Texture segmentation constitutes a task of utmost importance in statistical image processing. Focusing on the broad class of monofrac-tal textures characterized by piecewise constancy of the statistics of their multiscale representations, recently shown to be versatile enough for real-world texture modeling, the present work renews this recurrent topic by proposing an original approach enrolling jointly scale-free and local variance descriptors into a convex, but non smooth, minimization strategy. The performance of the proposed joint approach are compared against disjoint strategies working independently on scale-free features and on local variance on synthetic piecewise monofractal textures. Performance are also compared for multiphase flow image characterization, a topic of crucial importance in geophysics as well as in industrial processes. Applied to large-size images (above two million pixels), the proposed approach is shown to significantly improve state-of-the-art strategies by permitting the detection of the smallest gas bubbles and by offering a better understanding of multiphase flow structures.