Wavelets provide sparse representations for natural images. Yet, smooth edge contours are represented with as many wavelet coefficients as supported by the length of the contour. We believe this to be far from optimal for smooth edges in 2-D. This talk is about our current efforts for developing adaptive location -based representations that can capture the underlying edge-directed smoothness in as few dimensions as possible. In this context, we introduce the so called envelope+phase and contour+profile models to deal with the phase characteristics of wavelet coefficients around edges. Recent developments in the design of complex filter banks show that there exist nonredundant orthogonal representations with the desired linear and/or smooth phase properties. We conclude the talk by presenting some applications of these ideas in image coding and image interpolation.