Research And Application Of Finite Length Signal Boundary Process In Mallat Algorithm

Since wavelet analysis was proposed in 1970s,it attracted wide attention and developed fast. Wavelet transform is local transform of time (space) and frequency, able to effectively extract the useful signal. Due to its better property the wavelet transform is more useful than Fourier transform in application, especially when used to processing signal and image. In the image compression, wavelet image compression technology is better than Fourier transform and DCT image compression technology.Mallat algorithm proposed makes the wavelet transform by filter bank possible. However, Mallat algorithm assumes the input signal's length is infinite, but no matter one-dimension signals or two-dimensional image signal are all finite length in actual, which requires to do some boundary treatment of the input signals.Firstly, this paper briefly introduce the wavelet theory. Then, introduce sever extension methods, such as zero extension, smooth extension and the periodic extension. Discuss their analysis and reconstruction methods, compare their advantages and disadvantage. In a variety of algorithm, the paper focuses on the symmetric periodic extension algorithm, discusses detailedly the symmetry of finite length sequence, and symmetry of symmetric periodic sequences. Change the description method of the traditional symmetric extension, and unifies the symmetry center of the whole and half sample symmetric extension. Based on the unified symmetrical center, the paper deduce the symmetry and periodicity of symmetric periodic signal through two channel filter bank, and twice decimation. The paper presents the single level wavelet decomposition and reconstruction of two types of wavelet filter and find that with different filter banks the symmetry of signal after twice decimation is same as the input one. So when multilevel wavelet transform input signal of especially length its wavelet coefficients could be used for next level decompose directly, this method save a lot of repetitive movements. Traditional methods of multilevel wavelet decomposition is a repeat of one level, so after twice decimation the symmetric wavelet coefficient must be cut, after coder and ender the coefficient should extend. With the new method the cut and the extend are don't need. This method still adopt to signal without limit of length.Simulation results show that the symmetric periodic extension algorithm can achieve an accurate reconstruction without distortion. Symmetric periodic extension algorithm used for two-dimensional image wavelet decompose the reconstructed image has high quality and has obvious advantages compared to other extension method. In multi-level decomposition of wavelet transform, decomposition of special length and arbitrary length input signal are all able to realize accurate reconstruction, and the realize method is more convenience than the traditional method.