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

External energies of active contours are often formulated as Euclidean arc length integrals. In this paper, we show that such formulations are biased. By this we mean that the minimum of the external energy does not occur at an image edge. In addition, we also show that for certain forms of external energy the active contour is unstable—when initialized at the true edge, the contour drifts away and becomes jagged. Both of these phenomena are due to the use of Euclidean arc length integrals. We propose a non-Euclidean arc length which eliminates these problems. This requires a reformulation of active contours where a single external energy function is replaced by a sequence of energy functions and the contour evolves as an integral curve of the gradient of these energies. The resulting active contour not only has unbiased external energy, but is also more controllable. Experimental evidence is provided in support of the theoretical claims.