Compensated convex transforms are simple convexity based `tight’ approximations of functions. Applications so far include image processing, geometric singularity extraction, such as surface intersection, stable ridge transform, the medial axis map; interpolation and approximation of sampled functions, and smoothing in convex optimization.

An interesting feature of such transforms is the Hausdorff stability which provides robustness of the transforms. I will present a brief introduction to the theory and numerical methods, and describe some of the applications mentioned above.