Efficient energies and algorithms for parametric snakes - EPFL

18computed as the curve integral∫∮f u (x, y) dxdy = fu y (x, y) dx (31)SC∮= − fu x (x, y) dy, (32)wheref y u (x, y) =f x u (x, y) =∫ y−∞∫ x−∞Cf u (x, τ) dτ (33)f u (τ, y) dτ. (34)Applying the chain rule of differentiation on (32), we obtain ∂E image /∂c x,k as∂(E image ) = ∂∂c x,k ∂x (E ∂image) · (x (t))∂c x,k= −= −= −∫ M0∂f x u∂x }{{}f uϕ p (t − k)M∑c y,l ϕ ′ p (t − l)l=0} {{ }y ′ (t)M∑∫ Mc y,l f u (t) ϕ p (t − k) ϕ ′ p (t − l) dtl=0∞∑l=−∞c p y,l0∫ ∞f u (t) ϕ (t − k) ϕ ′ (t − l) dt . (35)−∞} {{ }Q fu (k,l)In the last step we expanded ϕ p (t − k) using (3) and made a change of variable, thus extendingthe integral from −∞ to ∞. We also transferred the periodicity of Q fu to the coefficient sequence.Since Q fu (k, l) is a finite sequence, the evaluation of (35) amounts to an appropriate finite sum.In a similar manner, using (31) we obtain∂E image∞∑= c p x,l∂c y,kl=−∞∫ ∞f u (t) ϕ (t − k) ϕ ′ (t − l) dt .−∞} {{ }Q fu (k,l)The main steps in the computation of the partial derivatives are:1) The evaluation of the sequence Q fu (k, l) ; |k − l| < N. (With a change of variables weobtain Q fu (k, l) = ∫ ∞−∞ f u (t + k) ϕ (t) ϕ ′ (t + k − l) dt. Since ϕ (t) is finitely supportedin the interval [0, N], Q fu (k, l) is zero if |k − l| ≥ N). Approximating the integral as aSubmitted to IEEE Trans. Image Processing, January 7, 2004.DRAFT