This article presents a new adaptive framework for locally parallel texture modeling. Oscillating patterns are modeled with functionals that constrain the local Fourier decomposition of the texture. We first introduce a convex texture functional which is a weighted Hilbert norm. The weights on the local Fourier atoms are optimized to match the local orientation and frequency of the texture. This adaptive convex model is used to solve image processing inverse problems, such as image decomposition and inpainting. The local orientation and frequency of the texture component are adaptively estimated during the minimization process. To improve inpainting performances over large missing regions, we introduce a non-convex generalization of our texture model. This new model constrains the amplitude of the texture and allows one to impose an arbitrary oscillation profile. This non-convex model bridges the gap between regularization methods for image restoration and patch-based synthesis approaches that are successful in texture synthesis. Numerical results show that our method improves state of the art algorithms for locally parallel textures.