Consistency and stability of active contours with Euclidean and non-Euclidean arc-lengths

External energies of active contours are often formulated as Euclidean are length integrals. Here, the authors show that such formulations are biased. By this they mean that the minimum of the external energy does not occur at an image edge. In addition, they also show that for certain forms of external energy the active contour is unstable-when initialized at the location where the first variation of the energy is zero, the contour drifts away and becomes jagged. Both of these phenomena are due to the use of Euclidean arc length. The authors propose a non-Euclidean arc length which eliminates this problem. This requires a reformulation of active contours where the global external energy function is replaced by a sequence of local external energy functions and the contour evolves as an integral curve of the gradient of the local energies. As a result, the authors present a new evolution equation that is simpler and more accurate. Experimental evidence is provided in support of the theoretical claims.