have set of polynomial vectors where are polynomials. The Module is analogous to the span of a set of vectors in linear algebra. The theory of Gröbner bases implies that the Module has a unique reduced Gröbner basis for a given order of power products in polynomials. If we define the Gröbner basis as , it can be obtained from by a finite sequence of reduction (division) steps. Using reverse engineering, we can compute the basis vectors in terms of the original vectors through a transformation matrix as: Mapping-based multidimensional filter banks Designing filters with good frequency responses is challenging via Gröbner bases approach. Mapping based design in popularly used to design nonseparable multidimensional filter banks with good frequency responses. The mapping approaches have certain restrictions on the kind of filters; however, it brings many important advantages, such as efficient implementation via lifting/ladder structures. Here we provide an example of two-channel filter banks in 2D with sampling matrix We would have several possible choices of ideal frequency responses of the channel filter and . (Note that the other two filters and are supported on complementary regions.) All the frequency regions in Figure can be critically sampled by the rectangular lattice spanned by . So imagine the filter bank achieves perfect reconstruction with FIR filters. Then from the polyphase domain