This paper deals with a new anisotropic discrete dual-tree wavelet transform (ADDTWT) to characterize the anisotropy of bone texture. More specifically, we propose to extend the conventional discrete dual-tree wavelet transform (DDTWT) by using the anisotropic basis functions associated with the hyperbolic wavelet transform instead of isotropic spectrum supports. A texture classification framework is adopted to assess the performance of the proposed transform. The generalized Gaussian distribution is used to model the distribution of the sub-band coefficients. The estimated vector of parameters for each image is then used as input for the support vector machine classifier. Experiments were conducted on synthesized anisotropic fractional Brownian motion fields and on a real database composed of osteoporotic patients and control cases. Results show that the ADDTWT outperforms most of the competing anisotropic transforms with an area under curve rate of 93%.