An active contour model for texture image segmentation using Rényi divergence measure
Abstract
This paper proposes an efficient method for active unsupervised texture segmentation. A new descriptor for texture features extractions based on Gaussian and mean curvature is constructed. Then the optimization of a functional who uses the R´enyi divergence measure and our descriptor is proposed in order to design an active contour model for texture segmentation. To get a global solution and efficient, fast algorithm, the optimization problem is redefined. The algorithm associated with this last optimization problem avoids local minimums and the run-time consuming compared to the level-set representation of our active contour model. In order to illustrate the performance of the technique, some results are presented showing the effectiveness and robustness of our approach.
Keyword : level set theory, texture segmentation, global minimization, differential geometry, Rényi divergence measure, shape optimization, partial differential equations
How to Cite
Idrissi, S. Y. (2022). An active contour model for texture image segmentation using Rényi divergence measure. Mathematical Modelling and Analysis, 27(3), 429–451. https://doi.org/10.3846/mma.2022.14060
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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A. Chambolle. An algorithm for total variation minimization and applications. Journal of Mathematical Imaging and Vision, 20(1-2):89–97, 2004. https://doi.org/10.1023/B:JMIV.0000011325.36760.1e
T.F. Chan, B.Y. Sandberg and L.A. Vese. Active contours without edges for vector valued images. Journal of Visual Communication Image Representation, 11(2):130–141, 2000. https://doi.org/10.1109/83.902291
T.F. Chan and L.A. Vese. Active contours without edges. IEEE Transactions on Image Processing, 10(2):266–277, 2001.
A. Cichocki and S. Amari. Families of alpha-beta-and gamma-divergences: Flexible and robust measures of similarities. Entropy, 12(6):1532–1568, 2010. https://doi.org/10.3390/e12061532
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T.V. Erven and P. Harremoes. Rényi divergence and Kullback-Leibler divergence. IEEE Transactions on Information Theory, 60(7):3797–3820, 2014. https://doi.org/10.1109/TIT.2014.2320500
L.C. Evans and G.F. Ronald. Measure Theory And Fine Proporties of Function. CRC Press, 2015. https://doi.org/10.1201/b18333
T. Goldstein, X. Bresson and S. Osher. Geometric applications of the split Bregman method: Segmentation and surface reconstruction. Journal of Scientific Computing, 45(1-3):272–293, 2010. https://doi.org/10.1007/s10915-009-9331-z
T. Goldstein and S. Osher. The split Bregman method for L1 regularized problems. SIAM Journal on Imaging Sciences, 2(2):323–343, 2009. https://doi.org/10.1137/080725891
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A.O. Hero, B. Ma, O. Michel and J.D. Gorman. Alpha-Divergence for Classification, Indexing and Retrieval. University of Michigan, 2002.
N. Houhou, J. Thiran and X. Bresson. Fast texture segmentation based on semi-local region descriptor and active contour. Numerical Mathematics: Theory, Methods and Applications, 2(4):445–468, 2020. https://doi.org/10.4208/nmtma.2009.m9007s
C. Hung, E. Song and Y. Lan. Image Texture Analysis: Foundations, Models and Algorithms. Springer International Publishing, 2019. https://doi.org/10.1007/978-3-030-13773-1
S. Jehan-Besson, M. Barlaud and G. Auber. Textures: A photographic album for artists and designers. Leonardo, 1(1):91–92, 1968. https://doi.org/10.2307/1571915
S. Jehan-Besson, M. Barlaud and G. Auber. Dream2s: Deformable regions driven by an Eulerian accurate minimization method for image and video segmentation. International Journal of Computer Vision, 53(1):365–380, 2003.
S. Kullback and R.A. Leibler. On information and sufficiency. Annals of Mathematical Statistics, 22(1):79–86, 1951. https://doi.org/10.1214/aoms/1177729694
R. Lerski. Texture Analysis for Magnetic Resonance Imaging. Med4 publishing, 2006.
C.M. Li, C.Y. Kao, J.C. Gore and Z.H. Ding. Implicit active contours driven by local binary fitting energy,. In IEEE Conference on Computer Vision and Pattern Recognition, pp. 1–7, Minneapolis, 2007. IEE. https://doi.org/10.1109/CVPR.2007.383014
X. Li, L. Hairong and Y. Xiaoping. Region intensity complexity active contours. Multidimensional Systems and Signal Processing, 31(2):1185–1206, 2020. https://doi.org/10.1007/s11045-020-00704-5
R. Malladi, J.A. Sethian and B.C. Vemuri. Shape modeling with front propagation: a level set approach. International Journal of Computer Vision, 17(2):158– 175, 1975. https://doi.org/10.1109/34.368173
S. Mallat. A theory for multiresolution signal decomposition: The wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11(7):674–693, 1989. https://doi.org/10.1109/34.192463
O.V. Michailovich, Y. Rathi and A. Tannenbaum. Image segmentation using active contours driven by the Bhattacharyya gradient flow. IEEE Transactions on Image Processing, 16(11):2787–2801, 2007. https://doi.org/10.1109/TIP.2007.908073
H. Min, W. Jia, X. Wang, Y. Zhao, R. Hu, Y. Luo, F. Xue and J. Lu. An intensity-texture model based level set method for image segmentation. Pattern Recognition, 48(4):1547–1562, 2015. https://doi.org/10.1016/j.patcog.2014.10.018
T. Minka. Divergence Measures and Message Passing. Microsoft Research Ltd, 2005.
A. Mitiche and I. Ben Ayedn. Variational and Level Set Methods in Image Segmentation. Springer, 2010. https://doi.org/10.1007/978-3-642-15352-5
D. Mumford and J. Shah. Optimal approximations by piecewise smooth functions and associated variational problems. Communication of Pure Applied Mathematics, 42(5):577–685, 1989. https://doi.org/10.1002/cpa.3160420503
M. Nixon and A.S. Aquado. Feature Extraction and Image Processing for Computer vision. Academic Press, 2020. https://doi.org/10.1016/B978-0-12-814976-8.00003-8
S. Osher and R. Fedkiw. Level Set Methods and Dynamic Implicit Surfaces. Springer, 2002. https://doi.org/10.1007/b98879
E. Parzen. On the estimation of a probability density function and the mode. Annals of Mathematical Statistics, 33(3):1065–1076, 1962. https://doi.org/10.1214/aoms/1177704472
C. Petitjean. Recalage Non Rigide d’Images par Approches Variationnelles Statistiques. Application à l’Analyse et à la Modélisation de la Fonction Myocardique en IRM. Université René Descartes -Paris V, 2003. (in French)
M. Petrou and P. Garcia Sevilla. Image Processing: Dealing with Texture. Wiley, 2006. https://doi.org/10.1002/047003534X
A. Ramola, A.K. Shakya and D.V. Pham. Study of statistical methods for texture analysis and their modern evolutions. Wiley, 2020. https://doi.org/10.1002/eng2.12149
A. R´enyi. On measures of entropy and information. In H. Ammann and V.A. Solonnikov(Eds.), Berkeley Symposium on Mathematical Statistics and Probability, pp. 547–5614, California, 1961. university of California press.
C. Sagiv, N. Sochen and Y. Zeevi. Integrated active contours for texture segmentation. IEEE Transactions on Image Processing, 15(6):1633–1646, 2006. https://doi.org/10.1109/TIP.2006.871133
J.A. Sethian. Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision and Material Sciences. Cambridge University Press, 1999.
N. Sochen, R. Kimmel and R. Malladi. A general framework for low level vision. IEEE Transaction on Image Processing, 7(3):310–318, 1998. https://doi.org/10.1109/83.661181
M. Sonka, V. Hlavac and R. Boyle. Image Processing, Analysis and Machine Vision. Thomson, 2007.
M. Spivak. A Comprehensive Introduction to Differential Geometry. Publish or Perish Press, 1999.
X.-C. Tai and C. Wu. Augmented Lagrangian method, dual methods and split Bregman iteration for ROF model. In H. Ammann and V.A. Solonnikov(Eds.), Scale Space and Variational Methods in Computer Vision, pp. 502–513, Norway, 2009. Springer. https://doi.org/10.1007/978-3-642-02256-2_42
C. Wu and X. Tai. Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models. SIAM Journal on Imaging Sciences, 3(3):300–339, 2010. https://doi.org/10.1137/090767558
S.Y. Yeo, X. Xie, I. Sazonov and P. Nithiarasu. Segmentation of biomedical images using active contour model with robust image feature and shape prior. International Journal for Numerical Methods in Biomedical Engineering, 6(5):232– 248, 2014. https://doi.org/10.1002/cnm.2600
X.H. Zhi and H.B. Shen. Saliency driven region-edge-based top down level set evolution reveals the asynchronous focus in image segmentation. Pattern Recognition, 80:241–255, 2018. https://doi.org/10.1016/j.patcog.2018.03.010
S.C. Zhu and A. Yuille. Region competition: Unifying snakes, region growing, and Bayes/MDL for multiband image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 18(9):884–900, 1996. https://doi.org/10.1109/34.537343
L. Antonelli, V. De Simone and D. Serafino. Spatially adaptive regularization in image segmentation. Algorithms, 13(9):226–240, 2020. https://doi.org/10.3390/a13090226
G. Aubert, M. Barlaud, O. Faugeras and S. Jehan-Besson. Image segmentation using active contours: Calculus of variations or shape gradients? SIAM Journal on Applied Mathematics, 63(6):2128–2154, 2003. https://doi.org/10.1137/S0036139902408928
G. Aubert and P. Kornprobst. Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations. Springer, 2006. https://doi.org/10.1007/978-0-387-44588-5
J.F. Aujol, G. Gilboa, T. Chan and S. Osher. Structure-texture image decomposition modeling, algorithms, and parameter selection. International Journal of Computer Vision, 67(1):111–136, 2006. https://doi.org/10.1007/s11263-006-4331-z
V. Caselles, R. Kimmel and G. Sapiro. Geodeisic active contours. International Journal of Computer Vision, 22(1):61–79, 1997. https://doi.org/10.1023/A:1007979827043
A. Chambolle. An algorithm for total variation minimization and applications. Journal of Mathematical Imaging and Vision, 20(1-2):89–97, 2004. https://doi.org/10.1023/B:JMIV.0000011325.36760.1e
T.F. Chan, B.Y. Sandberg and L.A. Vese. Active contours without edges for vector valued images. Journal of Visual Communication Image Representation, 11(2):130–141, 2000. https://doi.org/10.1109/83.902291
T.F. Chan and L.A. Vese. Active contours without edges. IEEE Transactions on Image Processing, 10(2):266–277, 2001.
A. Cichocki and S. Amari. Families of alpha-beta-and gamma-divergences: Flexible and robust measures of similarities. Entropy, 12(6):1532–1568, 2010. https://doi.org/10.3390/e12061532
D. Cremers, M. Rousson and R. Deriche. A review of statistical approaches to level set segmentation integrating color, texture, motion and shape. International Journal of Computer Vision, 72(2):195–215, 2007. https://doi.org/10.1007/s11263-006-8711-1
A.B. Dahl and V.A. Dahl. Dictionary snakes. In 22 nd International Conference on Pattern Recognition, pp. 142–147, Stockholm, Sweden, 2014. IEEE. https://doi.org/10.1109/ICPR.2014.34
A.B. Dahl and V.A. Dahl. Dictionary based image segmentation. In Scandinavian Conference on Image Analysis, pp. 416–423, Copenhagen, Denmark, 2015. Springer. https://doi.org/10.1007/978-3-319-19665-7_3
L. Devroye. A Course in Density Estimation. Springer-Verlag, 1987.
M.N. Do and M. Vetterli. Wavelet-based texture retrieval using generalized gaussian density and Kullback-Leibler distance. IEEE Transactions on Image Processing, 11(2):146–158, 2002. https://doi.org/10.1109/83.982822
D. Dunn and W.E. Higgins. Optimal gabor filters for texture segmentation. IEEE Transactions on Image Processing, 4(7):947–964, 1995. https://doi.org/10.1109/83.392336
T.V. Erven and P. Harremoes. Rényi divergence and Kullback-Leibler divergence. IEEE Transactions on Information Theory, 60(7):3797–3820, 2014. https://doi.org/10.1109/TIT.2014.2320500
L.C. Evans and G.F. Ronald. Measure Theory And Fine Proporties of Function. CRC Press, 2015. https://doi.org/10.1201/b18333
T. Goldstein, X. Bresson and S. Osher. Geometric applications of the split Bregman method: Segmentation and surface reconstruction. Journal of Scientific Computing, 45(1-3):272–293, 2010. https://doi.org/10.1007/s10915-009-9331-z
T. Goldstein and S. Osher. The split Bregman method for L1 regularized problems. SIAM Journal on Imaging Sciences, 2(2):323–343, 2009. https://doi.org/10.1137/080725891
A. Gray. Modern Differential Geometry of Curves and Surfaces with Mathematica. CRC Press, 1996.
G. Michaille H. Attouch, G. Buttazzo. Variational Analysis in Sobolev and BV Spaces: Applications to PDES and Optimization. SIAM, 2005.
A.O. Hero, B. Ma, O. Michel and J.D. Gorman. Alpha-Divergence for Classification, Indexing and Retrieval. University of Michigan, 2002.
N. Houhou, J. Thiran and X. Bresson. Fast texture segmentation based on semi-local region descriptor and active contour. Numerical Mathematics: Theory, Methods and Applications, 2(4):445–468, 2020. https://doi.org/10.4208/nmtma.2009.m9007s
C. Hung, E. Song and Y. Lan. Image Texture Analysis: Foundations, Models and Algorithms. Springer International Publishing, 2019. https://doi.org/10.1007/978-3-030-13773-1
S. Jehan-Besson, M. Barlaud and G. Auber. Textures: A photographic album for artists and designers. Leonardo, 1(1):91–92, 1968. https://doi.org/10.2307/1571915
S. Jehan-Besson, M. Barlaud and G. Auber. Dream2s: Deformable regions driven by an Eulerian accurate minimization method for image and video segmentation. International Journal of Computer Vision, 53(1):365–380, 2003.
S. Kullback and R.A. Leibler. On information and sufficiency. Annals of Mathematical Statistics, 22(1):79–86, 1951. https://doi.org/10.1214/aoms/1177729694
R. Lerski. Texture Analysis for Magnetic Resonance Imaging. Med4 publishing, 2006.
C.M. Li, C.Y. Kao, J.C. Gore and Z.H. Ding. Implicit active contours driven by local binary fitting energy,. In IEEE Conference on Computer Vision and Pattern Recognition, pp. 1–7, Minneapolis, 2007. IEE. https://doi.org/10.1109/CVPR.2007.383014
X. Li, L. Hairong and Y. Xiaoping. Region intensity complexity active contours. Multidimensional Systems and Signal Processing, 31(2):1185–1206, 2020. https://doi.org/10.1007/s11045-020-00704-5
R. Malladi, J.A. Sethian and B.C. Vemuri. Shape modeling with front propagation: a level set approach. International Journal of Computer Vision, 17(2):158– 175, 1975. https://doi.org/10.1109/34.368173
S. Mallat. A theory for multiresolution signal decomposition: The wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11(7):674–693, 1989. https://doi.org/10.1109/34.192463
O.V. Michailovich, Y. Rathi and A. Tannenbaum. Image segmentation using active contours driven by the Bhattacharyya gradient flow. IEEE Transactions on Image Processing, 16(11):2787–2801, 2007. https://doi.org/10.1109/TIP.2007.908073
H. Min, W. Jia, X. Wang, Y. Zhao, R. Hu, Y. Luo, F. Xue and J. Lu. An intensity-texture model based level set method for image segmentation. Pattern Recognition, 48(4):1547–1562, 2015. https://doi.org/10.1016/j.patcog.2014.10.018
T. Minka. Divergence Measures and Message Passing. Microsoft Research Ltd, 2005.
A. Mitiche and I. Ben Ayedn. Variational and Level Set Methods in Image Segmentation. Springer, 2010. https://doi.org/10.1007/978-3-642-15352-5
D. Mumford and J. Shah. Optimal approximations by piecewise smooth functions and associated variational problems. Communication of Pure Applied Mathematics, 42(5):577–685, 1989. https://doi.org/10.1002/cpa.3160420503
M. Nixon and A.S. Aquado. Feature Extraction and Image Processing for Computer vision. Academic Press, 2020. https://doi.org/10.1016/B978-0-12-814976-8.00003-8
S. Osher and R. Fedkiw. Level Set Methods and Dynamic Implicit Surfaces. Springer, 2002. https://doi.org/10.1007/b98879
E. Parzen. On the estimation of a probability density function and the mode. Annals of Mathematical Statistics, 33(3):1065–1076, 1962. https://doi.org/10.1214/aoms/1177704472
C. Petitjean. Recalage Non Rigide d’Images par Approches Variationnelles Statistiques. Application à l’Analyse et à la Modélisation de la Fonction Myocardique en IRM. Université René Descartes -Paris V, 2003. (in French)
M. Petrou and P. Garcia Sevilla. Image Processing: Dealing with Texture. Wiley, 2006. https://doi.org/10.1002/047003534X
A. Ramola, A.K. Shakya and D.V. Pham. Study of statistical methods for texture analysis and their modern evolutions. Wiley, 2020. https://doi.org/10.1002/eng2.12149
A. R´enyi. On measures of entropy and information. In H. Ammann and V.A. Solonnikov(Eds.), Berkeley Symposium on Mathematical Statistics and Probability, pp. 547–5614, California, 1961. university of California press.
C. Sagiv, N. Sochen and Y. Zeevi. Integrated active contours for texture segmentation. IEEE Transactions on Image Processing, 15(6):1633–1646, 2006. https://doi.org/10.1109/TIP.2006.871133
J.A. Sethian. Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision and Material Sciences. Cambridge University Press, 1999.
N. Sochen, R. Kimmel and R. Malladi. A general framework for low level vision. IEEE Transaction on Image Processing, 7(3):310–318, 1998. https://doi.org/10.1109/83.661181
M. Sonka, V. Hlavac and R. Boyle. Image Processing, Analysis and Machine Vision. Thomson, 2007.
M. Spivak. A Comprehensive Introduction to Differential Geometry. Publish or Perish Press, 1999.
X.-C. Tai and C. Wu. Augmented Lagrangian method, dual methods and split Bregman iteration for ROF model. In H. Ammann and V.A. Solonnikov(Eds.), Scale Space and Variational Methods in Computer Vision, pp. 502–513, Norway, 2009. Springer. https://doi.org/10.1007/978-3-642-02256-2_42
C. Wu and X. Tai. Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models. SIAM Journal on Imaging Sciences, 3(3):300–339, 2010. https://doi.org/10.1137/090767558
S.Y. Yeo, X. Xie, I. Sazonov and P. Nithiarasu. Segmentation of biomedical images using active contour model with robust image feature and shape prior. International Journal for Numerical Methods in Biomedical Engineering, 6(5):232– 248, 2014. https://doi.org/10.1002/cnm.2600
X.H. Zhi and H.B. Shen. Saliency driven region-edge-based top down level set evolution reveals the asynchronous focus in image segmentation. Pattern Recognition, 80:241–255, 2018. https://doi.org/10.1016/j.patcog.2018.03.010
S.C. Zhu and A. Yuille. Region competition: Unifying snakes, region growing, and Bayes/MDL for multiband image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 18(9):884–900, 1996. https://doi.org/10.1109/34.537343