# American Institute of Mathematical Sciences

August  2014, 8(3): 845-863. doi: 10.3934/ipi.2014.8.845

## Active arcs and contours

 1 Department of Mathematics, University of California, Irvine, Irvine, CA 92697, United States

Received  September 2012 Revised  November 2013 Published  August 2014

The level set method [33] is a commonly used framework for image segmentation algorithms. For edge detection and segmentation models, the standard level set method provides a flexible curve representation and implementation. However, one drawback has been in the types of curves that can be represented in this standard method. In the classical level set method, the curve must enclose an open set (i.e. loops or contours without endpoints). Thus the classical framework is limited to locating edge sets without endpoints. Using the curve representation from [37,36], we construct a segmentation and edge detection method which can locate arcs (i.e. curves with free endpoints) as well as standard contours. Within this new framework, the variational segmentation model presented here is able to detect general edge structures and linear objects. This energy is composed of two terms, an edge set regularizer and an edge attractor. Our variational model is related to the Mumford and Shah model [29] for joint segmentation and restoration in terms of an asymptotic limit, and in addition, is both general and flexible in terms of its uses and its applications. Numerical results are given on images with a variety of edge structures.
Citation: Hayden Schaeffer. Active arcs and contours. Inverse Problems and Imaging, 2014, 8 (3) : 845-863. doi: 10.3934/ipi.2014.8.845
##### References:
 [1] L. Alvarez, L. Baumela, P. Márquez-Neila and P. Henríquez, A Real Time Morphological Snakes Algorithm, Image Processing On Line, 2012. [2] L. Ambrosio and V. M. Tortorelli, Approximation of functional depending on jumps by elliptic functional via $\Gamma$-convergence, Communications on Pure and Applied Mathematics, 43 (1990), 999-1036. doi: 10.1002/cpa.3160430805. [3] D. Bao, S.-S. Chern and Z.Shen, An Introduction to Riemann-Finsler Geometry, {Vol.} 200, Springer, 2000. doi: 10.1007/978-1-4612-1268-3. [4] L. Bar and G. Sapiro, Generalized Newton-Type methods for energy formulations in image processing, SIAM Journal on Imaging Sciences, 2 (2009), 508-531. doi: 10.1137/080722436. [5] S. Basu, D. P. Mukherjee and S. T. Acton, Implicit evolution of open ended curves, In Image Processing, 2007. ICIP 2007. IEEE International Conference on, 1 (2007), I261-I264. doi: 10.1109/ICIP.2007.4378941. [6] V. Caselles, R. Kimmel and G. Sapiro, Geodesic active contours, International Journal of Computer Vision, 22 (1995), 694-699. doi: 10.1109/ICCV.1995.466871. [7] T. F. Chan, B. Y. Sandberg and L. A. Vese, Active contours without edges for vector-valued images, Journal of Visual Communication and Image Representation, 11 (2000), 130-141. doi: 10.1006/jvci.1999.0442. [8] T. F. Chan and L. A. Vese, Active contours without edges, IEEE Transactions on Image Processing, 10 (2001), 266-277. doi: 10.1109/83.902291. [9] G. Chung and L. A. Vese, Image segmentation using a multilayer level-set approach, Computing and Visualization in Science, 12 (2009), 267-285. doi: 10.1007/s00791-008-0113-1. [10] L. D. Cohen and R. Kimmel, Global minimum for active contour models: A minimal path approach, International Journal of Computer Vision, 24 (1997), 57-78. [11] M. G. Crandall, L. C. Evans and R. F. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian, Calculus of Variations and Partial Differential Equations, 13 (2001), 123-139. [12] G. Dal Maso, J. M. Morel and S. Solimini, A variational method in image segmentation: Existence and approximation results, Acta Mathematica, 168 (1992), 89-151. doi: 10.1007/BF02392977. [13] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, 1992. [14] L. C. Evans and Y. Yu, Various properties of solutions of the Infinity-Laplacian equation, Communications in Partial Differential Equations, 30 (2005), 1401-1428. doi: 10.1080/03605300500258956. [15] E. De Giorgi, M. Carriero and A. Leaci, Existence theorem for a minimum problem with free discontinuity set, Archive for Rational Mechanics and Analysis, 108 (1989), 195-218. doi: 10.1007/BF01052971. [16] M. Jung, G. Chung, G. Sundaramoorthi, L. A. Vese and A. L. Yuille, Sobolev gradients and joint variational image segmentation, denoising, and deblurring, IS&T/SPIE Electronic Imaging, 7246 (2009), 724601. doi: 10.1117/12.806067. [17] M. Kass, A. Witkin and D. Terzopoulos, Snakes: Active contour models, International Journal of Computer Vision, 1 (1988), 321-331. doi: 10.1007/BF00133570. [18] M. S. Keegan, B. Sandberg and T. F. Chan, A multiphase logic framework for multichannel image segmentation, Inverse Problems and Imaging, 6 (2012), 95-110. doi: 10.3934/ipi.2012.6.95. [19] S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum and A. Yezzi, Gradient flows and geometric active contour models, In Computer Vision, 1995. Proceedings., Fifth International Conference on, (1995), 810-815. doi: 10.1109/ICCV.1995.466855. [20] R. Kimmel and A. M. Bruckstein, Regularized Laplacian zero crossings as optimal edge integrators, International Journal of Computer Vision, 53 (2001), 225-243. [21] C. Lacoste, X. Descombes and J. Zerubia, Unsupervised line network extraction in remote sensing using a polyline process, Pattern Recognition, 43 (2010), 1631-1641. doi: 10.1016/j.patcog.2009.11.003. [22] C. Larsen, C. Richardson and M. Sarkis, A Level Set Method for the Mumford -Shah Functional and Fracture, Inst. Nacional de Matemática Pura e Aplicada, 2008. [23] S. Leung and H. Zhao, A grid based particle method for evolution of open curves and surfaces, Journal of Computational Physics, 228 (2009), 7706-7728. doi: 10.1016/j.jcp.2009.07.017. [24] W. H. Liao, L. Vese, S. C. Huang, M. Bergsneider and S. Osher, Computational anatomy and implicit object representation: A level set approach, In Biomedical Image Registration, Springer Berlin Heidelberg, 2717 (2003), 40-49. doi: 10.1007/978-3-540-39701-4_5. [25] G. Lu and P. Wang, Inhomogeneous infinity Laplace equation, Advances in Mathematics, 217 (2008), 1838-1868. doi: 10.1016/j.aim.2007.11.020. [26] J. Melonakos, E. Pichon, S. Angenent and A. Tannenbaum, Finsler active contours, Pattern Analysis and Machine Intelligence, IEEE Transactions on, 30 (2008), 412-423. doi: 10.1109/TPAMI.2007.70713. [27] J. M. Morel and S. Solimini, Segmentation of images by variational methods: A constructive approach, Revista Matemática de la Universidad Complutense de Madrid, 1 (1988), 169-182. [28] J. M. Morel and S. Solimini, Segmentation d'images par méthode variationnelle: Une preuve constructive d'existence, Comptes rendus de l'Académie des sciences. Série 1, Mathématique, 308 (1989), 465-470. [29] D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Communications on Pure and Applied Mathematics, 42 (1989), 577-685. doi: 10.1002/cpa.3160420503. [30] J. W. Neuberger, Sobolev Gradients and Differential Equations, Lecture Notes in Mathematics, 2010. doi: 10.1007/978-3-642-04041-2. [31] C. Niemann, A. S. Bondarenko, C. G. Constantin, E. T. Everson, K. A. Flippo, S. A. Gaillard, R. P. Johnson, S. A. Letzring, D. S. Montgomery, L. A. Morton, D. B. Schaeffer, T. Shimada and D. Winske, Collisionless shocks in a large magnetized laser-plasma plume, Plasma Science, IEEE Transactions on, 39 (2011), 2406-2407. doi: 10.1109/TPS.2011.2162007. [32] A. M. Oberman, A convergent difference scheme for the infinity laplacian: Construction of absolutely minimizing Lipschitz extensions, Mathematics of Computation, 74 (2005), 1217-1230. doi: 10.1090/S0025-5718-04-01688-6. [33] S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics, 79 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2. [34] R. J. Renka, A simple explanation of the Sobolev gradient method, Unpublished, University of North Texas, 2006. [35] W. B. Richardson, Sobolev gradient preconditioning for image-processing PDEs, Communications in Numerical Methods in Engineering, 24 (2008), 493-504. doi: 10.1002/cnm.951. [36] H. Schaeffer and L. Vese, Active contours with free endpoints, Journal of Mathematical Imaging and Vision, 49 (2014), 20-36. doi: 10.1007/s10851-013-0437-4. [37] P. Smereka, Spiral crystal growth, Physica D: Nonlinear Phenomena, 138 (2000), 282-301. doi: 10.1016/S0167-2789(99)00216-X. [38] G. Sundaramoorthi, A. Yezzi and A. C. Mennucci, Sobolev active contours, Lecture Notes in Computer Science, 3752 (2005), 109-120. doi: 10.1007/11567646_10. [39] A. Tsai, A. Yezzi Jr and A. S. Willsky, Curve evolution implementation of the Mumford-Shah functional for image segmentation, denoising, interpolation, and magnification, Image Processing, IEEE Transactions on, 10 (2001), 1169-1186. doi: 10.1109/83.935033. [40] L. A. Vese and T. F. Chan, A multiphase level set framework for image segmentation using the Mumford and Shah model, International Journal of Computer Vision, 50 (2002), 271-293. [41] C. Zach, L. Shan and M. Niethammer, Globally optimal Finsler active contours, In Pattern Recognition, Springer Berlin Heidelberg, 5748 (2009), 552-561. doi: 10.1007/978-3-642-03798-6_56.

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##### References:
 [1] L. Alvarez, L. Baumela, P. Márquez-Neila and P. Henríquez, A Real Time Morphological Snakes Algorithm, Image Processing On Line, 2012. [2] L. Ambrosio and V. M. Tortorelli, Approximation of functional depending on jumps by elliptic functional via $\Gamma$-convergence, Communications on Pure and Applied Mathematics, 43 (1990), 999-1036. doi: 10.1002/cpa.3160430805. [3] D. Bao, S.-S. Chern and Z.Shen, An Introduction to Riemann-Finsler Geometry, {Vol.} 200, Springer, 2000. doi: 10.1007/978-1-4612-1268-3. [4] L. Bar and G. Sapiro, Generalized Newton-Type methods for energy formulations in image processing, SIAM Journal on Imaging Sciences, 2 (2009), 508-531. doi: 10.1137/080722436. [5] S. Basu, D. P. Mukherjee and S. T. Acton, Implicit evolution of open ended curves, In Image Processing, 2007. ICIP 2007. IEEE International Conference on, 1 (2007), I261-I264. doi: 10.1109/ICIP.2007.4378941. [6] V. Caselles, R. Kimmel and G. Sapiro, Geodesic active contours, International Journal of Computer Vision, 22 (1995), 694-699. doi: 10.1109/ICCV.1995.466871. [7] T. F. Chan, B. Y. Sandberg and L. A. Vese, Active contours without edges for vector-valued images, Journal of Visual Communication and Image Representation, 11 (2000), 130-141. doi: 10.1006/jvci.1999.0442. [8] T. F. Chan and L. A. Vese, Active contours without edges, IEEE Transactions on Image Processing, 10 (2001), 266-277. doi: 10.1109/83.902291. [9] G. Chung and L. A. Vese, Image segmentation using a multilayer level-set approach, Computing and Visualization in Science, 12 (2009), 267-285. doi: 10.1007/s00791-008-0113-1. [10] L. D. Cohen and R. Kimmel, Global minimum for active contour models: A minimal path approach, International Journal of Computer Vision, 24 (1997), 57-78. [11] M. G. Crandall, L. C. Evans and R. F. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian, Calculus of Variations and Partial Differential Equations, 13 (2001), 123-139. [12] G. Dal Maso, J. M. Morel and S. Solimini, A variational method in image segmentation: Existence and approximation results, Acta Mathematica, 168 (1992), 89-151. doi: 10.1007/BF02392977. [13] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, 1992. [14] L. C. Evans and Y. Yu, Various properties of solutions of the Infinity-Laplacian equation, Communications in Partial Differential Equations, 30 (2005), 1401-1428. doi: 10.1080/03605300500258956. [15] E. De Giorgi, M. Carriero and A. Leaci, Existence theorem for a minimum problem with free discontinuity set, Archive for Rational Mechanics and Analysis, 108 (1989), 195-218. doi: 10.1007/BF01052971. [16] M. Jung, G. Chung, G. Sundaramoorthi, L. A. Vese and A. L. Yuille, Sobolev gradients and joint variational image segmentation, denoising, and deblurring, IS&T/SPIE Electronic Imaging, 7246 (2009), 724601. doi: 10.1117/12.806067. [17] M. Kass, A. Witkin and D. Terzopoulos, Snakes: Active contour models, International Journal of Computer Vision, 1 (1988), 321-331. doi: 10.1007/BF00133570. [18] M. S. Keegan, B. Sandberg and T. F. Chan, A multiphase logic framework for multichannel image segmentation, Inverse Problems and Imaging, 6 (2012), 95-110. doi: 10.3934/ipi.2012.6.95. [19] S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum and A. Yezzi, Gradient flows and geometric active contour models, In Computer Vision, 1995. Proceedings., Fifth International Conference on, (1995), 810-815. doi: 10.1109/ICCV.1995.466855. [20] R. Kimmel and A. M. Bruckstein, Regularized Laplacian zero crossings as optimal edge integrators, International Journal of Computer Vision, 53 (2001), 225-243. [21] C. Lacoste, X. Descombes and J. Zerubia, Unsupervised line network extraction in remote sensing using a polyline process, Pattern Recognition, 43 (2010), 1631-1641. doi: 10.1016/j.patcog.2009.11.003. [22] C. Larsen, C. Richardson and M. Sarkis, A Level Set Method for the Mumford -Shah Functional and Fracture, Inst. Nacional de Matemática Pura e Aplicada, 2008. [23] S. Leung and H. Zhao, A grid based particle method for evolution of open curves and surfaces, Journal of Computational Physics, 228 (2009), 7706-7728. doi: 10.1016/j.jcp.2009.07.017. [24] W. H. Liao, L. Vese, S. C. Huang, M. Bergsneider and S. Osher, Computational anatomy and implicit object representation: A level set approach, In Biomedical Image Registration, Springer Berlin Heidelberg, 2717 (2003), 40-49. doi: 10.1007/978-3-540-39701-4_5. [25] G. Lu and P. Wang, Inhomogeneous infinity Laplace equation, Advances in Mathematics, 217 (2008), 1838-1868. doi: 10.1016/j.aim.2007.11.020. [26] J. Melonakos, E. Pichon, S. Angenent and A. Tannenbaum, Finsler active contours, Pattern Analysis and Machine Intelligence, IEEE Transactions on, 30 (2008), 412-423. doi: 10.1109/TPAMI.2007.70713. [27] J. M. Morel and S. Solimini, Segmentation of images by variational methods: A constructive approach, Revista Matemática de la Universidad Complutense de Madrid, 1 (1988), 169-182. [28] J. M. Morel and S. Solimini, Segmentation d'images par méthode variationnelle: Une preuve constructive d'existence, Comptes rendus de l'Académie des sciences. Série 1, Mathématique, 308 (1989), 465-470. [29] D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Communications on Pure and Applied Mathematics, 42 (1989), 577-685. doi: 10.1002/cpa.3160420503. [30] J. W. Neuberger, Sobolev Gradients and Differential Equations, Lecture Notes in Mathematics, 2010. doi: 10.1007/978-3-642-04041-2. [31] C. Niemann, A. S. Bondarenko, C. G. Constantin, E. T. Everson, K. A. Flippo, S. A. Gaillard, R. P. Johnson, S. A. Letzring, D. S. Montgomery, L. A. Morton, D. B. Schaeffer, T. Shimada and D. Winske, Collisionless shocks in a large magnetized laser-plasma plume, Plasma Science, IEEE Transactions on, 39 (2011), 2406-2407. doi: 10.1109/TPS.2011.2162007. [32] A. M. Oberman, A convergent difference scheme for the infinity laplacian: Construction of absolutely minimizing Lipschitz extensions, Mathematics of Computation, 74 (2005), 1217-1230. doi: 10.1090/S0025-5718-04-01688-6. [33] S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics, 79 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2. [34] R. J. Renka, A simple explanation of the Sobolev gradient method, Unpublished, University of North Texas, 2006. [35] W. B. Richardson, Sobolev gradient preconditioning for image-processing PDEs, Communications in Numerical Methods in Engineering, 24 (2008), 493-504. doi: 10.1002/cnm.951. [36] H. Schaeffer and L. Vese, Active contours with free endpoints, Journal of Mathematical Imaging and Vision, 49 (2014), 20-36. doi: 10.1007/s10851-013-0437-4. [37] P. Smereka, Spiral crystal growth, Physica D: Nonlinear Phenomena, 138 (2000), 282-301. doi: 10.1016/S0167-2789(99)00216-X. [38] G. Sundaramoorthi, A. Yezzi and A. C. Mennucci, Sobolev active contours, Lecture Notes in Computer Science, 3752 (2005), 109-120. doi: 10.1007/11567646_10. [39] A. Tsai, A. Yezzi Jr and A. S. Willsky, Curve evolution implementation of the Mumford-Shah functional for image segmentation, denoising, interpolation, and magnification, Image Processing, IEEE Transactions on, 10 (2001), 1169-1186. doi: 10.1109/83.935033. [40] L. A. Vese and T. F. Chan, A multiphase level set framework for image segmentation using the Mumford and Shah model, International Journal of Computer Vision, 50 (2002), 271-293. [41] C. Zach, L. Shan and M. Niethammer, Globally optimal Finsler active contours, In Pattern Recognition, Springer Berlin Heidelberg, 5748 (2009), 552-561. doi: 10.1007/978-3-642-03798-6_56.
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