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The core of many approaches for the resolution of variational inverse problems arising in signal and image processing consists of promoting the sought solution to have a sparse representation in a well-suited space. A crucial task in this context is the choice of a good sparsity prior that can ensure a good trade-off between the quality of the solution and the resulting computational cost. The recently introduced Convex-Non-Convex (CNC) strategy appears as a great compromise, as it combines the high qualitative performance of non-convex sparsity-promoting functions with the convenience of dealing with convex optimization problems. This work proposes a new variational formulation to implement CNC approach in the context of image denoising. By suitably exploiting duality properties, our formulation allows to encompass sophisticated directional total variation (DTV) priors. We additionally propose an efficient optimisation strategy for the resulting convex minimisation problem. We illustrate on numerical examples the good performance of the resulting CNC-DTV method, when compared to the standard convex total variation denoiser.