The lifting factorization of 2D 4-channel nonseparable wavelet transforms

One-dimensional (1D) wavelet transform was lifted successfully using the division with remainder of univariate Laurent polynomials. However, division with remainder does not exist in the case of a bivariate polynomial. Thus, it makes sense that the polyphase matrix of a two-dimensional (2D) nonseparable wavelet transform cannot be decomposed into the lifting format using the same solution as that of the 1D wavelet transform. In this research, we present a new lifting factorization method of two-dimensional four-channel (2D 4-channel) nonseparable wavelet filter banks. According to the uniform constructing format of high-dimensional multivariate wavelet filter banks, the general form of 2D 4-channel nonseparable wavelet filter bank is given. With these filter banks, the polyphase matrix of 2D 4-channel nonseparable wavelet transform is found and proven. Then, we present the lifting factorization of the polyphase matrix and some examples are demonstrated in the factorization procedures. Finally, the lifting performances of the proposed method are analyzed. This lifting method factorizes the polyphase matrix into the product of a series of unit lower left triangular numerical matrices, unit upper right triangular numerical matrices, diagonal numerical matrices, and diagonal polynomial matrices whose elements on the diagonal line are 1, x, y, and xy. The original filter banks that all leading principal minors of the numerical matrices are not equal to zero can be factorized into lifting format. The proposed method transforms the lifting factorization of the polyphase matrix into the decompositions of the numerical matrices without Euclidian division. Thereby, only multiplication and addition operations are performed with no Fourier transformation involved. When compared with the lifting method of the tensor product lifting wavelet transform and the contourlet transform, the proposed lifting method can extract more edge information of images. The computational complexity of the original 2D 4-channel nonseparable wavelet transform for image decomposition is N+1 times as much as that of the proposed lifting factorization method and the original wavelet transform is accelerated. Furthermore, the proposed lifting factorization method is faster than the conventional 2D 4-channel nonseparable wavelet transform based on Fourier transformation theory and convolution operation when the size of each filter in the latter is greater than 4(N+1). The proposed lifting factorization has better sparsity than that of its original 2D 4-channel nonseparable wavelet transform and other typical 2D 4-channel nonseparable wavelet transforms.