The adaptive wavelet transforms via lifting is proposed. In the transform, update filter is selected by the signal's
character. Perfect reconstruction is possible without any overhead cost. To make sure the system's stability, in the lifting
scheme of adaptive wavelet, update step is placed before prediction step. The Adaptive wavelet transforms via lifting is
benefit for the image compression, because of the high stability, the small coefficients of high frequency parts, and the
perfect reconstruction. With the adaptive wavelet transforms via lifting and the SPIHT, the image compression is
realized in this paper, and the result is pleasant.