On smoothness measures of active contours and surfaces

We propose to study different smoothness measures of planar contours or surfaces. We first define a smoothness measure as a functional that follows three types of invariance, invariance to changes of contour parameterization, invariance to contour rotations and translations and invariance to the contour sizes. We then introduce different smoothness measures that can be classified into local or global functionals but that can also be of geometric or algebraic nature. We finally discuss their implementation by observing the advantages and disadvantages of explicit and implicit contour representations.