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Using optimal transport theory to optimize a deep convolutional neural network microscopic cell counting method

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Abstract

Medical image processing has become increasingly important in recent years, particularly in the field of microscopic cell imaging. However, accurately counting the number of cells in an image can be a challenging task due to the significant variations in cell size and shape. To tackle this problem, many existing methods rely on deep learning techniques, such as convolutional neural networks (CNNs), to count cells in an image or use regression counting methods to learn the similarities between an input image and a predicted cell image density map. In this paper, we propose a novel approach to monitor the cell counting process by optimizing the loss function using the optimal transport method, a rigorous measure to calculate the difference between the predicted count map and the dot annotation map generated by the CNN. We evaluated our algorithm on three publicly available cell count benchmarks: the synthetic fluorescence microscopy (VGG) dataset, the modified bone marrow (MBM) dataset, and the human subcutaneous adipose tissue (ADI) dataset. Our method outperforms other state-of-the-art methods, achieving a mean absolute error (MAE) of 2.3, 4.8, and 13.1 on the VGG, MBM, and ADI datasets, respectively, with smaller standard deviations. By using the optimal transport method, our approach provides a more accurate and reliable cell counting method for medical image processing.

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References

  1. Falk T, Mai D, Bensch R, Çiçek Ö, Abdulkadir A, Marrakchi Y, Böhm A, Deubner J, Jäckel Z, Seiwald K et al (2019) U-net: deep learning for cell counting, detection, and morphometry. Nat Methods 16(1):67–70

    Article  CAS  PubMed  Google Scholar 

  2. Zhang C, Li H, Wang X, Yang X (2015) Cross-scene crowd counting via deep convolutional neural networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp 833–841

  3. Laradji IH, Rostamzadeh N, Pinheiro PO, Vazquez D, Schmidt M (2018) Where are the blobs: counting by localization with point supervision. In Springer, Cham

    Google Scholar 

  4. Polley M-YC, Leung SC, McShane LM, Gao D, Hugh JC, Mastropasqua MG, Viale G, Zabaglo LA, Penault-Llorca F, Bartlett JM et al (2013) An international Ki67 reproducibility study. J Natl Cancer Inst 105(24):1897–1906

    Article  PubMed  PubMed Central  Google Scholar 

  5. Xie W, Noble JA, Zisserman A (2018) Microscopy cell counting and detection with fully convolutional regression networks. Comput Methods Biomech Biomed Eng: Imaging & Visualization 6(3):283–292

    Google Scholar 

  6. He S, Minn KT, Solnica-Krezel L, Anastasio M, Li H (2019) Automatic microscopic cell counting by use of deeply-supervised density regression model. In Medical Imaging 2019: Digital Pathology, vol 10956, pp 121–128 SPIE

  7. Guo Y, Stein J, Wu G, Krishnamurthy A (2019) SAU-Net: a universal deep network for cell counting. In Proceedings of the 10th ACM international conference on bioinformatics, computational biology and health informatics, pp 299–306

  8. Wang Z, Yin Z (2021) Cell counting by a location-aware network. In International workshop on machine learning in medical imaging, Springer pp 120–129

  9. Jiang N, Yu F (2020) A cell counting framework based on random forest and density map. Appl Sci 10(23):8346

    Article  CAS  Google Scholar 

  10. Barinova O, Lempitsky V, Kholi P (2012) On detection of multiple object instances using Hough transforms. IEEE Trans Pattern Anal Mach Intell 34(9):1773–1784

    Article  PubMed  Google Scholar 

  11. Arteta C, Lempitsky V, Noble JA, Zisserman A (2012) Learning to detect cells using non-overlapping extremal regions. In International conference on medical image computing and computer-assisted intervention, Springer pp 348–356

  12. Xing F, Su H, Neltner J, Yang L (2013) Automatic Ki-67 counting using robust cell detection and online dictionary learning. IEEE Transact Biomed Eng 61(3):859–870

    Article  Google Scholar 

  13. Arteta C, Lempitsky V, Noble JA, Zisserman A (2016) Detecting overlapping instances in microscopy images using extremal region trees. Med Image Anal 27:3–16

    Article  PubMed  Google Scholar 

  14. Lempitsky V, Zisserman A (2010) Learning to count objects in images. Adv Neural Inf Process Syst 23

  15. Xie Y, Xing F, Kong X, Su H, Yang L (2015) Beyond classification: structured regression for robust cell detection using convolutional neural network. In International conference on medical image computing and computer-assisted intervention, Springer pp 358–365

  16. Paul Cohen J, Boucher G, Glastonbury CA, Lo HZ, Bengio Y (2017) Count-ception: counting by fully convolutional redundant counting. In Proceedings of the IEEE international conference on computer vision workshops, pp 18–26

  17. Xu M, Hu M, Zhang Y, Zhou Y (2021) DAU-Net: a regression cell counting method. In ISCTT 2021; 6th International conference on information science, computer technology and transportation, VDE pp 1–6

  18. Cireşan DC, Giusti A, Gambardella LM, Schmidhuber J (2013) Mitosis detection in breast cancer histology images with deep neural networks. In International conference on medical image computing and computer-assisted intervention, Springer pp 411–418

  19. Fiaschi L, Köthe U, Nair R, Hamprecht FA (2012) Learning to count with regression forest and structured labels. In Proceedings of the 21st international conference on pattern recognition (ICPR2012), IEEE pp 2685–2688

  20. Arteta C, Lempitsky V, Noble JA, Zisserman A (2014) Interactive object counting. In European conference on computer vision, Springer pp 504–518

  21. Long J, Shelhamer E, Darrell T (2015) Fully convolutional networks for semantic segmentation. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp 3431–3440

  22. Seguí S, Pujol O, Vitria J (2015) Learning to count with deep object features. In Proceedings of the IEEE conference on computer vision and pattern recognition workshops, pp 90–96

  23. Ronneberger O, Fischer P, Brox T (2015) U-Net: convolutional networks for biomedical image segmentation. In International conference on medical image computing and computer-assisted intervention, Springer pp 234–241

  24. Wan J, Chan A (2019) Adaptive density map generation for crowd counting. In Proceedings of the IEEE/CVF international conference on computer vision, pp 1130–1139

  25. Ma Z, Wei X, Hong X, Gong Y (2019) Bayesian loss for crowd count estimation with point supervision. In Proceedings of the IEEE/CVF international conference on computer vision, pp 6142–6151

  26. Wang B, Liu H, Samaras D, Nguyen MH (2020) Distribution matching for crowd counting. Adv Neural Inf Process Syst 33:1595–1607

    Google Scholar 

  27. Wan J, Liu Z, Chan AB (2021) A generalized loss function for crowd counting and localization. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp 1974–1983

  28. Simonyan K, Zisserman A (2014) Very deep convolutional networks for large-scale image recognition. arXiv preprint arXiv:1409.1556

  29. Villani C (2009) Optimal transport: old and new, vol 338. Springer, Berlin

    Google Scholar 

  30. Rubner Y, Tomasi C, Guibas LJ (2000) The earth mover’s distance as a metric for image retrieval. Int J Comput Vision 40(2):99–21

    Article  Google Scholar 

  31. Chen P, Gui C (2013) Alpha divergences based mass transport models for image matching problems. Inverse Problems & Imaging 5(3):551–590

    Article  Google Scholar 

  32. Gibbs AL, Su FE (2010) On choosing and bounding probability metrics. Int Stat Rev 70(3):419–435

    Article  Google Scholar 

  33. Kantorovich LV (2006) On the translocation of masses. J Math Sci 133(4):1381–1382

    Article  Google Scholar 

  34. Bregman LM (1967) Proof of the convergence of Sheleikhovskii’s method for a problem with transportation constraints. USSR Comput Math Math Phys 7(1):191–204

    Article  Google Scholar 

  35. Sinkhorn R (1974) Diagonal equivalence to matrices with prescribed row and column sums. ii. Proc Am Math Soc 45(2):195–198

  36. Kalantari B, Khachiyan L (1993) On the rate of convergence of deterministic and randomized RAS matrix scaling algorithms. Oper Res Lett 14(5):237–244

    Article  Google Scholar 

  37. Cuturi M (2013) Sinkhorn distances: lightspeed computation of optimal transport. Adv Neural Inf Process Syst 26

  38. Shalev-Shwartz S, Ben-David S (2014) Understanding machine learning: from theory to algorithms. Cambridge University Press, Cambridgeshire

    Book  Google Scholar 

  39. Jones DR, Perttunen CD, Stuckman BE (1993) Lipschitzian optimization without the Lipschitz constant. J Optim Theory Appl 79(1):157–181

    Article  Google Scholar 

  40. Peyré G, Cuturi M et al (2019) Computational optimal transport: with applications to data science. Found Trends ® Mach Learn 11(5–6):355–607

    Article  Google Scholar 

  41. Chambolle A (2004) An algorithm for total variation minimization and applications. J Math Imaging Vision 20(1):89–97

  42. Bartlett PL, Mendelson S (2002) Rademacher and Gaussian complexities: risk bounds and structural results. J Mach Learn Res 3(Nov):463–482

  43. Lehmussola A, Ruusuvuori P, Selinummi J, Huttunen H, Yli-Harja O (2007) Computational framework for simulating fluorescence microscope images with cell populations. IEEE Trans Med Imaging 26(7):1010–1016

    Article  PubMed  Google Scholar 

  44. Kainz P, Urschler M, Schulter S, Wohlhart P, Lepetit V (2015) You should use regression to detect cells. In International Conference on Medical Image Computing and Computer-Assisted Intervention, Springer pp 276–283

  45. Lonsdale J, Thomas J, Salvatore M, Phillips R, Lo E, Shad S, Hasz R, Walters G, Garcia F, Young N et al (2013) The genotype-tissue expression (GTEx) project. Nature Genetics 45(6):580–585

    Article  CAS  Google Scholar 

Download references

Funding

This work is supported by the National Natural Science Foundation of China (81871508, 61773246), the Major Program of Shandong Province Natural Science Foundation (ZR2018ZB0419), the China Postdoctoral Science Foundation (No. 2021M691982), the Taishan Scholar Program of Shandong Province of China (TSHW201502038), and its 2nd round of support.

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Correspondence to Yuanjie Zheng.

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Ding, Y., Zheng, Y., Han, Z. et al. Using optimal transport theory to optimize a deep convolutional neural network microscopic cell counting method. Med Biol Eng Comput 61, 2939–2950 (2023). https://doi.org/10.1007/s11517-023-02862-7

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