Iterative decomposition and reconstruction are needed in Mallat algorithm. In order to realize perfect reconstruction

finite-length signals must be extended to some extent before they can be transformed. The algorithm based on periodic boundary-extension always can be seen in the literature. Symmetric boundary-extension has better performance than periodic method in image processing

whereas the matrix transform method based on symmetric boundary-extension is seldom mentioned in the literature. A method of constructing decomposition and reconstruction matrices with arbitrary wavelet transform depth in mirror-symmetric boundary-extension is proposed for wavelet transform in matrix-vector multiplication

and the condition for perfect reconstruction of Mallat algorithm is proved. As an example

the base vectors and base graphs of Bior33 wavelet were given. The application of wavelet transform matrices in the wavelet-based image processing can avoid iterative operation

simplify the calculation and meanwhile reduce the edge effect evidently.