2-D wavelet transforms in the form of matrices and application in compressed sensing

As a signal analysis and processing method, wavelet transform (WT) plays an important role in almost all the areas in engineering today. However, compared to other traditional orthogonal transforms, such as DFT and DCT, The usually used fast wavelet transform (FWT) has its inconvenience in application. One frequently met problem is that FWT is rarely realized in the form of linear transformation by matrix and vector multiplication, which is the form that almost all the other existing orthogonal transforms take. That is because FWT dose not usually have an explicit transform matrix. As a result, FWT cannot be used in some cases where an explicit transform matrix is required. In this paper, we explore the matrix forms of 2-D discrete wavelet transform (DWT) and apply one of them in compressed sensing (CS). Our contribution is in two aspects: we give the equivalent 2-D DWT matrix that can be used to perform the 2-D DWT in the matrix form of 1-D DWT; meanwhile, we propose a separable 2-D DWT that is different from the traditional one and has some good properties.