Surface metrology is the science of measuring small-scale features on surfaces, which is important to many disciplines including tribology, fluid mechanics, optics, and manufacturing. Applications of wavelets on functional surfaces have become an increasing interest. Unfortunately, the traditional wavelets have good performance only at representing point singularities because they ignore the geometric properties of surfaces with scratches and do not exploit the regularity of curves. In this letter, a second-generation curvelet transform combined with total variation minimization is proposed to characterize surfaces with general scratches along C2 singularities. The ability of the developed geometric multiscale method is demonstrated on bioengineering surfaces.