This article presents a new adaptive texture model. Locally parallel oscillating patterns are modeled with a weighted Hilbert space defined over local Fourier coefficients. The weights on the local Fourier atoms are optimized to match the local orientation and frequency of the texture. We propose an adaptive method to decompose an image into a cartoon layer and a locally parallel texture layer using this model and a total variation cartoon model. This decomposition method is then used to denoise an image containing oscillating patterns. Finally we show how to take advantage of such a separation framework to simultaneously inpaint the structure and texture components of an image with missing parts. Numerical results show that our method improves state of the art algorithms for directional and complex textures.