Abstract: In many applications of curve and surface evolution, the curve or surface is required to have a particular topology. We present a new method for controlling the topology of a curve or surface within the level set framework. We use the concept of the total robustness of the level set function, which is based on ideas from persistent homology, and we show how to minimize this total robustness by computing its functional derivative with respect to the level set function. This method differs from existing methods in that it is inherently continuous and not digital; and in the way that total robustness directly relates to the topology of the underlying curve or surface, versus existing knot-based measures which are related in a more indirect fashion. The proposed method is validated empirically on segmentation examples.