Efficient energies and algorithms for parametric snakes - EPFL

17(a) Initialization (b) No curvilinear energy (c) With curvilinear energyFig. 5. Without the curvilinear energy, the parametric representation cannot guarantee low curvature curves. Note that for thesame initialization, the curve with the curvilinear reparametrization energy leads to smoother curves. Without the energy, thecurve knots accumulate at some regions of the curve, thus leading to sharp edges; low energy curves are ensured only if the arclength is constant on the curve.We consider the integral in (6) and differentiate it with respect to the coefficients using thechain rule (using (1)):⎡⎤⎣ ∂E mag/∂c x,k⎦ =∂E mag /∂c y,k∫ M0∇g (t) ϕ p (t − k) dt (29)where g = |∇f| 2 . We approximate the inner-product as a discrete sum:⎡⎤⎣ ∂E mag/∂c x,k⎦ ≈ 1 ∑NR( ) ( [k R + i]MR i∇g ϕ , (30)∂E mag /∂c y,kRR R)i=0where R is the sampling rate and [k] M stands for k mod M. In the above expression, we used thefinite support of the scaling function to limit the range of the summation. Also note that we havetransferred the periodicity from the kernel to ∇g; this means that the summation is evaluatedassuming periodic boundary conditions on ∇g. Thus, if ∇g and ϕ ( iR − k) are precomputed, theevaluation of the partial derivatives just involves a weighted sum. The computational complexityis therefore proportional to RMN.2) Partial derivatives of the unified image energy: For closed curves, we preferentially usethe unified energy to optimize the curve. In line with the work of [10], [11], [37], we now useGreen’s theorem (12) to convert region integrals (over the region bounded by a closed curve) tointegrals over the curve; our main motivation is computational efficiency. (20) can be efficientlySubmitted to IEEE Trans. Image Processing, January 7, 2004.DRAFT