American Institute of Mathematical Sciences

Advanced
May  2013, 7(2): 499-521. doi: 10.3934/ipi.2013.7.499

A geometry guided image denoising scheme

Weihong Guo 1, and  Jing Qin 1,

1. 

10900 Euclid Avenue, Cleveland, OH 44106-7058, United States, United States

Received  August 2011 Revised  January 2013 Published  May 2013

During image denoising, it is often difficult to balance between the removal of noise and the preservation of contrast and fine features, especially when the noise is excessive. We propose to efficiently balance the two using segmentation and more general geometry extraction transforms. Explained in the nonlocal-means (NL-means) framework, we introduce a mutual position function to ensure the averaging is only taken over pixels in the same segmentation phase, and provide selection schemes for convolution kernel and weight function to further improve the performance. To address unreliable segmentation due to more excessive noise, we use a feature extraction transform that is more general than segmentation and less sensitive to noise. Unlike most denoising approaches that only work for one type of noise and/or involve heuristic parameter tuning, the proposed method comes with an automatic parameter selection scheme, and can be easily adapted for various types of noise, ranging from Gaussian, Poisson, Rician to ultrasound noise. Comparison with the original NL-means as well as ROF, BM3D, and K-SVD on various simulated data, MRI and SEM images, indicates potentials of the proposed method.
Keywords: segmentation, nonlocal means., Image denoising.
Mathematics Subject Classification: 68U10, 65D18, 65D1.
Citation: Weihong Guo, Jing Qin. A geometry guided image denoising scheme. Inverse Problems & Imaging, 2013, 7 (2) : 499-521. doi: 10.3934/ipi.2013.7.499
References:
[1]

D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math., 42 (1989), 577-685. doi: 10.1002/cpa.3160420503.  Google Scholar

[2]

P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Transactions on Pattern Analysis and Machine Intelligence, 12 (1990), 629-639. doi: 10.1109/34.56205.  Google Scholar

[3]

L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithm, Physica D: Nonlinear Phenomena, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F.  Google Scholar

[4]

M. Bertalmio, V. Caselles, B. Rougé and A. Solé, TV based image restoration with local constraints, Journal of Scientific Computing, 19 (2003), 95-122. doi: 10.1023/A:1025391506181.  Google Scholar

[5]

S. Geman and D. Geman, Stochastic relaxation, gibbs distributions, and the bayesian restoration of images, IEEE Transactions on Pattern Analysis and Machine Intelligence, 6 (1984), 721-741. Google Scholar

[6]

P. Saint-Marc, J. S. Chen and G. Medioni, Adaptive smoothing: A general tool for early vision, IEEE Transactions on Pattern Analysis and Machine Intelligence, 13 (1991), 514-529. Google Scholar

[7]

S. M. Smith and J. M. Brady, SUSAN - A new approach to low level image processing, International Journal of Computer Vision, 23 (1995), 45-78. Google Scholar

[8]

C. Tomasi and R. Manduchi, Bilateral filtering for gray and color images, in "Proceedings of Sixth International Conference on Computer Vision," (1998), 839-846. doi: 10.1109/ICCV.1998.710815.  Google Scholar

[9]

M. Elad, On the origin of the bilateral filter and ways to improve it, IEEE Transactions on Image Processing, 11 (2002), 1141-1151. doi: 10.1109/TIP.2002.801126.  Google Scholar

[10]

S. Durand and J. Froment, Reconstruction of wavelet coefficients using total variation minimization, SIAM J. Sci. Comput., 24 (2003), 1754-1767. doi: 10.1137/S1064827501397792.  Google Scholar

[11]

A. Buades, B. Coll and J. M. Morel, A non-local algorithm for image denoising, IEEE Computer Society, 2 (2005), 60-65. doi: 10.1109/CVPR.2005.38.  Google Scholar

[12]

S. Kindermann, S. Osher and P. W. Jones, Deblurring and Denoising of Images by Nonlocal Functionals, Multiscale Modelling and Simulation, 4 (2005), 1091-1115. doi: 10.1137/050622249.  Google Scholar

[13]

T. Brox and D. Cremers, Iterated nonlocal means for texture restoration, Scale Space and Variational Methods in Computer Vision, (2008), 13-24. doi: 10.1007/978-3-540-72823-8_2.  Google Scholar

[14]

C. Kervrann, J. Boulanger and P. Coupé, Bayesian non-local means filter, image redundancy and adaptive dictionaries for noise removal, Scale Space and Variational Methods in Computer Vision, (2007), 520-532. doi: 10.1007/978-3-540-72823-8_45.  Google Scholar

[15]

B. Goossens, Q. Luong, A. Pizurica and W. Philips, An improved non-local denoising algorithm, in "Local and Non-Local Approximation in Image Processing, International Workshop, Proceedings," (2008). Google Scholar

[16]

G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Modeling and Simulation, 7 (2008), 1005-1028. doi: 10.1137/070698592.  Google Scholar

[17]

C-A. Deledalle, L. Denis and F. Tupin, Iterative wieghted maximum likelihood denoising with probabilistic patch-based weights, Transactions on Image Processing, 18 (2009), 2661-2672. doi: 10.1109/TIP.2009.2029593.  Google Scholar

[18]

D. Peter, V. Govindan and A. Mathew, Nonlocal-means image denoising techonology using robust M-estimator, Journal of Computer Science and Technology, 25 (2010), 623-631. Google Scholar

[19]

N. Wiest-Daesslé, S. Prima, P. Coupé, S. P. Morrissey and C. Barillot, Rician noise removal by non-local means filtering for low signal-to-noise ratio MRI: Applications to DT-MRI, in "Proceedings of the 11th International Conference on Medical Image Computing and Computer-Assisted Intervention," (2008), 171-179. doi: 10.1007/978-3-540-85990-1_21.  Google Scholar

[20]

P. Coupé, P. Yger, S. Prima, P. Hellier, C. Kervrann and C. Barillot, An optimized blockwise nonlocal means denoising filter for 3-D magnetic resonance images, Medical Imaging, IEEE Transactions on, 27 (2008), 425-441. doi: 10.1109/TMI.2007.906087.  Google Scholar

[21]

C-A. Deledalle, F. Tupin and L. Denis, Poisson NL means: Unsupervised non local means for Poisson noise, in "Proc. of ICIP", Hongkong, (2010). doi: 10.1109/ICIP.2010.5653394.  Google Scholar

[22]

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.  Google Scholar

[23]

T. F. Chan and L. A. Vese, Active contours without edges, Image Processing, IEEE Transactions on, 10 (2001), 266-277. doi: 10.1109/83.902291.  Google Scholar

[24]

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. Google Scholar

[25]

K. Krishnamoorthy, "Handbook of Statistical Distributions with Applications," Chapman and Hall/CRC Press, London/Boca Raton, 2006. doi: 10.1201/9781420011371.  Google Scholar

[26]

T. Gasser, L. Sroka and C. Jennen-Steinmetz, Residual variance and residual pattern in nonlinear regression, Biometrika, 73 (1986), 625-633. doi: 10.1093/biomet/73.3.625.  Google Scholar

[27]

M. N. Do and M. Vetterli, The contourlet transform: an efficient directional multiresolution image representation, IEEE Transactions on Image Processing, 14 (2005), 2091-2106. doi: 10.1109/TIP.2005.859376.  Google Scholar

[28]

Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: From error visibility to structural similarity, Image Processing, IEEE Transactions on, 13 (2004), 600-612. doi: 10.1109/TIP.2003.819861.  Google Scholar

[29]

E. Candés and D. Donoho, "Curvelets: A Surprisingly Effective Nonadaptive Representation of Objects with Edges,", in, ().   Google Scholar

[30]

K. Guo, G. Kutyniok and D. Labate, "Sparse Multidimensional Representations using Anisotropic Dilation and Shear Operators," Wavelets and Splines (Athens, GA, 2005), Nashboro Press, Nashville, TN.  Google Scholar

[31]

E. A. Nadaraya, On estimating regression, Theory Probab. Appl., 9 (1964), 141-142. doi: 10.1137/1109020.  Google Scholar

[32]

F. J. Anscombe, The transformation of Poisson, binomial and negative-binomial data, Biometrika, 35 (1948), 246-254.  Google Scholar

[33]

M. D. DeVore, A. D. Lanterman and J. A. O'Sullivan, ATR performance of a rician model for SAR images, in "Automatic Target Recognition X, Proc. SPIE 4050," (2000), 34-45. doi: 10.1117/12.395589.  Google Scholar

[34]

J. Sijbers, A. J. Den Dekker, P. Scheunders and D. Van Dyck, Maximum-likelihood estimation of Rician distribution parameters, Medical Imaging, IEEE Transactions on, 17 (1998), 357-361. doi: 10.1109/42.712125.  Google Scholar

[35]

T. Loupas, W. N. McDicken and P. L. Allan, An adaptive weighted median filter for speckle suppression in medical ultrasonic images, Circuits and Systems, IEEE Transactions on, 36 (1989), 129-135. doi: 10.1109/31.16577.  Google Scholar

[36]

K. Krissian, Speckle-constrained anisotropic diffusion for ultrasound images, in "Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition," (2005), 547-552. Google Scholar

[37]

K. Dabov, A. Foi, V. Katkovnik and K. Egiazarian, Image denoising with block-matching and 3D filtering, in "Electronic Imaging" 06, Proc. SPIE 6064", (2006). doi: 10.1117/12.643267.  Google Scholar

[38]

M. Elad and M. Aharon, Image denoising via sparse and redundant representations over learned dictionaries, Image Processing, IEEE Transactions on, 15 (2006), 3736-3745. doi: 10.1109/TIP.2006.881969.  Google Scholar

[39]

P. Coupé, P. Hellier, C. Kervrann and C. Barillot, Bayesian non local means-based speckle filtering, in "Proceedings of ISBI," (2008), 1291-1294. doi: 10.1109/ISBI.2008.4541240.  Google Scholar

show all references

References:
[1]

D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math., 42 (1989), 577-685. doi: 10.1002/cpa.3160420503.  Google Scholar

[2]

P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Transactions on Pattern Analysis and Machine Intelligence, 12 (1990), 629-639. doi: 10.1109/34.56205.  Google Scholar

[3]

L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithm, Physica D: Nonlinear Phenomena, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F.  Google Scholar

[4]

M. Bertalmio, V. Caselles, B. Rougé and A. Solé, TV based image restoration with local constraints, Journal of Scientific Computing, 19 (2003), 95-122. doi: 10.1023/A:1025391506181.  Google Scholar

[5]

S. Geman and D. Geman, Stochastic relaxation, gibbs distributions, and the bayesian restoration of images, IEEE Transactions on Pattern Analysis and Machine Intelligence, 6 (1984), 721-741. Google Scholar

[6]

P. Saint-Marc, J. S. Chen and G. Medioni, Adaptive smoothing: A general tool for early vision, IEEE Transactions on Pattern Analysis and Machine Intelligence, 13 (1991), 514-529. Google Scholar

[7]

S. M. Smith and J. M. Brady, SUSAN - A new approach to low level image processing, International Journal of Computer Vision, 23 (1995), 45-78. Google Scholar

[8]

C. Tomasi and R. Manduchi, Bilateral filtering for gray and color images, in "Proceedings of Sixth International Conference on Computer Vision," (1998), 839-846. doi: 10.1109/ICCV.1998.710815.  Google Scholar

[9]

M. Elad, On the origin of the bilateral filter and ways to improve it, IEEE Transactions on Image Processing, 11 (2002), 1141-1151. doi: 10.1109/TIP.2002.801126.  Google Scholar

[10]

S. Durand and J. Froment, Reconstruction of wavelet coefficients using total variation minimization, SIAM J. Sci. Comput., 24 (2003), 1754-1767. doi: 10.1137/S1064827501397792.  Google Scholar

[11]

A. Buades, B. Coll and J. M. Morel, A non-local algorithm for image denoising, IEEE Computer Society, 2 (2005), 60-65. doi: 10.1109/CVPR.2005.38.  Google Scholar

[12]

S. Kindermann, S. Osher and P. W. Jones, Deblurring and Denoising of Images by Nonlocal Functionals, Multiscale Modelling and Simulation, 4 (2005), 1091-1115. doi: 10.1137/050622249.  Google Scholar

[13]

T. Brox and D. Cremers, Iterated nonlocal means for texture restoration, Scale Space and Variational Methods in Computer Vision, (2008), 13-24. doi: 10.1007/978-3-540-72823-8_2.  Google Scholar

[14]

C. Kervrann, J. Boulanger and P. Coupé, Bayesian non-local means filter, image redundancy and adaptive dictionaries for noise removal, Scale Space and Variational Methods in Computer Vision, (2007), 520-532. doi: 10.1007/978-3-540-72823-8_45.  Google Scholar

[15]

B. Goossens, Q. Luong, A. Pizurica and W. Philips, An improved non-local denoising algorithm, in "Local and Non-Local Approximation in Image Processing, International Workshop, Proceedings," (2008). Google Scholar

[16]

G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Modeling and Simulation, 7 (2008), 1005-1028. doi: 10.1137/070698592.  Google Scholar

[17]

C-A. Deledalle, L. Denis and F. Tupin, Iterative wieghted maximum likelihood denoising with probabilistic patch-based weights, Transactions on Image Processing, 18 (2009), 2661-2672. doi: 10.1109/TIP.2009.2029593.  Google Scholar

[18]

D. Peter, V. Govindan and A. Mathew, Nonlocal-means image denoising techonology using robust M-estimator, Journal of Computer Science and Technology, 25 (2010), 623-631. Google Scholar

[19]

N. Wiest-Daesslé, S. Prima, P. Coupé, S. P. Morrissey and C. Barillot, Rician noise removal by non-local means filtering for low signal-to-noise ratio MRI: Applications to DT-MRI, in "Proceedings of the 11th International Conference on Medical Image Computing and Computer-Assisted Intervention," (2008), 171-179. doi: 10.1007/978-3-540-85990-1_21.  Google Scholar

[20]

P. Coupé, P. Yger, S. Prima, P. Hellier, C. Kervrann and C. Barillot, An optimized blockwise nonlocal means denoising filter for 3-D magnetic resonance images, Medical Imaging, IEEE Transactions on, 27 (2008), 425-441. doi: 10.1109/TMI.2007.906087.  Google Scholar

[21]

C-A. Deledalle, F. Tupin and L. Denis, Poisson NL means: Unsupervised non local means for Poisson noise, in "Proc. of ICIP", Hongkong, (2010). doi: 10.1109/ICIP.2010.5653394.  Google Scholar

[22]

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.  Google Scholar

[23]

T. F. Chan and L. A. Vese, Active contours without edges, Image Processing, IEEE Transactions on, 10 (2001), 266-277. doi: 10.1109/83.902291.  Google Scholar

[24]

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. Google Scholar

[25]

K. Krishnamoorthy, "Handbook of Statistical Distributions with Applications," Chapman and Hall/CRC Press, London/Boca Raton, 2006. doi: 10.1201/9781420011371.  Google Scholar

[26]

T. Gasser, L. Sroka and C. Jennen-Steinmetz, Residual variance and residual pattern in nonlinear regression, Biometrika, 73 (1986), 625-633. doi: 10.1093/biomet/73.3.625.  Google Scholar

[27]

M. N. Do and M. Vetterli, The contourlet transform: an efficient directional multiresolution image representation, IEEE Transactions on Image Processing, 14 (2005), 2091-2106. doi: 10.1109/TIP.2005.859376.  Google Scholar

[28]

Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: From error visibility to structural similarity, Image Processing, IEEE Transactions on, 13 (2004), 600-612. doi: 10.1109/TIP.2003.819861.  Google Scholar

[29]

E. Candés and D. Donoho, "Curvelets: A Surprisingly Effective Nonadaptive Representation of Objects with Edges,", in, ().   Google Scholar

[30]

K. Guo, G. Kutyniok and D. Labate, "Sparse Multidimensional Representations using Anisotropic Dilation and Shear Operators," Wavelets and Splines (Athens, GA, 2005), Nashboro Press, Nashville, TN.  Google Scholar

[31]

E. A. Nadaraya, On estimating regression, Theory Probab. Appl., 9 (1964), 141-142. doi: 10.1137/1109020.  Google Scholar

[32]

F. J. Anscombe, The transformation of Poisson, binomial and negative-binomial data, Biometrika, 35 (1948), 246-254.  Google Scholar

[33]

M. D. DeVore, A. D. Lanterman and J. A. O'Sullivan, ATR performance of a rician model for SAR images, in "Automatic Target Recognition X, Proc. SPIE 4050," (2000), 34-45. doi: 10.1117/12.395589.  Google Scholar

[34]

J. Sijbers, A. J. Den Dekker, P. Scheunders and D. Van Dyck, Maximum-likelihood estimation of Rician distribution parameters, Medical Imaging, IEEE Transactions on, 17 (1998), 357-361. doi: 10.1109/42.712125.  Google Scholar

[35]

T. Loupas, W. N. McDicken and P. L. Allan, An adaptive weighted median filter for speckle suppression in medical ultrasonic images, Circuits and Systems, IEEE Transactions on, 36 (1989), 129-135. doi: 10.1109/31.16577.  Google Scholar

[36]

K. Krissian, Speckle-constrained anisotropic diffusion for ultrasound images, in "Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition," (2005), 547-552. Google Scholar

[37]

K. Dabov, A. Foi, V. Katkovnik and K. Egiazarian, Image denoising with block-matching and 3D filtering, in "Electronic Imaging" 06, Proc. SPIE 6064", (2006). doi: 10.1117/12.643267.  Google Scholar

[38]

M. Elad and M. Aharon, Image denoising via sparse and redundant representations over learned dictionaries, Image Processing, IEEE Transactions on, 15 (2006), 3736-3745. doi: 10.1109/TIP.2006.881969.  Google Scholar

[39]

P. Coupé, P. Hellier, C. Kervrann and C. Barillot, Bayesian non local means-based speckle filtering, in "Proceedings of ISBI," (2008), 1291-1294. doi: 10.1109/ISBI.2008.4541240.  Google Scholar

[1]

Fan Jia, Xue-Cheng Tai, Jun Liu. Nonlocal regularized CNN for image segmentation. Inverse Problems & Imaging, 2020, 14 (5) : 891-911. doi: 10.3934/ipi.2020041
[2]

Haijuan Hu, Jacques Froment, Baoyan Wang, Xiequan Fan. Spatial-Frequency domain nonlocal total variation for image denoising. Inverse Problems & Imaging, 2020, 14 (6) : 1157-1184. doi: 10.3934/ipi.2020059
[3]

Baoli Shi, Zhi-Feng Pang, Jing Xu. Image segmentation based on the hybrid total variation model and the K-means clustering strategy. Inverse Problems & Imaging, 2016, 10 (3) : 807-828. doi: 10.3934/ipi.2016022
[4]

Sung Ha Kang, Berta Sandberg, Andy M. Yip. A regularized k-means and multiphase scale segmentation. Inverse Problems & Imaging, 2011, 5 (2) : 407-429. doi: 10.3934/ipi.2011.5.407
[5]

Ye Yuan, Yan Ren, Xiaodong Liu, Jing Wang. Approach to image segmentation based on interval neutrosophic set. Numerical Algebra, Control & Optimization, 2020, 10 (1) : 1-11. doi: 10.3934/naco.2019028
[6]

Dominique Zosso, Jing An, James Stevick, Nicholas Takaki, Morgan Weiss, Liane S. Slaughter, Huan H. Cao, Paul S. Weiss, Andrea L. Bertozzi. Image segmentation with dynamic artifacts detection and bias correction. Inverse Problems & Imaging, 2017, 11 (3) : 577-600. doi: 10.3934/ipi.2017027
[7]

Matthew S. Keegan, Berta Sandberg, Tony F. Chan. A multiphase logic framework for multichannel image segmentation. Inverse Problems & Imaging, 2012, 6 (1) : 95-110. doi: 10.3934/ipi.2012.6.95
[8]

Mujibur Rahman Chowdhury, Jun Zhang, Jing Qin, Yifei Lou. Poisson image denoising based on fractional-order total variation. Inverse Problems & Imaging, 2020, 14 (1) : 77-96. doi: 10.3934/ipi.2019064
[9]

Zhiguang Zhang, Qiang Liu, Tianling Gao. A fast explicit diffusion algorithm of fractional order anisotropic diffusion for image denoising. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021018
[10]

Fangfang Dong, Yunmei Chen. A fractional-order derivative based variational framework for image denoising. Inverse Problems & Imaging, 2016, 10 (1) : 27-50. doi: 10.3934/ipi.2016.10.27
[11]

Rongliang Chen, Jizu Huang, Xiao-Chuan Cai. A parallel domain decomposition algorithm for large scale image denoising. Inverse Problems & Imaging, 2019, 13 (6) : 1259-1282. doi: 10.3934/ipi.2019055
[12]

Wei Zhu, Xue-Cheng Tai, Tony Chan. Augmented Lagrangian method for a mean curvature based image denoising model. Inverse Problems & Imaging, 2013, 7 (4) : 1409-1432. doi: 10.3934/ipi.2013.7.1409
[13]

Qiang Liu, Zhichang Guo, Chunpeng Wang. Renormalized solutions to a reaction-diffusion system applied to image denoising. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1839-1858. doi: 10.3934/dcdsb.2016025
[14]

Feishe Chen, Lixin Shen, Yuesheng Xu, Xueying Zeng. The Moreau envelope approach for the L1/TV image denoising model. Inverse Problems & Imaging, 2014, 8 (1) : 53-77. doi: 10.3934/ipi.2014.8.53
[15]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 4963-4998. doi: 10.3934/dcdsb.2020321
[16]

Shi Yan, Jun Liu, Haiyang Huang, Xue-Cheng Tai. A dual EM algorithm for TV regularized Gaussian mixture model in image segmentation. Inverse Problems & Imaging, 2019, 13 (3) : 653-677. doi: 10.3934/ipi.2019030
[17]

Jianping Zhang, Ke Chen, Bo Yu, Derek A. Gould. A local information based variational model for selective image segmentation. Inverse Problems & Imaging, 2014, 8 (1) : 293-320. doi: 10.3934/ipi.2014.8.293
[18]

Lu Tan, Ling Li, Senjian An, Zhenkuan Pan. Nonlinear diffusion based image segmentation using two fast algorithms. Mathematical Foundations of Computing, 2019, 2 (2) : 149-168. doi: 10.3934/mfc.2019011
[19]

Ruiliang Zhang, Xavier Bresson, Tony F. Chan, Xue-Cheng Tai. Four color theorem and convex relaxation for image segmentation with any number of regions. Inverse Problems & Imaging, 2013, 7 (3) : 1099-1113. doi: 10.3934/ipi.2013.7.1099
[20]

Balázs Kósa, Karol Mikula, Markjoe Olunna Uba, Antonia Weberling, Neophytos Christodoulou, Magdalena Zernicka-Goetz. 3D image segmentation supported by a point cloud. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 971-985. doi: 10.3934/dcdss.2020351

2020 Impact Factor: 1.639

Tools

Metrics

  • PDF downloads (36)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences