Summary form only given. Image compression relies on efficient representations of images, and within smooth image regions, the wavelet transform provides such a representation. However, near edges, wavelet coefficients decay slowly and are expensive to code. We focus on improving the transform by incorporating adaptivity. Construction of nonlinear filter banks has been discussed, but the question of how to utilize the nonlinearities remained. We answer this question by describing our transform via lifting. Lifting provides a spatial domain framework for the wavelet transform. In the lifting formalism, wavelet coefficients are seen as prediction residuals from a linear prediction operation. Wavelet coefficients are large near edges because the linear predictors are built to interpolate low order polynomials. Our goal is to avoid this problem by adapting the predictor based on local image properties. In smooth regions of the image, we use high order polynomial predictors. We adaptively reduce the prediction order to avoid attempting to predict values across discontinuities.