Efficient energies and algorithms for parametric snakes - EPFL

29APPENDIX - D: INTEGRAL OF THE TANGENTIAL ANGLE.We start by observing that the integral (46) can be expressed as(∫ M)d (x ′ (t) + j y ′ (t))θ total = Imdt , (64)x ′ (t) + jy ′ (t)0where Im (z) gives the imaginary part of z and j = √ −1. This can be rewritten as the curveintegral(∮ )dzθ total = Im , (65)C z ′where C ′ is the curve described (x ′ (t) , y ′ (t)) and z = x ′ +iy ′ . Using Cauchy’s integral formula,we obtain the value of this integral as 2π times the winding number 8 of the contour C ′ aboutthe origin. Since each loop in C corresponds to one in C ′ in the same direction, but around theorigin, the winding number of C ′ is (m − n), where m and n are the number of times C loopsin the anticlockwise and clockwise direction respectively.REFERENCES[1] A.K. Jain, Y. Zhong, and M.P.D. Jolly, “Deformable template models: A review,” Signal Processing, vol. 76, pp. 109–129,1998.[2] T. McInerney and D. Terzopoulos, “Deformable models in medical image analysis: a survey,” Medical Image Analysis,vol. 1, pp. 91–108, 1996.[3] M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: Active contour models,” International Journal of Computer Vision,vol. 1, pp. 321–332, 1988.[4] C. Xu and J. L. Prince, “Snakes, shapes and gradient vector flow,” IEEE Transactions on Image Processing, vol. 7, no.3, pp. 359–369, 1998.[5] J. Gao, A. Kosaka, and A. Kak, “A deformable model for human organ extraction,” in ICIP, Chicago, 1998, pp. 323–327.[6] S. Menet, P. Saint-Mark, and G. Medioni, “B-snakes: implementation and application to stereo,” in Image Understandingworkshop, 1990, pp. 720–726.[7] M. Gebhard, J. Mattes, and R. Eils, “An active contour model for segmentation based on cubic B-splines and gradientvector flow,” in MICCAI, 2001.[8] M. A. Figueiredo and J. M. N. Leitao, “Unsupervised contour representation and estimation using B-splines and a minimumdescription length criterion,” IEEE Transactions on Image Processing, vol. 9, no. 6, pp. 1075–1087, 2000.[9] P. Brigger, J. Hoeg, and M. Unser, “B-spline snakes: A flexible tool for parametric contour detection,” IEEE Transactionson Image Processing, vol. 9, no. 9, pp. 1484–1496, 2000.[10] L. H. Staib and J. S. Duncan, “Boundary fitting with parametrically deformable models,” IEEE Transactions on PatternAnalysis and Machine Intelligence, vol. 14, no. 11, pp. 1061–1075, 1992.8 The winding number of a contour about a point z 0 is the number of times the contour passes around z 0 in the counterclockwisedirection [39].Submitted to IEEE Trans. Image Processing, January 7, 2004.DRAFT