We propose an adaptive lifted discrete wavelet transform to locally adapt the ltering direction to the geometric ow in the image. The proposed approach re nes previous directional lifting approaches to achieve superior compression performance with less demand for computation. Additionally, a bandeletization procedure is combined with directional lifting in a uni ed framework to further remove the correlation in the wavelet coef cients. Up to 2.8 dB improvement in PSNR over the conventional 2-D CDF 9/7 wavelet transform for natural images is reported. Signi cant improvement in subjective quality is also observed.