The geometry-texture decomposition of images produced by X-Ray Computed Tomography (CT) is a challenging inverse problem which is usually performed in two steps: reconstruction and decomposition. Decomposition can be used for instance to produce an approximate segmentation of the image, but this one can be compromised by artifacts and noise arising from the acquisition and reconstruction processes. We propose a geometry-texture decomposition based on a TV-Laplacian model, well-suited for segmentation and edge detection. The corresponding joint reconstruction and decomposition task from CT data is then formulated as a convex constrained minimization problem. We use our recently introduced proximal interior point method to solve this inverse problem in a reliable manner. Numerical experiments on realistic images of material samples illustrate the practical efficiency of the proposed approach. Our algorithm indeed compares favorably with a state-of-the-art method.