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 & Imaging, 2014, 8 (3) : 845-863. doi: 10.3934/ipi.2014.8.845
References:
[1]

Image Processing On Line, 2012. Google Scholar

[2]

Communications on Pure and Applied Mathematics, 43 (1990), 999-1036. doi: 10.1002/cpa.3160430805.  Google Scholar

[3]

Springer, 2000. doi: 10.1007/978-1-4612-1268-3.  Google Scholar

[4]

SIAM Journal on Imaging Sciences, 2 (2009), 508-531. doi: 10.1137/080722436.  Google Scholar

[5]

In Image Processing, 2007. ICIP 2007. IEEE International Conference on, 1 (2007), I261-I264. doi: 10.1109/ICIP.2007.4378941.  Google Scholar

[6]

International Journal of Computer Vision, 22 (1995), 694-699. doi: 10.1109/ICCV.1995.466871.  Google Scholar

[7]

Journal of Visual Communication and Image Representation, 11 (2000), 130-141. doi: 10.1006/jvci.1999.0442.  Google Scholar

[8]

IEEE Transactions on Image Processing, 10 (2001), 266-277. doi: 10.1109/83.902291.  Google Scholar

[9]

Computing and Visualization in Science, 12 (2009), 267-285. doi: 10.1007/s00791-008-0113-1.  Google Scholar

[10]

International Journal of Computer Vision, 24 (1997), 57-78. Google Scholar

[11]

Calculus of Variations and Partial Differential Equations, 13 (2001), 123-139.  Google Scholar

[12]

Acta Mathematica, 168 (1992), 89-151. doi: 10.1007/BF02392977.  Google Scholar

[13]

CRC Press, 1992.  Google Scholar

[14]

Communications in Partial Differential Equations, 30 (2005), 1401-1428. doi: 10.1080/03605300500258956.  Google Scholar

[15]

Archive for Rational Mechanics and Analysis, 108 (1989), 195-218. doi: 10.1007/BF01052971.  Google Scholar

[16]

IS&T/SPIE Electronic Imaging, 7246 (2009), 724601. doi: 10.1117/12.806067.  Google Scholar

[17]

International Journal of Computer Vision, 1 (1988), 321-331. doi: 10.1007/BF00133570.  Google Scholar

[18]

Inverse Problems and Imaging, 6 (2012), 95-110. doi: 10.3934/ipi.2012.6.95.  Google Scholar

[19]

In Computer Vision, 1995. Proceedings., Fifth International Conference on, (1995), 810-815. doi: 10.1109/ICCV.1995.466855.  Google Scholar

[20]

International Journal of Computer Vision, 53 (2001), 225-243. Google Scholar

[21]

Pattern Recognition, 43 (2010), 1631-1641. doi: 10.1016/j.patcog.2009.11.003.  Google Scholar

[22]

Inst. Nacional de Matemática Pura e Aplicada, 2008. Google Scholar

[23]

Journal of Computational Physics, 228 (2009), 7706-7728. doi: 10.1016/j.jcp.2009.07.017.  Google Scholar

[24]

In Biomedical Image Registration, Springer Berlin Heidelberg, 2717 (2003), 40-49. doi: 10.1007/978-3-540-39701-4_5.  Google Scholar

[25]

Advances in Mathematics, 217 (2008), 1838-1868. doi: 10.1016/j.aim.2007.11.020.  Google Scholar

[26]

Pattern Analysis and Machine Intelligence, IEEE Transactions on, 30 (2008), 412-423. doi: 10.1109/TPAMI.2007.70713.  Google Scholar

[27]

Revista Matemática de la Universidad Complutense de Madrid, 1 (1988), 169-182.  Google Scholar

[28]

Comptes rendus de l'Académie des sciences. Série 1, Mathématique, 308 (1989), 465-470.  Google Scholar

[29]

Communications on Pure and Applied Mathematics, 42 (1989), 577-685. doi: 10.1002/cpa.3160420503.  Google Scholar

[30]

Lecture Notes in Mathematics, 2010. doi: 10.1007/978-3-642-04041-2.  Google Scholar

[31]

Plasma Science, IEEE Transactions on, 39 (2011), 2406-2407. doi: 10.1109/TPS.2011.2162007.  Google Scholar

[32]

Mathematics of Computation, 74 (2005), 1217-1230. doi: 10.1090/S0025-5718-04-01688-6.  Google Scholar

[33]

Journal of Computational Physics, 79 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2.  Google Scholar

[34]

Unpublished, University of North Texas, 2006. Google Scholar

[35]

Communications in Numerical Methods in Engineering, 24 (2008), 493-504. doi: 10.1002/cnm.951.  Google Scholar

[36]

Journal of Mathematical Imaging and Vision, 49 (2014), 20-36. doi: 10.1007/s10851-013-0437-4.  Google Scholar

[37]

Physica D: Nonlinear Phenomena, 138 (2000), 282-301. doi: 10.1016/S0167-2789(99)00216-X.  Google Scholar

[38]

Lecture Notes in Computer Science, 3752 (2005), 109-120. doi: 10.1007/11567646_10.  Google Scholar

[39]

Image Processing, IEEE Transactions on, 10 (2001), 1169-1186. doi: 10.1109/83.935033.  Google Scholar

[40]

International Journal of Computer Vision, 50 (2002), 271-293. Google Scholar

[41]

In Pattern Recognition, Springer Berlin Heidelberg, 5748 (2009), 552-561. doi: 10.1007/978-3-642-03798-6_56.  Google Scholar

show all references

References:
[1]

Image Processing On Line, 2012. Google Scholar

[2]

Communications on Pure and Applied Mathematics, 43 (1990), 999-1036. doi: 10.1002/cpa.3160430805.  Google Scholar

[3]

Springer, 2000. doi: 10.1007/978-1-4612-1268-3.  Google Scholar

[4]

SIAM Journal on Imaging Sciences, 2 (2009), 508-531. doi: 10.1137/080722436.  Google Scholar

[5]

In Image Processing, 2007. ICIP 2007. IEEE International Conference on, 1 (2007), I261-I264. doi: 10.1109/ICIP.2007.4378941.  Google Scholar

[6]

International Journal of Computer Vision, 22 (1995), 694-699. doi: 10.1109/ICCV.1995.466871.  Google Scholar

[7]

Journal of Visual Communication and Image Representation, 11 (2000), 130-141. doi: 10.1006/jvci.1999.0442.  Google Scholar

[8]

IEEE Transactions on Image Processing, 10 (2001), 266-277. doi: 10.1109/83.902291.  Google Scholar

[9]

Computing and Visualization in Science, 12 (2009), 267-285. doi: 10.1007/s00791-008-0113-1.  Google Scholar

[10]

International Journal of Computer Vision, 24 (1997), 57-78. Google Scholar

[11]

Calculus of Variations and Partial Differential Equations, 13 (2001), 123-139.  Google Scholar

[12]

Acta Mathematica, 168 (1992), 89-151. doi: 10.1007/BF02392977.  Google Scholar

[13]

CRC Press, 1992.  Google Scholar

[14]

Communications in Partial Differential Equations, 30 (2005), 1401-1428. doi: 10.1080/03605300500258956.  Google Scholar

[15]

Archive for Rational Mechanics and Analysis, 108 (1989), 195-218. doi: 10.1007/BF01052971.  Google Scholar

[16]

IS&T/SPIE Electronic Imaging, 7246 (2009), 724601. doi: 10.1117/12.806067.  Google Scholar

[17]

International Journal of Computer Vision, 1 (1988), 321-331. doi: 10.1007/BF00133570.  Google Scholar

[18]

Inverse Problems and Imaging, 6 (2012), 95-110. doi: 10.3934/ipi.2012.6.95.  Google Scholar

[19]

In Computer Vision, 1995. Proceedings., Fifth International Conference on, (1995), 810-815. doi: 10.1109/ICCV.1995.466855.  Google Scholar

[20]

International Journal of Computer Vision, 53 (2001), 225-243. Google Scholar

[21]

Pattern Recognition, 43 (2010), 1631-1641. doi: 10.1016/j.patcog.2009.11.003.  Google Scholar

[22]

Inst. Nacional de Matemática Pura e Aplicada, 2008. Google Scholar

[23]

Journal of Computational Physics, 228 (2009), 7706-7728. doi: 10.1016/j.jcp.2009.07.017.  Google Scholar

[24]

In Biomedical Image Registration, Springer Berlin Heidelberg, 2717 (2003), 40-49. doi: 10.1007/978-3-540-39701-4_5.  Google Scholar

[25]

Advances in Mathematics, 217 (2008), 1838-1868. doi: 10.1016/j.aim.2007.11.020.  Google Scholar

[26]

Pattern Analysis and Machine Intelligence, IEEE Transactions on, 30 (2008), 412-423. doi: 10.1109/TPAMI.2007.70713.  Google Scholar

[27]

Revista Matemática de la Universidad Complutense de Madrid, 1 (1988), 169-182.  Google Scholar

[28]

Comptes rendus de l'Académie des sciences. Série 1, Mathématique, 308 (1989), 465-470.  Google Scholar

[29]

Communications on Pure and Applied Mathematics, 42 (1989), 577-685. doi: 10.1002/cpa.3160420503.  Google Scholar

[30]

Lecture Notes in Mathematics, 2010. doi: 10.1007/978-3-642-04041-2.  Google Scholar

[31]

Plasma Science, IEEE Transactions on, 39 (2011), 2406-2407. doi: 10.1109/TPS.2011.2162007.  Google Scholar

[32]

Mathematics of Computation, 74 (2005), 1217-1230. doi: 10.1090/S0025-5718-04-01688-6.  Google Scholar

[33]

Journal of Computational Physics, 79 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2.  Google Scholar

[34]

Unpublished, University of North Texas, 2006. Google Scholar

[35]

Communications in Numerical Methods in Engineering, 24 (2008), 493-504. doi: 10.1002/cnm.951.  Google Scholar

[36]

Journal of Mathematical Imaging and Vision, 49 (2014), 20-36. doi: 10.1007/s10851-013-0437-4.  Google Scholar

[37]

Physica D: Nonlinear Phenomena, 138 (2000), 282-301. doi: 10.1016/S0167-2789(99)00216-X.  Google Scholar

[38]

Lecture Notes in Computer Science, 3752 (2005), 109-120. doi: 10.1007/11567646_10.  Google Scholar

[39]

Image Processing, IEEE Transactions on, 10 (2001), 1169-1186. doi: 10.1109/83.935033.  Google Scholar

[40]

International Journal of Computer Vision, 50 (2002), 271-293. Google Scholar

[41]

In Pattern Recognition, Springer Berlin Heidelberg, 5748 (2009), 552-561. doi: 10.1007/978-3-642-03798-6_56.  Google Scholar

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