1-D and 2-D transforms from integers to integers

Substituting a real valued linear transform with an integer-to-integer mapping has become very important in lots of applications. This paper introduces a new kind of matrix decomposition method called lifting-like factorization, which leads to a theorem: every 2ⁿ-order real matrix with determinant norm 1 can be expressed as the product of one permutation matrix and at most three unit triangular matrices. Rounding error of this method is analyzed. Realization of 2D integer transform is also studied and it is shown that a 2D integer-to-integer transform cannot be realized by performing two 1D integer transforms separately. Left and right permutation matrices are introduced to reduce rounding error and an application of this method to intDCT is discussed.