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Euler characteristic surfaces
1. | School of Mathematical Sciences, Queen Mary University of London, London, E1 4NS, UK |
2. | School of Informatics, University of Edinburgh, Edinburgh, EH8 9AB, UK |
3. | Centre for Medical Informatics, Usher Institute, University of Edinburgh, Edinburgh, EH16 4UX, UK |
In this paper, we investigate the use of the Euler characteristic for the topological data analysis, particularly over higher dimensional parameter spaces. The Euler characteristic is a classical, well-understood topological invariant that has appeared in numerous applications, primarily in the context of random fields. The goal of this paper, is to present the extension of using the Euler characteristic in higher dimensional parameter spaces. The topological data analysis of higher dimensional parameter spaces using stronger invariants such as homology, has and continues to be the subject of intense research. However, as important theoretical and computational obstacles remain, the use of the Euler characteristic represents an important intermediary step toward multi-parameter topological data analysis. We show the usefulness of the techniques using generated examples as well as a real world dataset of detecting diabetic retinopathy in retinal images.
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show all references
References:
[1] |
R. J. Adler, E. Subag and J. E. Taylor,
Rotation and scale space random fields and the gaussian kinematic formula, Ann. Statist., 40 (2012), 2910-2942.
doi: 10.1214/12-AOS1055. |
[2] |
R. J. Adler and J. E. Taylor, Random Fields and Geometry, Springer, New York, 2007. |
[3] |
M. N. Alam, T. Son, D. Toslak, J. I. Lim and X. Yao, Quantitative artery-vein analysis in optical coherence tomography angiography of diabetic retinopathy, in Ophthalmic Technologies XXIX, vol. 10858, International Society for Optics and Photonics, (2019), 1085802.
doi: 10.1117/12.2510213. |
[4] |
M. Alam, Y. Zhang, J. I. Lim, R. V. Chan, M. Yang and X. Yao,
Quantitative optical coherence tomography angiography features for objective classification and staging of diabetic retinopathy, Retina, 40 (2020), 322-332.
doi: 10.1097/IAE.0000000000002373. |
[5] |
R. Andreeva, A. Fontanella, Y. Giarratano and M. O. Bernabeu, Dr detection using optical coherence tomography angiography (octa): A transfer learning approach with robustness analysis, in International Workshop on Ophthalmic Medical Image Analysis, Springer, (2020), 11–20. |
[6] |
Y. Baryshnikov and R. Ghrist,
Target enumeration via Euler characteristic integrals, SIAM J. Appl. Math., 70 (2009), 825-844.
doi: 10.1137/070687293. |
[7] |
Y. Baryshnikov and R. Ghrist,
Euler integration over definable functions, Proc. Natl. Acad. Sci. USA, 107 (2010), 9525-9530.
doi: 10.1073/pnas.0910927107. |
[8] |
Y. Baryshnikov, R. Ghrist and D. Lipsky, Inversion of Euler integral transforms with applications to sensor data, Inverse Problems, 27 (2011), 124001, 10 pp.
doi: 10.1088/0266-5611/27/12/124001. |
[9] |
O. Bobrowski and P. Skraba, Homological percolation and the Euler characteristic, Phys. Rev. E, 101 (2020), 032304, 16 pp.
doi: 10.1103/physreve.101.032304. |
[10] |
F. Cagliari and C. Landi,
Finiteness of rank invariants of multidimensional persistent homology groups, Appl. Math. Lett., 24 (2011), 516-518.
doi: 10.1016/j.aml.2010.11.004. |
[11] |
G. Carlsson and A. Zomorodian,
The theory of multidimensional persistence, Discrete Comput. Geom., 42 (2009), 71-93.
doi: 10.1007/s00454-009-9176-0. |
[12] |
F. Chazal, L. J. Guibas, S. Y. Oudot and P. Skraba, Persistence-based clustering in Riemannian manifolds, J. ACM, 60 (2013), Art. 41, 38 pp.
doi: 10.1145/2535927. |
[13] |
L. Crawford, A. Monod, A. X. Chen, S. Mukherjee and R. Rabadán,
Predicting clinical outcomes in glioblastoma: an application of topological and functional data analysis, J. Amer. Statist. Assoc., 115 (2020), 1139-1150.
doi: 10.1080/01621459.2019.1671198. |
[14] |
J. Curry, R. Ghrist and M. Robinson, Euler calculus with applications to signals and sensing, in Proc. Sympos. Appl. Math., vol. 70, (2012), 75–146.
doi: 10.1090/psapm/070/589. |
[15] |
J. Curry, S. Mukherjee and K. Turner, How many directions determine a shape and other sufficiency results for two topological transforms, arXiv preprint, arXiv: 1805.09782. |
[16] |
M. Díaz, J. Novo, P. Cutrín, F. Gómez-Ulla, M. G. Penedo and M. Ortega, Automatic segmentation of the foveal avascular zone in ophthalmological OCT-A images, PloS One, 14. |
[17] |
P. J. Diggle, Statistical Analysis of Spatial and Spatio-Temporal Point Patterns, CRC press, 2014.
![]() ![]() |
[18] |
H. Edelsbrunner and J. L. Harer, Computational Topology: An Introduction, American Mathematical Society, 2010.
doi: 10.1090/mbk/069. |
[19] |
H. Edelsbrunner and E. P. Mücke, Three-dimensional alpha shapes, VVS '92: Proceedings of the 1992 workshop on Volume visualization, (1992), 75–82.
doi: 10.1145/147130.147153. |
[20] |
J. M. Ekoé, M. Rewers, R. Williams and P. Zimmet, The Epidemiology of Diabetes Mellitus, John Wiley & Sons, 2008. |
[21] |
B. T. Fasy, S. Micka, D. L. Millman, A. Schenfisch and L. Williams, Challenges in reconstructing shapes from euler characteristic curves, arXiv preprint, arXiv: 1811.11337. |
[22] |
F. J. Freiberg, M. Pfau, J. Wons, M. A. Wirth, M. D. Becker and S. Michels,
Optical coherence tomography angiography of the foveal avascular zone in diabetic retinopathy, Graefe's Archive for Clinical and Experimental Ophthalmology, 254 (2016), 1051-1058.
doi: 10.1007/s00417-015-3148-2. |
[23] |
R. Ghrist, R. Levanger and H. Mai,
Persistent homology and Euler integral transforms, J. Appl. Comput. Topol., 2 (2018), 55-60.
doi: 10.1007/s41468-018-0017-1. |
[24] |
R. Ghrist and M. Robinson, Euler–Bessel and Euler–Fourier transforms, Inverse Problems, 27 (2011), 124006, 12 pp.
doi: 10.1088/0266-5611/27/12/124006. |
[25] |
Y. Giarratano, E. Bianchi, C. Gray, A. Morris, T. MacGillivray, B. Dhillon and M. O. Bernabeu,
Automated segmentation of optical coherence tomography angiography images: Benchmark data and clinically relevant metrics, Translational Vision Science & Technology, 9 (2020), 5-5.
|
[26] |
Y. Giarratano, A. Pavel, J. Lian, R. Andreeva, A. Fontanella, R. Sarkar, L. Reid, S. Forbes, D. Pugh, T. E. Farrah, N. Dhaun, B. Dhillon, T. MacGillivray and M. O. Bernabeu, A framework for the discovery of retinal biomarkers in Optical Coherence Tomography Angiography (OCTA), MICCAI Workshop on Ophthalmic Medical Image Analysis – OMIA 2020. |
[27] |
Y. Giarratano, A. Pavel, J. Lian, R. Andreeva, A. Fontanella, R. Sarkar, L. J. Reid, S. Forbes, D. Pugh, T. E. Farrah et al., A framework for the discovery of retinal biomarkers in Optical Coherence Tomography Angiography (OCTA), in International Workshop on Ophthalmic Medical Image Analysis, Springer, (2020), 155–164. |
[28] |
V. Guillemin and A. Pollack, Differential Topology, vol. 370, American Mathematical Soc., 2010.
doi: 10.1090/chel/370. |
[29] |
H. A. Harrington, N. Otter, H. Schenck and U. Tillmann,
Stratifying multiparameter persistent homology, SIAM J. Appl. Algebra Geom., 3 (2019), 439-471.
doi: 10.1137/18M1224350. |
[30] |
A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002.
![]() ![]() |
[31] |
T. Heiss and H. Wagner,
Streaming algorithm for Euler characteristic curves of multidimensional images, CAIP 2017: Computer Analysis of Images and Patterns, 10424 (2017), 397-409.
doi: 10.1007/978-3-319-64689-3_32. |
[32] |
M. Hofert, I. Kojadinovic, M. Mächler and J. Yan, Elements of Copula Modeling with R, Springer, 2018.
doi: 10.1007/978-3-319-89635-9. |
[33] |
Y. Jia, O. Tan, J. Tokayer, B. Potsaid, Y. Wang, J. J. Liu, M. F. Kraus, H. Subhash, J. G. Fujimoto and J. Hornegger,
Split-spectrum amplitude-decorrelation angiography with optical coherence tomography, Optics Express, 20 (2012), 4710-4725.
doi: 10.1364/OE.20.004710. |
[34] |
M. Kahle,
Topology of random clique complexes, Discrete Math., 309 (2009), 1658-1671.
doi: 10.1016/j.disc.2008.02.037. |
[35] |
M. Kahle,
Topology of random simplicial complexes: A survey, Algebraic Topology: Applications and New Directions, 620 (2014), 201-221.
doi: 10.1090/conm/620/12367. |
[36] |
M. Kashiwara and P. Schapira,
Integral transforms with exponential kernels and laplace transform, J. Amer. Math. Soc., 10 (1997), 939-972.
doi: 10.1090/S0894-0347-97-00245-2. |
[37] |
M. Kashiwara and P. Schapira,
Persistent homology and microlocal sheaf theory, J. Appl. Comput. Topol., 2 (2018), 83-113.
doi: 10.1007/s41468-018-0019-z. |
[38] |
J. Khadamy, K. A. Aghdam and K. G. Falavarjani,
An update on optical coherence tomography angiography in diabetic retinopathy, Journal of Ophthalmic & Vision Research, 13 (2018), 487.
|
[39] |
D. P. Kroese and Z. I. Botev, Spatial process generation, arXiv preprint, arXiv: 1308.0399. |
[40] |
D. Le, M. Alam, B. A. Miao, J. I. Lim and X. Yao,
Fully automated geometric feature analysis in optical coherence tomography angiography for objective classification of diabetic retinopathy, Biomedical Optics Express, 10 (2019), 2493-2503.
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Features | Avg test accuracy |
Euler curve - pixel intensity | |
Euler surface - pixel intensity, Laplacian | |
MNIST | |
Features | Test accuracy |
Euler curve - pixel intensity | |
Euler surface - pixel intensity, top/bottom gradient |
Features | Avg test accuracy |
Euler curve - pixel intensity | |
Euler surface - pixel intensity, Laplacian | |
MNIST | |
Features | Test accuracy |
Euler curve - pixel intensity | |
Euler surface - pixel intensity, top/bottom gradient |
NHS Lothian, Control vs. DR | |||
Baseline | Our approach | VGG16 | |
Overall Acc | 0.84 |
||
Sen (Control) | 0.94 |
0.94 |
|
Spe (Control) | 0.77 |
||
AUC | 0.88 |
0.88 |
|
OCTAGON, Control vs. DR | |||
Baseline | Our approach | VGG16 | |
Overall Acc | 0.87 |
||
Sen (Control) | 0.96 |
1.00 |
|
Spe (Control) | 0.71 |
0.71 |
|
AUC | 0.91 |
0.94 |
NHS Lothian, Control vs. DR | |||
Baseline | Our approach | VGG16 | |
Overall Acc | 0.84 |
||
Sen (Control) | 0.94 |
0.94 |
|
Spe (Control) | 0.77 |
||
AUC | 0.88 |
0.88 |
|
OCTAGON, Control vs. DR | |||
Baseline | Our approach | VGG16 | |
Overall Acc | 0.87 |
||
Sen (Control) | 0.96 |
1.00 |
|
Spe (Control) | 0.71 |
0.71 |
|
AUC | 0.91 |
0.94 |
Controls | NoDR | DR | ||||
Our approach | VGG16 | Our approach | VGG16 | Our approach | VGG16 | |
ACC | 0.68 |
0.78 |
0.81 |
0.76 |
0.77 |
|
SEN | 0.90 |
0.56 |
0.26 |
0.55 |
||
SPE | 0.57 |
0.67 |
0.90 |
0.88 |
0.90 |
|
AUC | 0.80 |
0.90 |
0.70 |
0.67 |
0.86 |
Controls | NoDR | DR | ||||
Our approach | VGG16 | Our approach | VGG16 | Our approach | VGG16 | |
ACC | 0.68 |
0.78 |
0.81 |
0.76 |
0.77 |
|
SEN | 0.90 |
0.56 |
0.26 |
0.55 |
||
SPE | 0.57 |
0.67 |
0.90 |
0.88 |
0.90 |
|
AUC | 0.80 |
0.90 |
0.70 |
0.67 |
0.86 |
EC ( |
EC ( |
|
(VD, EC(VD)) | ||
(FAZ, EC(FAZ)) |
EC ( |
EC ( |
|
(VD, EC(VD)) | ||
(FAZ, EC(FAZ)) |
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