American Institute of Mathematical Sciences

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October  2020, 14(5): 891-911. doi: 10.3934/ipi.2020041

Nonlocal regularized CNN for image segmentation

Fan Jia 1, , Xue-Cheng Tai 1, and  Jun Liu 2,,

1. 

Department of Mathematics, Hong Kong Baptist University, Hong Kong, China
2. 

Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing, China

* Corresponding author: Jun Liu

Received  January 2020 Revised  April 2020 Published  July 2020

Non-local dependency is a very important prior for many image segmentation tasks. Generally, convolutional operations are building blocks that process one local neighborhood at a time which means the convolutional neural networks(CNNs) usually do not explicitly make use of the non-local prior on image segmentation tasks. Though the pooling and dilated convolution techniques can enlarge the receptive field to use some nonlocal information during the feature extracting step, there is no nonlocal priori for feature classification step in the current CNNs' architectures. In this paper, we present a non-local total variation (TV) regularized softmax activation function method for semantic image segmentation tasks. The proposed method can be integrated into the architecture of CNNs. To handle the difficulty of back-propagation for CNNs due to the non-smoothness of nonlocal TV, we develop a primal-dual hybrid gradient method to realize the back-propagation of nonlocal TV in CNNs. Experimental evaluations of the non-local TV regularized softmax layer on a series of image segmentation datasets showcase its good performance. Many CNNs can benefit from our proposed method on image segmentation tasks.

Keywords: CNN, nonlocal TV, regularization, duality, image segmentation.
Mathematics Subject Classification: 68U10, 68T07, 65K10.
Citation: 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
References:
[1]

R. Adams and L. Bischof, Seeded region growing, IEEE Transactions on Pattern Analysis and Machine Intelligence, 16 (1994), 641-647.  doi: 10.1109/34.295913.  Google Scholar

[2]

M. Z. Alom, M. Hasan, C. Yakopcic, T. M. Taha and V. K. Asari, Recurrent residual convolutional neural network based on u-net (r2u-net) for medical image segmentation, arXiv: 1802.06955. Google Scholar

[3]

V. Badrinarayanan, A. Kendall and R. Cipolla, Segnet: A deep convolutional encoder-decoder architecture for image segmentation, arXiv: 1511.00561. doi: 10.1109/TPAMI.2016.2644615.  Google Scholar

[4]

L. Barghout and L. Lee, Perceptual information processing system, US Patent App. 10/618,543, (2004). Google Scholar

[5]

M. Benning, C. Brune, M. Burger and J. Müller, Higher-order tv methods–enhancement via bregman iteration, Journal of Scientific Computing, 54 (2013), 269-310.  doi: 10.1007/s10915-012-9650-3.  Google Scholar

[6]

H. Birkholz, A unifying approach to isotropic and anisotropic total variation denoising models, Journal of Computational and Applied Mathematics, 235 (2011), 2502-2514.  doi: 10.1016/j.cam.2010.11.003.  Google Scholar

[7]

J. Canny, A computational approach to edge detection, IEEE Transactions on Pattern Analysis and Machine Intelligence, 8 (1986), 679-698.  doi: 10.1016/B978-0-08-051581-6.50024-6.  Google Scholar

[8]

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

[9]

K. He, X. Zhang, S. Ren and J. Sun, Delving deep into rectifiers: Surpassing human-level performance on imagenet classification, in Proceedings of the IEEE International Conference on Computer Vision, IEEE, 2015, 1026–1034. doi: 10.1109/ICCV.2015.123.  Google Scholar

[10]

F. Jia, J. Liu and X. Tai, A regularized convolutional neural network for semantic image segmentation, Analysis and Applications, (2020) 1–19. Google Scholar

[11]

M. Johnson-Roberson, C. Barto, R. Mehta, S. N. Sridhar, K. Rosaen and R. Vasudevan, Driving in the matrix: Can virtual worlds replace human-generated annotations for real world tasks?, preprint, arXiv: 1610.01983. doi: 10.1109/ICRA.2017.7989092.  Google Scholar

[12]

M. Kass, A. Witkin and D. Terzopoulos, Snakes: Active contour models, International Journal of Computer Vision, 1, (1988) 321–331. doi: 10.1007/BF00133570.  Google Scholar

[13]

P. Krähenbühl and V. Koltun, Efficient inference in fully connected crfs with gaussian edge potentials., Advances in Neural Information Processing Systems, (2011), 109–117. Google Scholar

[14]

A. Krizhevsky, I. Sutskever and G. E. Hinton, Imagenet classification with deep convolutional neural networks, Advances in Neural Information Processing Systems, (2012), 1097–1105. doi: 10.1145/3065386.  Google Scholar

[15]

Y. LeCun, B. Boser, J. S. Denker, D. Henderson, R. E. Howard, W. Hubbard and L. D. Jackel, Backpropagation applied to handwritten zip code recognition, Neural Computation, 1 (1989), 541-551.  doi: 10.1162/neco.1989.1.4.541.  Google Scholar

[16]

G. Lin, C. Shen, A. V. D. Hengel and I. Reid, Efficient piecewise training of deep structured models for semantic segmentation, in Proceedings of the IEEE Conference on Computer Cision and Pattern Recognition, IEEE, 2016, 3194–3203. doi: 10.1109/CVPR.2016.348.  Google Scholar

[17]

J. Long, E. Shelhamer, and T. Darrell, Fully convolutional networks for semantic segmentation, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, IEEE, 2015, 3431–3440. doi: 10.1109/CVPR.2015.7298965.  Google Scholar

[18]

M. Lysaker, A. Lundervold and X.-C. Tai, Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time, IEEE Transactions on Image Processing, 12, (2003), 1579–1590. doi: 10.1109/TIP.2003.819229.  Google Scholar

[19]

D. R. Martin, C. C. Fowlkes and and J. Malik, Learning to detect natural image boundaries using local brightness, color, and texture cues, IEEE Transactions on Pattern Analysis and Machine Intelligence, 26 (2004), 530-549.  doi: 10.1109/TPAMI.2004.1273918.  Google Scholar

[20]

K. Mikula, A. Sarti and F. Sgallari, Co-volume level set method in subjective surface based medical image segmentation, in Handbook of Biomedical Image Analysis, Springer, (2005), 583–626. doi: 10.1007/0-306-48551-6_11.  Google Scholar

[21]

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.  Google Scholar

[22]

H. Noh, S. Hong and B. Han, Learning deconvolution network for semantic segmentation, in Proceedings of the IEEE International Conference on Computer Vision, IEEE, 2015, 1520–1528. doi: 10.1109/ICCV.2015.178.  Google Scholar

[23]

O. Oktay, et al., Attention u-net: Learning where to look for the pancreas, preprint, arXiv: 1804.03999. Google Scholar

[24]

N. Otsu, A threshold selection method from gray-level histograms, IEEE Transactions on Systems, Man and Cybernetics, 9 (1979), 62-66.  doi: 10.1109/TSMC.1979.4310076.  Google Scholar

[25]

O. Ronneberger, P. Fischer and T. Brox, U-net: Convolutional networks for biomedical image segmentation, in International Conference on Medical Image Computing and Computer-Assisted Intervention, Springer, 2015,234–241. doi: 10.1007/978-3-319-24574-4_28.  Google Scholar

[26]

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

[27] B. Schölkopf, K. Tsuda and J.-P. Vert, Support Vector Machine Applications in Computational Biology, MIT press, 2004.   Google Scholar
[28]

L. Shapiro and G. C. Stockman, Computer Vision, Prentice Hall, 2001. Google Scholar

[29]

J. Shi and J. Malik, Normalized cuts and image segmentation, IEEE Transactions on Pattern Analysis and Machine Intelligence, 22 (2000), 888-908.   Google Scholar

[30]

K. Simonyan and A. Zisserman, Very deep convolutional networks for large-scale image recognition, preprint, arXiv: 1409.1556. Google Scholar

[31]

M. Unger, T. Mauthner, T. Pock and H. Bischof, Tracking as segmentation of spatial-temporal volumes by anisotropic weighted tv, in International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition, Springer 2009,193–206. doi: 10.1007/978-3-642-03641-5_15.  Google Scholar

[32]

P. Wang, P. Chen, Y. Yuan, D. Liu, Z. Huang, X. Hou, and G. Cottrell, Understanding convolution for semantic segmentation, in 2018 IEEE Winter Conference on Applications of Computer Vision (WACV), IEEE, 2018, 1451–1460. doi: 10.1109/WACV.2018.00163.  Google Scholar

[33]

K. Wei, K. Yin, X.-C. Tai and T. F. Chan, New region force for variational models in image segmentation and high dimensional data clustering, preprint, arXiv: 1704.08218. doi: 10.4310/AMSA.2018.v3.n1.a8.  Google Scholar

[34]

K. Yin and X.-C. Tai, An effective region force for some variational models for learning and clustering, Journal of Scientific Computing, 74 (2018), 175-196.  doi: 10.1007/s10915-017-0429-4.  Google Scholar

[35]

F. Yu and V. Koltun, Multi-scale context aggregation by dilated convolutions, preprint, arXiv: 1511.07122. Google Scholar

[36]

L. Zelnik-Manor and P. Perona, Self-tuning spectral clustering, Advances in Neural Information Processing Systems, (2005), 1601–1608. Google Scholar

[37]

X. Zheng, Y. Wang, G. Wang and J. Liu, Fast and robust segmentation of white blood cell images by self-supervised learning, Micron, 107 (2018), 55-71.  doi: 10.1016/j.micron.2018.01.010.  Google Scholar

show all references

References:
[1]

R. Adams and L. Bischof, Seeded region growing, IEEE Transactions on Pattern Analysis and Machine Intelligence, 16 (1994), 641-647.  doi: 10.1109/34.295913.  Google Scholar

[2]

M. Z. Alom, M. Hasan, C. Yakopcic, T. M. Taha and V. K. Asari, Recurrent residual convolutional neural network based on u-net (r2u-net) for medical image segmentation, arXiv: 1802.06955. Google Scholar

[3]

V. Badrinarayanan, A. Kendall and R. Cipolla, Segnet: A deep convolutional encoder-decoder architecture for image segmentation, arXiv: 1511.00561. doi: 10.1109/TPAMI.2016.2644615.  Google Scholar

[4]

L. Barghout and L. Lee, Perceptual information processing system, US Patent App. 10/618,543, (2004). Google Scholar

[5]

M. Benning, C. Brune, M. Burger and J. Müller, Higher-order tv methods–enhancement via bregman iteration, Journal of Scientific Computing, 54 (2013), 269-310.  doi: 10.1007/s10915-012-9650-3.  Google Scholar

[6]

H. Birkholz, A unifying approach to isotropic and anisotropic total variation denoising models, Journal of Computational and Applied Mathematics, 235 (2011), 2502-2514.  doi: 10.1016/j.cam.2010.11.003.  Google Scholar

[7]

J. Canny, A computational approach to edge detection, IEEE Transactions on Pattern Analysis and Machine Intelligence, 8 (1986), 679-698.  doi: 10.1016/B978-0-08-051581-6.50024-6.  Google Scholar

[8]

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

[9]

K. He, X. Zhang, S. Ren and J. Sun, Delving deep into rectifiers: Surpassing human-level performance on imagenet classification, in Proceedings of the IEEE International Conference on Computer Vision, IEEE, 2015, 1026–1034. doi: 10.1109/ICCV.2015.123.  Google Scholar

[10]

F. Jia, J. Liu and X. Tai, A regularized convolutional neural network for semantic image segmentation, Analysis and Applications, (2020) 1–19. Google Scholar

[11]

M. Johnson-Roberson, C. Barto, R. Mehta, S. N. Sridhar, K. Rosaen and R. Vasudevan, Driving in the matrix: Can virtual worlds replace human-generated annotations for real world tasks?, preprint, arXiv: 1610.01983. doi: 10.1109/ICRA.2017.7989092.  Google Scholar

[12]

M. Kass, A. Witkin and D. Terzopoulos, Snakes: Active contour models, International Journal of Computer Vision, 1, (1988) 321–331. doi: 10.1007/BF00133570.  Google Scholar

[13]

P. Krähenbühl and V. Koltun, Efficient inference in fully connected crfs with gaussian edge potentials., Advances in Neural Information Processing Systems, (2011), 109–117. Google Scholar

[14]

A. Krizhevsky, I. Sutskever and G. E. Hinton, Imagenet classification with deep convolutional neural networks, Advances in Neural Information Processing Systems, (2012), 1097–1105. doi: 10.1145/3065386.  Google Scholar

[15]

Y. LeCun, B. Boser, J. S. Denker, D. Henderson, R. E. Howard, W. Hubbard and L. D. Jackel, Backpropagation applied to handwritten zip code recognition, Neural Computation, 1 (1989), 541-551.  doi: 10.1162/neco.1989.1.4.541.  Google Scholar

[16]

G. Lin, C. Shen, A. V. D. Hengel and I. Reid, Efficient piecewise training of deep structured models for semantic segmentation, in Proceedings of the IEEE Conference on Computer Cision and Pattern Recognition, IEEE, 2016, 3194–3203. doi: 10.1109/CVPR.2016.348.  Google Scholar

[17]

J. Long, E. Shelhamer, and T. Darrell, Fully convolutional networks for semantic segmentation, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, IEEE, 2015, 3431–3440. doi: 10.1109/CVPR.2015.7298965.  Google Scholar

[18]

M. Lysaker, A. Lundervold and X.-C. Tai, Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time, IEEE Transactions on Image Processing, 12, (2003), 1579–1590. doi: 10.1109/TIP.2003.819229.  Google Scholar

[19]

D. R. Martin, C. C. Fowlkes and and J. Malik, Learning to detect natural image boundaries using local brightness, color, and texture cues, IEEE Transactions on Pattern Analysis and Machine Intelligence, 26 (2004), 530-549.  doi: 10.1109/TPAMI.2004.1273918.  Google Scholar

[20]

K. Mikula, A. Sarti and F. Sgallari, Co-volume level set method in subjective surface based medical image segmentation, in Handbook of Biomedical Image Analysis, Springer, (2005), 583–626. doi: 10.1007/0-306-48551-6_11.  Google Scholar

[21]

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.  Google Scholar

[22]

H. Noh, S. Hong and B. Han, Learning deconvolution network for semantic segmentation, in Proceedings of the IEEE International Conference on Computer Vision, IEEE, 2015, 1520–1528. doi: 10.1109/ICCV.2015.178.  Google Scholar

[23]

O. Oktay, et al., Attention u-net: Learning where to look for the pancreas, preprint, arXiv: 1804.03999. Google Scholar

[24]

N. Otsu, A threshold selection method from gray-level histograms, IEEE Transactions on Systems, Man and Cybernetics, 9 (1979), 62-66.  doi: 10.1109/TSMC.1979.4310076.  Google Scholar

[25]

O. Ronneberger, P. Fischer and T. Brox, U-net: Convolutional networks for biomedical image segmentation, in International Conference on Medical Image Computing and Computer-Assisted Intervention, Springer, 2015,234–241. doi: 10.1007/978-3-319-24574-4_28.  Google Scholar

[26]

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

[27] B. Schölkopf, K. Tsuda and J.-P. Vert, Support Vector Machine Applications in Computational Biology, MIT press, 2004.   Google Scholar
[28]

L. Shapiro and G. C. Stockman, Computer Vision, Prentice Hall, 2001. Google Scholar

[29]

J. Shi and J. Malik, Normalized cuts and image segmentation, IEEE Transactions on Pattern Analysis and Machine Intelligence, 22 (2000), 888-908.   Google Scholar

[30]

K. Simonyan and A. Zisserman, Very deep convolutional networks for large-scale image recognition, preprint, arXiv: 1409.1556. Google Scholar

[31]

M. Unger, T. Mauthner, T. Pock and H. Bischof, Tracking as segmentation of spatial-temporal volumes by anisotropic weighted tv, in International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition, Springer 2009,193–206. doi: 10.1007/978-3-642-03641-5_15.  Google Scholar

[32]

P. Wang, P. Chen, Y. Yuan, D. Liu, Z. Huang, X. Hou, and G. Cottrell, Understanding convolution for semantic segmentation, in 2018 IEEE Winter Conference on Applications of Computer Vision (WACV), IEEE, 2018, 1451–1460. doi: 10.1109/WACV.2018.00163.  Google Scholar

[33]

K. Wei, K. Yin, X.-C. Tai and T. F. Chan, New region force for variational models in image segmentation and high dimensional data clustering, preprint, arXiv: 1704.08218. doi: 10.4310/AMSA.2018.v3.n1.a8.  Google Scholar

[34]

K. Yin and X.-C. Tai, An effective region force for some variational models for learning and clustering, Journal of Scientific Computing, 74 (2018), 175-196.  doi: 10.1007/s10915-017-0429-4.  Google Scholar

[35]

F. Yu and V. Koltun, Multi-scale context aggregation by dilated convolutions, preprint, arXiv: 1511.07122. Google Scholar

[36]

L. Zelnik-Manor and P. Perona, Self-tuning spectral clustering, Advances in Neural Information Processing Systems, (2005), 1601–1608. Google Scholar

[37]

X. Zheng, Y. Wang, G. Wang and J. Liu, Fast and robust segmentation of white blood cell images by self-supervised learning, Micron, 107 (2018), 55-71.  doi: 10.1016/j.micron.2018.01.010.  Google Scholar

Figure 1.  An example of segmentation results by applying the algorithm of [34] and our proposed method on an image from BSD500. When using 4 geometrical nearest neighbors, the weights are set to 1. The segmentation is quite smooth and missing details (Figure 1(b)). When we use Eq. (11) to compute W, the segmentation results are with more details and better accuracy
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Figure 2.  Given an input $ O $, $ \lambda = 3 $ and $ \tau = 0.03 $, we perform algorithms for regularized softmax with local operator and non-local operator, respectively. Figure 2(a) is the convergence of softmax with local operate, the primal energy curve has a peak during the iteration. While in Figure 2(b), the energy curve drops rapidly at first and finally converges smoothly
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Figure 3.  Segmentation results predicted by Unet, RUnet and NLUnet on images from testing dataset of White Blood Cell. From row 2 to row 5, The black regions are background, the gray regions are cell sap, the white regions are nucleus
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Figure 4.  An enlarged view of segmentation results from Figure 3
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Figure 5.  Segmentation results predicted by AUnet, RAUnet and NLAUnet on images from testing dataset of White Blood Cell. From row 2 to row 5, The black regions are background, the gray regions are cell sap, the white regions are nucleus
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Figure 6.  Segmentation results predicted by Segnet, RSegnet and NLSegnet trained on CamVid Dataset
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Figure 7.  An enlarged view of segmentation results from Figure 6 column 2
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Figure 8.  An enlarged view of segmentation results from Figure 6 column 1
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Table 1.  Results of Unet, RUnet and NLUnet trained on WBC Dataset
Method Unet [25] RUnet [10] NLUnet
mIoU 89.79 90.15 90.80
Accuracy 97.04 97.13 97.42
RE 1.82 1.30 1.59
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Method Unet [25] RUnet [10] NLUnet
mIoU 89.79 90.15 90.80
Accuracy 97.04 97.13 97.42
RE 1.82 1.30 1.59
Table 2.  Results of AUnet, RAUnet and NLAUnet trained on WBC Dataset
Method AUnet [23] RAUnet NLAUnet
mIoU 90.75 91.01 91.69
Accuracy 97.35 97.40 97.57
RE 1.43 1.41 1.43
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Method AUnet [23] RAUnet NLAUnet
mIoU 90.75 91.01 91.69
Accuracy 97.35 97.40 97.57
RE 1.43 1.41 1.43
Table 3.  Results of Segnet, RSegnet, NLSegnet trained on CamVid Dataset
Method Segnet [3] RSegnet[10] NLSegnet
mIoU 57.35 57.79 59.84
Accuracy 87.74 88.01 88.59
RE 4.10 2.43 3.40
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Method Segnet [3] RSegnet[10] NLSegnet
mIoU 57.35 57.79 59.84
Accuracy 87.74 88.01 88.59
RE 4.10 2.43 3.40
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