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Efficient variational segmentation with local intensity fitting for noisy and inhomogeneous images

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Abstract

This paper introduces a novel local intensity fitting energy model for segmenting noisy and intensity inhomogeneous images. A notable feature of the proposed model is its ability to simultaneously segment the image while obtaining a denoised and inhomogeneity-corrected result. The model integrates a local clustering criterion function with a denoising mechanism, in which the total energy functional comprises three key components: a local fitting energy on the denoised image, which generates a local force to attract the segmentation contour towards the expected object boundary; an edge detector-dependent smoothing term to denoise the source image, and a length regularization ensuring precise wrapping of the segmentation contour around the target object. In addition, we employ an efficient iterative convolution-thresholding method to solve the associated energy minimization problem, ensuring energy decay at each iteration. We demonstrate the efficacy and efficiency of our proposed variational image segmentation model through numerical experiments conducted on both synthetic and real images.

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Data availability

The datasets generated during and analyzed during the current study are available from the corresponding author upon reasonable request.

References

  1. Alberti, G., Bellettini, G.: A non-local anisotropic model for phase transitions: asymptotic behaviour of rescaled energies. Eur. J. Appl. Math. 9, 261–284 (1998)

    Article  MathSciNet  Google Scholar 

  2. Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, 2nd edn. Springer, New York (2006)

    Book  Google Scholar 

  3. Bowden, A., Sirakov, N.M.: Active contour directed by the Poisson gradient vector field and edge tracking. J. Math. Imaging Vis. 63, 665–680 (2021)

    Article  MathSciNet  Google Scholar 

  4. Cai, X., Chan, R., Nikolova, M., Zeng, T.: A three-stage approach for segmenting degraded color images: smoothing, lifting and thresholding (SLaT). J. Sci. Comput. 72, 1313–1332 (2017)

    Article  MathSciNet  Google Scholar 

  5. Cai, X., Chan, R., Zeng, T.: An overview of SaT segmentation methodology and its applications in image processing. In: Chen, K., et al. (eds.) Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging, pp. 1385–1411. Springer Nature Switzerland AG (2023)

    Chapter  Google Scholar 

  6. Cai, Q., Qian, Y., Zhou, S., Li, J., Yang, Y.-H., Wu, F., Zhang, D.: AVLSM: adaptive variational level set model for image segmentation in the presence of severe intensity inhomogeneity and high noise. IEEE Trans. Image Process. 31, 43–57 (2022)

    Article  Google Scholar 

  7. Chan, T.F., Shen, J.: Image Processing and Analysis: Variational, PDE, Wavelet and Stochastic Methods. Society for Industrial and Applied Mathematics, Philadelphia (2005)

    Book  Google Scholar 

  8. Chan, T.F., Vese, L.A.: An active contour model without edges. Lect. Notes Comput. Sci. 1682, 141–151 (1999)

    Article  Google Scholar 

  9. Chan, T.F., Vese, L.A.: Active contours without edges. IEEE Trans. Image Process. 10, 266–277 (2001)

    Article  Google Scholar 

  10. Chen, K., Schönlieb, C.-B., Tai, X.-C., Younes, L.: Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging. Springer Nature Switzerland AG (2023)

    Book  Google Scholar 

  11. Esedoḡlu, S., Otto, F.: Threshold dynamics for networks with arbitrary surface tensions. Commun. Pure Appl. Math. 68, 808–864 (2015)

    Article  MathSciNet  Google Scholar 

  12. Getreuer, P.: Chan-Vese segmentation. Image Process. On Line 2, 214–224 (2012)

    Article  Google Scholar 

  13. Gonzalez, R.C., Woods, R.E.: Digital Image Processing, 4th edn. Pearson Education Limited, New York (2018)

    Google Scholar 

  14. Gui, L., Ma, J., Yang, X.: Variational models and their combinations with deep learning in medical image segmentation: a survey. In: Chen, K., et al. (eds.) Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging, pp. 1001–1022. Springer Nature Switzerland AG (2023)

    Chapter  Google Scholar 

  15. Guo, X., Xue, Y., Wu, C.: Effective two-stage image segmentation: a new non-Lipschitz decomposition approach with convergent algorithm. J. Math. Imaging Vis. 63, 356–379 (2021)

    Article  MathSciNet  Google Scholar 

  16. Hou, Z.: A review on MR image intensity inhomogeneity correction. Int. J. Biomed. Imaging 2006, 1–11 (2006)

    Article  Google Scholar 

  17. Hunter, J.K., Nachtergaele, B.: Applied Analysis. World Scientific Publishing, Singapore (2001)

    Book  Google Scholar 

  18. Lambert, Z., Le Guyader, C.: About the incorporation of topological prescriptions in CNNs for medical image semantic segmentation. J. Math. Imaging Vis. (2024). https://doi.org/10.1007/s10851-024-01172-3

    Article  MathSciNet  Google Scholar 

  19. Li, C., Huang, R., Ding, Z., Gatenby, J.C., Metaxas, D.N., Gore, J.C.: A level set method for image segmentation in the presence of intensity inhomogeneities with application to MRI. IEEE Trans. Image Process. 20, 2007–2016 (2011)

    Article  MathSciNet  Google Scholar 

  20. Li, C., Kao, C.-Y., Gore, J.C., Ding, Z.: Implicit active contours driven by local binary fitting energy. In: IEEE Conference on Computer Vision and Pattern Recognition, pp. 1–7 (2007)

  21. Li, C., Kao, C.-Y., Gore, J.C., Ding, Z.: Minimization of region-scalable fitting energy for image segmentation. IEEE Trans. Image Process. 17, 1940–1949 (2008)

    Article  MathSciNet  Google Scholar 

  22. Li, X., Yang, X., Zeng, T.: A three-stage variational image segmentation framework incorporating intensity inhomogeneity information. SIAM J. Imaging Sci. 13, 1692–1715 (2020)

    Article  MathSciNet  Google Scholar 

  23. Liao, W.-T., Yang, S.-Y., You, C.-S.: An entropy-weighted local intensity clustering-based model for segmenting intensity inhomogeneous images. Multimed. Syst. 30, 49 (2024)

    Article  Google Scholar 

  24. Liu, C., Qiao, Z., Zhang, Q.: Two-phase segmentation for intensity inhomogeneous images by the Allen-Cahn local binary fitting model. SIAM J. Sci. Comput. 44, B177–B196 (2022)

    Article  MathSciNet  Google Scholar 

  25. Liu, C., Qiao, Z., Zhang, Q.: An active contour model with local variance force term and its efficient minimization solver for multiphase image segmentation. SIAM J. Imaging Sci. 16, 144–168 (2023)

    Article  MathSciNet  Google Scholar 

  26. Liu, J., Wang, X., Tai, X.-C.: Deep convolutional neural networks with spatial regularization, volume and star-shape priors for image segmentation. J. Math. Imaging Vis. 64, 625–645 (2022)

    Article  MathSciNet  Google Scholar 

  27. Min, L., Lian, X., Jin, Z., Zheng, M.: A Retinex-based selective segmentation model for inhomogeneous images. J. Math. Imaging Vis. 65, 437–452 (2023)

    Article  MathSciNet  Google Scholar 

  28. Miranda, M., Pallara, D., Paronetto, F., Preunkert, M.: Short-time heat flow and functions of bounded variation in \(\varvec {R}^N\). Annales de la Faculté des Sciences de Toulouse 16, 125–145 (2007)

    MathSciNet  Google Scholar 

  29. Mitiche, A., Ayed, I.B.: Variational and Level Set Methods in Image Segmentation, vol. 5. Springer, Berlin (2010)

    Google Scholar 

  30. Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79, 12–49 (1988)

    Article  MathSciNet  Google Scholar 

  31. Ren, Y., Li, D., Tang, L.: A variational level set model based on additive decomposition for segmenting noisy images with intensity inhomogeneity. Signal Process. 212, 109169 (2023)

    Article  Google Scholar 

  32. Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)

    Article  MathSciNet  Google Scholar 

  33. Song, S., Zheng, Y., He, Y.: A review of methods for bias correction in medical images. Biomed. Eng. Rev. 1, 10 (2017)

    Google Scholar 

  34. Vese, L., Chan, T.: A multiphase level set framework for image segmentation using the Mumford and Shah model. Int. J. Comput. Vis. 50, 271–293 (2002)

    Article  Google Scholar 

  35. Vovk, U., Pernuš, F., Likar, B.: A review of methods for correction of intensity inhomogeneity in MRI. IEEE Trans. Med. Imaging 26, 405–421 (2007)

    Article  Google Scholar 

  36. Wang, D., Wang, X.-P.: The iterative convolution-thresholding method (ICTM) for image segmentation. Pattern Recognit. 130, 108794 (2022)

    Article  Google Scholar 

  37. Wang, D., Li, H., Wei, X., Wang, X.-P.: An efficient iterative thresholding method for image segmentation. J. Comput. Phys. 350, 657–667 (2017)

    Article  MathSciNet  Google Scholar 

  38. Wang, L., Li, C., Sun, Q., Xia, D., Kao, C.-Y.: Active contours driven by local and global intensity fitting energy with application to brain MR image segmentation. Comput. Med. Imaging Graph. 33, 520–531 (2009)

    Article  Google Scholar 

  39. Wang, X.-F., Huang, D.-S., Xu, H.: An efficient local Chan-Vese model for image segmentation. Pattern Recognit. 43, 603–618 (2010)

    Article  Google Scholar 

  40. Wu, T., Gu, X., Shao, J., Zhou, R., Li, Z.: Colour image segmentation based on a convex K-means approach. IET Image Proc. 15, 1596–1606 (2021)

    Article  Google Scholar 

  41. Zhang, K., Song, H., Zhang, L.: Active contours driven by local image fitting energy. Pattern Recognit. 43, 1199–1206 (2010)

    Article  Google Scholar 

  42. Zhang, K., Zhang, L., Lam, K.-M., Zhang, D.: A level set approach to image segmentation with intensity inhomogeneity. IEEE Trans. Cybern. 46, 546–557 (2016)

    Article  Google Scholar 

Download references

Acknowledgements

The authors thank the anonymous referees for their valuable comments and suggestions, which improved the quality of the paper. This work was partially funded by the National Science and Technology Council, Taiwan, under grants NSTC 112-2115-M-005-006-MY2 and NSTC 112-2811-M-008-043, and by the Ministry of Science and Technology, Taiwan, under grant MOST 111-2115-M-008-006-MY3.

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Authors and Affiliations

  1. Department of Applied Mathematics, National Chung Hsing University, South District, Taichung City, 402202, Taiwan

    Po-Wen Hsieh

  2. Department of Mathematics, National Central University, Jhongli District, Taoyuan City, 320317, Taiwan

    Chung-Lin Tseng & Suh-Yuh Yang

Authors
  1. Po-Wen Hsieh
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  2. Chung-Lin Tseng
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  3. Suh-Yuh Yang
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Contributions

PWH: methodology, analysis; CLT: methodology, analysis, coding; SYY: conceptualization, methodology, analysis, writing; All authors reviewed the manuscript.

Corresponding author

Correspondence to Suh-Yuh Yang.

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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Communicated by Bing-kun Bao.

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Hsieh, PW., Tseng, CL. & Yang, SY. Efficient variational segmentation with local intensity fitting for noisy and inhomogeneous images. Multimedia Systems 30, 277 (2024). https://doi.org/10.1007/s00530-024-01487-6

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