doi: 10.3934/fods.2021027
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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

* Corresponding author

Received  February 2021 Revised  September 2021 Early access November 2021

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.

Citation: Gabriele Beltramo, Primoz Skraba, Rayna Andreeva, Rik Sarkar, Ylenia Giarratano, Miguel O. Bernabeu. Euler characteristic surfaces. Foundations of Data Science, doi: 10.3934/fods.2021027
References:
[1]

R. J. AdlerE. 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.  Google Scholar

[2]

R. J. Adler and J. E. Taylor, Random Fields and Geometry, Springer, New York, 2007.  Google Scholar

[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.  Google Scholar

[4]

M. AlamY. ZhangJ. I. LimR. V. ChanM. 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.  Google Scholar

[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. Google Scholar

[6]

Y. Baryshnikov and R. Ghrist, Target enumeration via Euler characteristic integrals, SIAM J. Appl. Math., 70 (2009), 825-844.  doi: 10.1137/070687293.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

L. CrawfordA. MonodA. X. ChenS. 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[17] P. J. Diggle, Statistical Analysis of Spatial and Spatio-Temporal Point Patterns, CRC press, 2014.   Google Scholar
[18]

H. Edelsbrunner and J. L. Harer, Computational Topology: An Introduction, American Mathematical Society, 2010. doi: 10.1090/mbk/069.  Google Scholar

[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.  Google Scholar

[20]

J. M. Ekoé, M. Rewers, R. Williams and P. Zimmet, The Epidemiology of Diabetes Mellitus, John Wiley & Sons, 2008. Google Scholar

[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. Google Scholar

[22]

F. J. FreibergM. PfauJ. WonsM. A. WirthM. 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.  Google Scholar

[23]

R. GhristR. 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.  Google Scholar

[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.  Google Scholar

[25]

Y. GiarratanoE. BianchiC. GrayA. MorrisT. MacGillivrayB. 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.   Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

V. Guillemin and A. Pollack, Differential Topology, vol. 370, American Mathematical Soc., 2010. doi: 10.1090/chel/370.  Google Scholar

[29]

H. A. HarringtonN. OtterH. Schenck and U. Tillmann, Stratifying multiparameter persistent homology, SIAM J. Appl. Algebra Geom., 3 (2019), 439-471.  doi: 10.1137/18M1224350.  Google Scholar

[30] A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[33]

Y. JiaO. TanJ. TokayerB. PotsaidY. WangJ. J. LiuM. F. KrausH. SubhashJ. 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.  Google Scholar

[34]

M. Kahle, Topology of random clique complexes, Discrete Math., 309 (2009), 1658-1671.  doi: 10.1016/j.disc.2008.02.037.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

J. KhadamyK. A. Aghdam and K. G. Falavarjani, An update on optical coherence tomography angiography in diabetic retinopathy, Journal of Ophthalmic & Vision Research, 13 (2018), 487.   Google Scholar

[39]

D. P. Kroese and Z. I. Botev, Spatial process generation, arXiv preprint, arXiv: 1308.0399. Google Scholar

[40]

D. LeM. AlamB. A. MiaoJ. 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.  doi: 10.1364/BOE.10.002493.  Google Scholar

[41]

Y. LecunL. BottouY. Bengio and P. Haffner, Gradient-based learning applied to document recognition, Proceedings of the IEEE, 86 (1998), 2278-2324.  doi: 10.1109/5.726791.  Google Scholar

[42]

T. Leinster, The Euler characteristic of a category, Doc. Math., 13 (2008), 21-49.   Google Scholar

[43]

M. P. Lesnick, Multidimensional Interleavings and Applications to Topological Inference, Stanford University, 2012.  Google Scholar

[44]

X.-X. LiW. WuH. ZhouJ.-J. DengM.-Y. ZhaoT.-W. QianC. YanX. Xu and S.-Q. Yu, A quantitative comparison of five optical coherence tomography angiography systems in clinical performance, International Journal of Ophthalmology, 11 (2018), 1784.   Google Scholar

[45]

N. Linial and Y. Peled, On the phase transition in random simplicial complexes, Ann. of Math., 184 (2016), 745-773.  doi: 10.4007/annals.2016.184.3.3.  Google Scholar

[46]

A. McCleary and A. Patel, Multiparameter persistence diagrams, arXiv preprint. Google Scholar

[47]

National Health Service, Diabetic retinopathy, Available from: https://www.nhs.uk/conditions/diabetic-retinopathy, 2020, [Accessed on 1 August 2020]. Google Scholar

[48]

R. B. Nelsen, An Introduction to Copulas, Second edition. Springer Series in Statistics. Springer, New York, 2006.  Google Scholar

[49]

T. Ojala, T. Mäenpää, M. Pietikäinen, J. Viertola, J. Kyllönen and S. Huovinen, Outex-new framework for empirical evaluation of texture analysis algorithms, in Proceedings of the 16th International Conference on Pattern Recognition, (2002), 701–706. doi: 10.1109/ICPR.2002.1044854.  Google Scholar

[50]

S. Oudot and E. Solomon, Inverse problems in topological persistence, in Topological Data Analysis, Springer, (2020), 405–433. Google Scholar

[51]

F. PedregosaG. VaroquauxA. GramfortV. MichelB. ThirionO. GriselM. BlondelP. PrettenhoferR. Weiss and V. Dubourg, Scikit-learn: Machine learning in Python, J. Mach. Learn. Res., 12 (2011), 2825-2830.   Google Scholar

[52]

W. D. Penny, K. J. Friston, J. T. Ashburner, S. J. Kiebel and T. E. Nichols, Statistical Parametric Mapping: The Analysis of Functional Brain Images, Elsevier, 2011. Google Scholar

[53]

P. PranavR. Van de WeygaertG. VegterB. J. JonesR. J. AdlerJ. FeldbruggeC. ParkT. Buchert and M. Kerber, Topology and geometry of Gaussian random fields I: On Betti numbers, Euler characteristic, and Minkowski functionals, Monthly Notices of the Royal Astronomical Society, 485 (2019), 4167-4208.  doi: 10.1093/mnras/stz541.  Google Scholar

[54]

E. Richardson and M. Werman, Efficient classification using Euler characteristic, Pattern Recognition Letters, 49 (2014), 99-106.  doi: 10.1016/j.patrec.2014.07.001.  Google Scholar

[55]

E. Richardson and M. Werman, Efficient classification using the Euler characteristic, Pattern Recognition Letters, 49 (2014), 99-106.  doi: 10.1016/j.patrec.2014.07.001.  Google Scholar

[56]

H. S. SandhuN. EladawiM. ElmogyR. KeyntonO. HelmyS. Schaal and A. El-Baz, Automated diabetic retinopathy detection using optical coherence tomography angiography: A pilot study, British Journal of Ophthalmology, 102 (2018), 1564-1569.   Google Scholar

[57]

M. SasongkoT. WongT. NguyenC. CheungJ. Shaw and J. Wang, Retinal vascular tortuosity in persons with diabetes and diabetic retinopathy, Diabetologia, 54 (2011), 2409-2416.  doi: 10.1007/s00125-011-2200-y.  Google Scholar

[58]

P. Schapira, Tomography of constructible functions, in International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes, Springer, (1995), 427–435. doi: 10.1007/3-540-60114-7_33.  Google Scholar

[59]

M. ScolamieroW. ChachólskiA. LundmanR. Ramanujam and S. Öberg, Multidimensional persistence and noise, Found. Comput. Math., 17 (2017), 1367-1406.  doi: 10.1007/s10208-016-9323-y.  Google Scholar

[60]

S. Stolte and R. Fang, A survey on medical image analysis in diabetic retinopathy, Medical Image Analysis, 64 (2020), 101742.  doi: 10.1016/j.media.2020.101742.  Google Scholar

[61]

N. TakaseM. NozakiA. KatoH. OzekiM. Yoshida and Y. Ogura, Enlargement of foveal avascular zone in diabetic eyes evaluated by en face optical coherence tomography angiography, Retina, 35 (2015), 2377-2383.  doi: 10.1097/IAE.0000000000000849.  Google Scholar

[62]

K. Y. TeyK. TeoA. C. TanK. DevarajanB. TanJ. TanL. Schmetterer and M. Ang, Optical coherence tomography angiography in diabetic retinopathy: A review of current applications, Eye and Vision, 6 (2019), 1-10.  doi: 10.1186/s40662-019-0160-3.  Google Scholar

[63]

I. A. ThompsonA. K. Durrani and S. Patel, Optical coherence tomography angiography characteristics in diabetic patients without clinical diabetic retinopathy, Eye, 33 (2019), 648-652.  doi: 10.1038/s41433-018-0286-x.  Google Scholar

[64]

K. TurnerS. Mukherjee and D. M. Boyer, Persistent homology transform for modeling shapes and surfaces, Inf. Inference, 3 (2014), 310-344.  doi: 10.1093/imaiai/iau011.  Google Scholar

[65]

R. Van De Weygaert, G. Vegter, H. Edelsbrunner, B. J. Jones, P. Pranav, C. Park, W. A. Hellwing, B. Eldering, N. Kruithof, E. P. Bos et al., Alpha, Betti and the Megaparsec Universe: On the topology of the cosmic web, in Transactions on Computational Science XIV, Springer, (2011), 60–101. doi: 10.1007/978-3-642-25249-5_3.  Google Scholar

[66] L. van den Dries, Tame Topology and O-Minimal Structures, vol. 248, Cambridge university press, 1998.  doi: 10.1017/CBO9780511525919.  Google Scholar
[67]

K. J. Worsley, Detecting activation in fMRI data, Stat. Methods Med. Res., 12 (2003), 401-418.  doi: 10.1191/0962280203sm340ra.  Google Scholar

[68]

K. J. Worsley, J. E. Taylor, F. Tomaiuolo and J. Lerch, Unified univariate and multivariate random field theory, Neuroimage, 23 (2004), S189–S195. doi: 10.1016/j.neuroimage.2004.07.026.  Google Scholar

[69]

X. YaoM. N. AlamD. Le and D. Toslak, Quantitative optical coherence tomography angiography: A review, Experimental Biology and Medicine, 245 (2020), 301-312.  doi: 10.1177/1535370219899893.  Google Scholar

show all references

References:
[1]

R. J. AdlerE. 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.  Google Scholar

[2]

R. J. Adler and J. E. Taylor, Random Fields and Geometry, Springer, New York, 2007.  Google Scholar

[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.  Google Scholar

[4]

M. AlamY. ZhangJ. I. LimR. V. ChanM. 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.  Google Scholar

[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. Google Scholar

[6]

Y. Baryshnikov and R. Ghrist, Target enumeration via Euler characteristic integrals, SIAM J. Appl. Math., 70 (2009), 825-844.  doi: 10.1137/070687293.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

L. CrawfordA. MonodA. X. ChenS. 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[17] P. J. Diggle, Statistical Analysis of Spatial and Spatio-Temporal Point Patterns, CRC press, 2014.   Google Scholar
[18]

H. Edelsbrunner and J. L. Harer, Computational Topology: An Introduction, American Mathematical Society, 2010. doi: 10.1090/mbk/069.  Google Scholar

[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.  Google Scholar

[20]

J. M. Ekoé, M. Rewers, R. Williams and P. Zimmet, The Epidemiology of Diabetes Mellitus, John Wiley & Sons, 2008. Google Scholar

[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. Google Scholar

[22]

F. J. FreibergM. PfauJ. WonsM. A. WirthM. 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.  Google Scholar

[23]

R. GhristR. 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.  Google Scholar

[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.  Google Scholar

[25]

Y. GiarratanoE. BianchiC. GrayA. MorrisT. MacGillivrayB. 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.   Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

V. Guillemin and A. Pollack, Differential Topology, vol. 370, American Mathematical Soc., 2010. doi: 10.1090/chel/370.  Google Scholar

[29]

H. A. HarringtonN. OtterH. Schenck and U. Tillmann, Stratifying multiparameter persistent homology, SIAM J. Appl. Algebra Geom., 3 (2019), 439-471.  doi: 10.1137/18M1224350.  Google Scholar

[30] A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[33]

Y. JiaO. TanJ. TokayerB. PotsaidY. WangJ. J. LiuM. F. KrausH. SubhashJ. 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.  Google Scholar

[34]

M. Kahle, Topology of random clique complexes, Discrete Math., 309 (2009), 1658-1671.  doi: 10.1016/j.disc.2008.02.037.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

J. KhadamyK. A. Aghdam and K. G. Falavarjani, An update on optical coherence tomography angiography in diabetic retinopathy, Journal of Ophthalmic & Vision Research, 13 (2018), 487.   Google Scholar

[39]

D. P. Kroese and Z. I. Botev, Spatial process generation, arXiv preprint, arXiv: 1308.0399. Google Scholar

[40]

D. LeM. AlamB. A. MiaoJ. 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.  doi: 10.1364/BOE.10.002493.  Google Scholar

[41]

Y. LecunL. BottouY. Bengio and P. Haffner, Gradient-based learning applied to document recognition, Proceedings of the IEEE, 86 (1998), 2278-2324.  doi: 10.1109/5.726791.  Google Scholar

[42]

T. Leinster, The Euler characteristic of a category, Doc. Math., 13 (2008), 21-49.   Google Scholar

[43]

M. P. Lesnick, Multidimensional Interleavings and Applications to Topological Inference, Stanford University, 2012.  Google Scholar

[44]

X.-X. LiW. WuH. ZhouJ.-J. DengM.-Y. ZhaoT.-W. QianC. YanX. Xu and S.-Q. Yu, A quantitative comparison of five optical coherence tomography angiography systems in clinical performance, International Journal of Ophthalmology, 11 (2018), 1784.   Google Scholar

[45]

N. Linial and Y. Peled, On the phase transition in random simplicial complexes, Ann. of Math., 184 (2016), 745-773.  doi: 10.4007/annals.2016.184.3.3.  Google Scholar

[46]

A. McCleary and A. Patel, Multiparameter persistence diagrams, arXiv preprint. Google Scholar

[47]

National Health Service, Diabetic retinopathy, Available from: https://www.nhs.uk/conditions/diabetic-retinopathy, 2020, [Accessed on 1 August 2020]. Google Scholar

[48]

R. B. Nelsen, An Introduction to Copulas, Second edition. Springer Series in Statistics. Springer, New York, 2006.  Google Scholar

[49]

T. Ojala, T. Mäenpää, M. Pietikäinen, J. Viertola, J. Kyllönen and S. Huovinen, Outex-new framework for empirical evaluation of texture analysis algorithms, in Proceedings of the 16th International Conference on Pattern Recognition, (2002), 701–706. doi: 10.1109/ICPR.2002.1044854.  Google Scholar

[50]

S. Oudot and E. Solomon, Inverse problems in topological persistence, in Topological Data Analysis, Springer, (2020), 405–433. Google Scholar

[51]

F. PedregosaG. VaroquauxA. GramfortV. MichelB. ThirionO. GriselM. BlondelP. PrettenhoferR. Weiss and V. Dubourg, Scikit-learn: Machine learning in Python, J. Mach. Learn. Res., 12 (2011), 2825-2830.   Google Scholar

[52]

W. D. Penny, K. J. Friston, J. T. Ashburner, S. J. Kiebel and T. E. Nichols, Statistical Parametric Mapping: The Analysis of Functional Brain Images, Elsevier, 2011. Google Scholar

[53]

P. PranavR. Van de WeygaertG. VegterB. J. JonesR. J. AdlerJ. FeldbruggeC. ParkT. Buchert and M. Kerber, Topology and geometry of Gaussian random fields I: On Betti numbers, Euler characteristic, and Minkowski functionals, Monthly Notices of the Royal Astronomical Society, 485 (2019), 4167-4208.  doi: 10.1093/mnras/stz541.  Google Scholar

[54]

E. Richardson and M. Werman, Efficient classification using Euler characteristic, Pattern Recognition Letters, 49 (2014), 99-106.  doi: 10.1016/j.patrec.2014.07.001.  Google Scholar

[55]

E. Richardson and M. Werman, Efficient classification using the Euler characteristic, Pattern Recognition Letters, 49 (2014), 99-106.  doi: 10.1016/j.patrec.2014.07.001.  Google Scholar

[56]

H. S. SandhuN. EladawiM. ElmogyR. KeyntonO. HelmyS. Schaal and A. El-Baz, Automated diabetic retinopathy detection using optical coherence tomography angiography: A pilot study, British Journal of Ophthalmology, 102 (2018), 1564-1569.   Google Scholar

[57]

M. SasongkoT. WongT. NguyenC. CheungJ. Shaw and J. Wang, Retinal vascular tortuosity in persons with diabetes and diabetic retinopathy, Diabetologia, 54 (2011), 2409-2416.  doi: 10.1007/s00125-011-2200-y.  Google Scholar

[58]

P. Schapira, Tomography of constructible functions, in International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes, Springer, (1995), 427–435. doi: 10.1007/3-540-60114-7_33.  Google Scholar

[59]

M. ScolamieroW. ChachólskiA. LundmanR. Ramanujam and S. Öberg, Multidimensional persistence and noise, Found. Comput. Math., 17 (2017), 1367-1406.  doi: 10.1007/s10208-016-9323-y.  Google Scholar

[60]

S. Stolte and R. Fang, A survey on medical image analysis in diabetic retinopathy, Medical Image Analysis, 64 (2020), 101742.  doi: 10.1016/j.media.2020.101742.  Google Scholar

[61]

N. TakaseM. NozakiA. KatoH. OzekiM. Yoshida and Y. Ogura, Enlargement of foveal avascular zone in diabetic eyes evaluated by en face optical coherence tomography angiography, Retina, 35 (2015), 2377-2383.  doi: 10.1097/IAE.0000000000000849.  Google Scholar

[62]

K. Y. TeyK. TeoA. C. TanK. DevarajanB. TanJ. TanL. Schmetterer and M. Ang, Optical coherence tomography angiography in diabetic retinopathy: A review of current applications, Eye and Vision, 6 (2019), 1-10.  doi: 10.1186/s40662-019-0160-3.  Google Scholar

[63]

I. A. ThompsonA. K. Durrani and S. Patel, Optical coherence tomography angiography characteristics in diabetic patients without clinical diabetic retinopathy, Eye, 33 (2019), 648-652.  doi: 10.1038/s41433-018-0286-x.  Google Scholar

[64]

K. TurnerS. Mukherjee and D. M. Boyer, Persistent homology transform for modeling shapes and surfaces, Inf. Inference, 3 (2014), 310-344.  doi: 10.1093/imaiai/iau011.  Google Scholar

[65]

R. Van De Weygaert, G. Vegter, H. Edelsbrunner, B. J. Jones, P. Pranav, C. Park, W. A. Hellwing, B. Eldering, N. Kruithof, E. P. Bos et al., Alpha, Betti and the Megaparsec Universe: On the topology of the cosmic web, in Transactions on Computational Science XIV, Springer, (2011), 60–101. doi: 10.1007/978-3-642-25249-5_3.  Google Scholar

[66] L. van den Dries, Tame Topology and O-Minimal Structures, vol. 248, Cambridge university press, 1998.  doi: 10.1017/CBO9780511525919.  Google Scholar
[67]

K. J. Worsley, Detecting activation in fMRI data, Stat. Methods Med. Res., 12 (2003), 401-418.  doi: 10.1191/0962280203sm340ra.  Google Scholar

[68]

K. J. Worsley, J. E. Taylor, F. Tomaiuolo and J. Lerch, Unified univariate and multivariate random field theory, Neuroimage, 23 (2004), S189–S195. doi: 10.1016/j.neuroimage.2004.07.026.  Google Scholar

[69]

X. YaoM. N. AlamD. Le and D. Toslak, Quantitative optical coherence tomography angiography: A review, Experimental Biology and Medicine, 245 (2020), 301-312.  doi: 10.1177/1535370219899893.  Google Scholar

Figure 1.  From left to right: $ 3\times 3 $ gray-scale image, full cubical complex $ Q $ of the image in a), finite point set in $ \mathbb{R}^2 $, and Delaunay complex $ D $ of the points in (c)
Figure 1, for two given set of values $ \{ a_s \}_{s = 1}^m $. In both (a) and (b) the filtration parameters $ a_s $ are displayed above each subcomplex. The last subcomplexes in the sequence are the full cubical complex $ Q $ and the Delaunay complex $ D $">Figure 2.  Filtrations of the example data in Figure 1, for two given set of values $ \{ a_s \}_{s = 1}^m $. In both (a) and (b) the filtration parameters $ a_s $ are displayed above each subcomplex. The last subcomplexes in the sequence are the full cubical complex $ Q $ and the Delaunay complex $ D $
Figure 1a, where sublevel sets are taken at all values between $ 0 $ and $ 255 $. The plot in (b) is the Euler characteristic curve of the points in Figure 1c, where the sublevel sets are taken at values $ a_s $ such that there is a one simplex difference between $ Q_{a_{i-1}} $ and $ Q_{a_s} $ for each $ i \in [1, m] $">Figure 3.  The plot in (a) is the Euler characteristic curve of the image in Figure 1a, where sublevel sets are taken at all values between $ 0 $ and $ 255 $. The plot in (b) is the Euler characteristic curve of the points in Figure 1c, where the sublevel sets are taken at values $ a_s $ such that there is a one simplex difference between $ Q_{a_{i-1}} $ and $ Q_{a_s} $ for each $ i \in [1, m] $
Figure 4.  Contour plot of $ S_{(h_{M_1}, h_{M_2})} $ of a pair of random images $ M_1 $, $ M_2 $ with correlation coefficient equal to $ 0.1 $ in (a) and equal to $ 0.8 $ in (b). The difference between these two Euler surfaces is the Euler terrain in (c)
$ \mathbb{S}^1\times \mathbb{R} $. The 2-dimensional Euler surface we consider is a direction and the function sublevel sets, so the Euler surface can be thought of as representing the slice shown">Figure 5.  The realtionship between the ECT and the ECS for functions on $ \mathbb{R}^2 $. For every direction $ \alpha $, the ECT computes the Euler characteristic curve based on the filtration arising from the height function in that direction. In the standard setup, it would be a direction on the sphere $ \mathbb{S}^2 $, but it could equivalently be all "directions'' in $ \mathbb{S}^1\times \mathbb{R} $. The 2-dimensional Euler surface we consider is a direction and the function sublevel sets, so the Euler surface can be thought of as representing the slice shown
Figure 6.  Euler characteristic changes produced by adding an elementary cube of maximal dimension in a two-dimensional cubical complex $ Q_{s-1, t} $. In (a) the change is equal to $ -3 $, while in (b) it is $ +1 $
Figure 7.  The images in (a) are eight of the twenty-four patterns in the $ \texttt{OUTEX_TC_00000} $ test suite. The images in (b) are a random selection of the $ 70,000 $ handwritten digits from the MNIST database
Figure 8.  Points sampled from two Clayton copula distributions. In (a) $ \theta = 1 $, while in (b) $ \theta = 5 $
Figure 8. Each point represents the function values of a voxel (in a 3D image) generated from Clayton copula distributions to have uniform marginal distributions but different joint distributions. In (a), the surface was obtained with $ \theta = 1 $, and in (b) with $ \theta = 5 $. The absolute value of the difference of the surfaces in (a) and (b) is in (c)">Figure 9.  The average Euler characteristic surfaces of random images generated with random points as in Figure 8. Each point represents the function values of a voxel (in a 3D image) generated from Clayton copula distributions to have uniform marginal distributions but different joint distributions. In (a), the surface was obtained with $ \theta = 1 $, and in (b) with $ \theta = 5 $. The absolute value of the difference of the surfaces in (a) and (b) is in (c)
Figure 10.  Point processes, corresponding to a Poisson point process in (a) and Hawkes cluster process in (b)
Figure 10. In (a) and (b) the average surfaces of points obtained from a Poisson and Hawkes cluster process respectively. In (c) the Euler terrain representing their difference. In (d) the black area represent regions of the parameter space where the two average surfaces in (a) and (b) are significantly different">Figure 11.  Average Euler characteristic surfaces obtained from the point processes in Figure 10. In (a) and (b) the average surfaces of points obtained from a Poisson and Hawkes cluster process respectively. In (c) the Euler terrain representing their difference. In (d) the black area represent regions of the parameter space where the two average surfaces in (a) and (b) are significantly different
Figure 11(c) for a cluster process (a, b) and the Poisson process (c, d)">Figure 12.  Examples of the complex built at the radius and density corresponding to Region A in Figure 11(c) for a cluster process (a, b) and the Poisson process (c, d)
Figure 13.  OCTA scans from Control, DR and NoDR patients. Changes to the microvasculature are apparent with disease progression. For example, the vessel density reduces and the foveal avascular zone (FAZ), which is the black regions in the middles of the image, is enlarged and distorted with less circular shape
Figure 14.  Example of the ECS with $ M_2 $ being the complement image
Figure 15.  Example of the ECS with $ M_2 $ being the radial gradient image
Figure 16.  Examples of regions of interest in the terrains of OCTAGON
Figure 17.  Examples of level-sets of images of the control (a, b) and DR (c, d) corresponding to Region B
Figure 18.  Examples of sublevel sets with the radial mask shown in yellow with the control shown in (a, b) and DR (c, d) corresponding to Region A
Table 1.  Classification results obtained with Euler characteristic based feature vectors and logistic regression
$ \texttt{Outex_TC_00000} $
Features Avg test accuracy
Euler curve - pixel intensity $ 91.08\pm 1.57 $ %
Euler surface - pixel intensity, Laplacian $ \mathbf{96.29}\pm 1.24 $ %
MNIST
Features Test accuracy
Euler curve - pixel intensity $ 33.63 $ %
Euler surface - pixel intensity, top/bottom gradient $ \mathbf{71.79} $ %
$ \texttt{Outex_TC_00000} $
Features Avg test accuracy
Euler curve - pixel intensity $ 91.08\pm 1.57 $ %
Euler surface - pixel intensity, Laplacian $ \mathbf{96.29}\pm 1.24 $ %
MNIST
Features Test accuracy
Euler curve - pixel intensity $ 33.63 $ %
Euler surface - pixel intensity, top/bottom gradient $ \mathbf{71.79} $ %
Table 2.  Tables of classification performances in the Control vs. DR study
NHS Lothian, Control vs. DR
Baseline Our approach VGG16
Overall Acc $0.72 \pm 0.03$ $0.81 \pm 0.04$ 0.84 $\pm$ 0.07
Sen (Control) 0.94 $\pm$ 0.05 0.94 $\pm$ 0.05 $0.88 \pm 0.07$
Spe (Control) $0.30 \pm 0.06$ $0.60 \pm 0.06$ 0.77 $\pm$ 0.09
AUC $0.75 \pm 0.06$ 0.88 $\pm$ 0.03 0.88 $\pm$ 0.12
 
OCTAGON, Control vs. DR
Baseline Our approach VGG16
Overall Acc $0.82 \pm 0.04$ 0.87 $\pm$ 0.04 $0.84 \pm 0.07$
Sen (Control) $0.87 \pm 0.04$ 0.96 $\pm$ 0.04 1.00 $\pm$ 0.00
Spe (Control) 0.71 $\pm$ 0.08 0.71 $\pm$ 0.08 $0.53 \pm 0.20$
AUC $0.87 \pm 0.04$ 0.91 $\pm$ 0.03 0.94 $\pm$ 0.06
NHS Lothian, Control vs. DR
Baseline Our approach VGG16
Overall Acc $0.72 \pm 0.03$ $0.81 \pm 0.04$ 0.84 $\pm$ 0.07
Sen (Control) 0.94 $\pm$ 0.05 0.94 $\pm$ 0.05 $0.88 \pm 0.07$
Spe (Control) $0.30 \pm 0.06$ $0.60 \pm 0.06$ 0.77 $\pm$ 0.09
AUC $0.75 \pm 0.06$ 0.88 $\pm$ 0.03 0.88 $\pm$ 0.12
 
OCTAGON, Control vs. DR
Baseline Our approach VGG16
Overall Acc $0.82 \pm 0.04$ 0.87 $\pm$ 0.04 $0.84 \pm 0.07$
Sen (Control) $0.87 \pm 0.04$ 0.96 $\pm$ 0.04 1.00 $\pm$ 0.00
Spe (Control) 0.71 $\pm$ 0.08 0.71 $\pm$ 0.08 $0.53 \pm 0.20$
AUC $0.87 \pm 0.04$ 0.91 $\pm$ 0.03 0.94 $\pm$ 0.06
Table 3.  Table of classification performances with transfer learning
Controls NoDR DR
Our approach VGG16 Our approach VGG16 Our approach VGG16
ACC 0.68 $ \pm $ 0.05 0.78 $ \pm $ 0.05 0.81 $ \pm $ 0.04 $ 0.72 \pm 0.04 $ 0.76 $ \pm $ 0.02 0.77 $ \pm $ 0.04
SEN $ 0.79 \pm 0.05 $ 0.90 $ \pm $ 0.05 0.56 $ \pm $ 0.06 $ 0.20 \pm 0.13 $ 0.26 $ \pm $ 0.14 0.55 $ \pm $ 0.11
SPE 0.57 $ \pm $ 0.09 0.67 $ \pm $ 0.11 0.90 $ \pm $ 0.04 0.88$ \pm $ 0.05 0.90 $ \pm $ 0.02 $ 0.86 \pm 0.05 $
AUC 0.80 $ \pm $ 0.04 0.90 $ \pm $ 0.15 0.70 $ \pm $ 0.04 0.67$ \pm $ 0.28 0.86 $ \pm $ 0.06 $ 0.75 \pm 0.22 $
Controls NoDR DR
Our approach VGG16 Our approach VGG16 Our approach VGG16
ACC 0.68 $ \pm $ 0.05 0.78 $ \pm $ 0.05 0.81 $ \pm $ 0.04 $ 0.72 \pm 0.04 $ 0.76 $ \pm $ 0.02 0.77 $ \pm $ 0.04
SEN $ 0.79 \pm 0.05 $ 0.90 $ \pm $ 0.05 0.56 $ \pm $ 0.06 $ 0.20 \pm 0.13 $ 0.26 $ \pm $ 0.14 0.55 $ \pm $ 0.11
SPE 0.57 $ \pm $ 0.09 0.67 $ \pm $ 0.11 0.90 $ \pm $ 0.04 0.88$ \pm $ 0.05 0.90 $ \pm $ 0.02 $ 0.86 \pm 0.05 $
AUC 0.80 $ \pm $ 0.04 0.90 $ \pm $ 0.15 0.70 $ \pm $ 0.04 0.67$ \pm $ 0.28 0.86 $ \pm $ 0.06 $ 0.75 \pm 0.22 $
Table 4.  Correlation results for NHS Lothian and OCTAGON
EC ($ p $-value), NHS Lothian EC ($ p $-value), OCTAGON
(VD, EC(VD)) $ 0.86 (1.65 \times 10^{-12}) $ $ 0.20 (0.20) $
(FAZ, EC(FAZ)) $ 0.55 (2.52 \times 10^{-4}) $ $ 0.57 (7.13 \times 10^{-5}) $
EC ($ p $-value), NHS Lothian EC ($ p $-value), OCTAGON
(VD, EC(VD)) $ 0.86 (1.65 \times 10^{-12}) $ $ 0.20 (0.20) $
(FAZ, EC(FAZ)) $ 0.55 (2.52 \times 10^{-4}) $ $ 0.57 (7.13 \times 10^{-5}) $
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