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An Application of Quadratic Measure Filters to the Segmentation of Chorio-Retinal OCT Data

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Abstract

As an application of Walther’s convergence theorem for quadratic measure filters, a novel approach for the approximate detection of the jump set of functions \(x \in BV (\Omega , \mathbbm {R}^{})\,\cap \,{{ PC }}^{1}_{} (\Omega , \mathbbm {R}^{})\) has been established. The method has been successfully applied to the segmentation of medical image data which have been generated by optical coherence tomography of the human retina and choroid. There is an excellent correspondence between the automated segmentations and manual delineations, while standard edge detectors fail completely in the recognition of the low-contrasted outer choroid boundary.

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Notes

  1. To the best of the author’s knowledge, the only attempts of numerical application of Walther’s theorem were made in [7] and [53].

  2. See, for example, [4, 46] and [48] .

  3. We mention exemplarily [3] and [5], pp. 65 ff., [41] (variational methods), [12, 13] (level-set methods), [19] and [45] (graph-theoretical methods).

  4. We refer to [22] .

  5. See the literature cited in Sect. 3.

  6. The following summary is based on [18] , pp. 166 ff.

  7. [23] , p. 9, Remark 1.12.

  8. [18] , p. 189, Theorem 1, (i).

  9. Ibid., p. 183, Theorem 1.

  10. [18], p. 210, Lemma 1.

  11. Ibid., p. 211, Theorem 2.

  12. Ibid., p. 210.

  13. Ibid., p. 210, Theorem 1.

  14. Ibid., p. 213, Theorem 3, (ii).

  15. [2] , p. 129 f., Theorem 1.

  16. [47], p. 36, Lemma 6.2.

  17. [52], p. 2, Theorem 1; independently proven in [21], p. 701 f., Theorem 3.1. (\(p=2\), \(G(\nu ) = { const.}\,\)).

  18. [28].

  19. For more details, see [26], pp. 26–31, and [51].

  20. Cf. [31], pp. 3–18, as well as [15], p. 104, Fig. 3–37.

  21. Structure and function of the choroid are summarized in [38].

  22. See [17], p. 17, Fig. 11, and [25].

  23. [44], p. 1208.

  24. [31], pp. 121 ff. and 137.

  25. Cf. [42] and [43].

  26. [20] .

  27. [37].

  28. [36] .

  29. See e.g., [14, 17] and [32].

  30. [49].

  31. [29].

  32. [50], p. 025004-15, Table 3.6.

  33. We mention [1, 27, 54] and [55]. In [30], the statistical segmentation method from [29] has been extended to the OCB detection.

  34. [33] and [34].

  35. For more details, see [40].

  36. The plot has been realized using a HSI color model where every color is represented by the three coordinates hue, saturation and intensity, cf. [10], p. 1197 and [39], pp. 25 ff. Since we need only two coordinates for the visualization of the normalized gradient field, the saturation has been left constant.

  37. [24] , p. 11.

  38. [11].

  39. Cf. [35]. The edge sketches have been created using the following parameters: grad win. 3; min. length 10; nonmaxima suppr.: arc/0.95/0.85; hysteresis thresh. low: arc/0.8/0.8; hysteresis thresh. high: box/0.9/0.9.

  40. See [3] and [5], pp. 166–173.

  41. [6], p. 205 f., Theorem 2.1., for \(p=2\).

  42. Cf. [8].

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Acknowledgements

The author would like to express sincere thanks to Prof. S. Luckhaus (Leipzig) for bringing to his attention Walther’s theses and pointing out the possible application of quadratic measure filters in ophthalmology. Further, he thanks Prof. W. Kiess (Leipzig) for granting the use of OCT data from the LIFE Child study, Dr. F. Rauscher (Leipzig) and P. Scheibe (Leipzig) for introduction into the medical and technical background of OCT imaging and many helpful discussions, A. X. Bestehorn (Münster), Dr. M. Francke (Leipzig), S. Scheibe (Leipzig), A. Schirmer (Berlin) and B. Zimmerling (Leipzig) for carrying out manual segmentations. The work has been supported by the Max-Planck-Gesellschaft granting a stay at the MPI for Mathematics in the Sciences, Leipzig, in 2015 and 2016.

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    Marcus Wagner

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Wagner, M. An Application of Quadratic Measure Filters to the Segmentation of Chorio-Retinal OCT Data. J Math Imaging Vis 60, 216–231 (2018). https://doi.org/10.1007/s10851-017-0752-2

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