SpringerLink
Log in

Analyzing Eye Paths Using Fractals

  • Chapter
  • First Online:
The Fractal Geometry of the Brain

Part of the book series: Advances in Neurobiology ((NEUROBIOL,volume 36))

  • 383 Accesses

Abstract

Visual patterns reflect the anatomical and cognitive background underlying process governing how we perceive information, influenced by stimulus characteristics and our own visual perception. These patterns are both spatially complex and display self-similarity seen in fractal geometry at different scales, making them challenging to measure using the traditional topological dimensions used in Euclidean geometry.

However, methods for measuring eye gaze patterns using fractals have shown success in quantifying geometric complexity, matchability, and implementation into machine learning methods. This success is due to the inherent capabilities that fractals possess when reducing dimensionality using Hilbert curves, measuring temporal complexity using the Higuchi fractal dimension (HFD), and determining geometric complexity using the Minkowski–Bouligand dimension.

Understanding the many applications of fractals when measuring and analyzing eye gaze patterns can extend the current growing body of knowledge by identifying markers tied to neurological pathology. Additionally, in future work, fractals can facilitate defining imaging modalities in eye tracking diagnostics by exploiting their capability to acquire multiscale information, including complementary functions, structures, and dynamics.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
GBP 19.95
Price includes VAT (United Kingdom)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
GBP 159.50
Price includes VAT (United Kingdom)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
GBP 199.99
Price includes VAT (United Kingdom)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Alamudun F, Yoon H-J, Hudson K, Morin-Ducote G, Hammond T, Tourassi G. Fractal analysis of visual search activity for mass detection during mammographic screening. Med Phys. 2017;44(January) https://doi.org/10.1002/mp.12100.

  2. Ashraf H, Sodergren MH, Merali N, Mylonas G, Singh H, Darzi A. Eye-tracking Technology in medical education: a systematic review. Med Teach. 2018;40(1):62–9. https://doi.org/10.1080/0142159X.2017.1391373.

    Article  PubMed  Google Scholar 

  3. Baker DB. The Oxford handbook of the history of psychology: global perspectives. Oxford University Press; 2012. https://doi.org/10.1093/oxfordhb/9780195366556.001.0001.

    Book  Google Scholar 

  4. Bell C, Davy H. XV. On the motions of the eye, in illustration of the uses of the muscles and nerves of the orbit. Philos Trans R Soc Lond. 1823;113:166–86. https://doi.org/10.1098/rstl.1823.0017.

    Article  Google Scholar 

  5. Bhattacharya N, Rakshit S, Gwizdka J, Kogut P. Relevance prediction from eye-movements using semi-interpretable convolutional neural networks. CoRR abs/2001.05152; 2020. https://arxiv.org/abs/2001.05152

    Book  Google Scholar 

  6. Brandt SA, Stark LW. Spontaneous eye movements during visual imagery reflect the content of the visual scene. J Cogn Neurosci. 1997;9(1):27–38. https://doi.org/10.1162/jocn.1997.9.1.27.

    Article  CAS  PubMed  Google Scholar 

  7. Brown C. A lecture on the relation between the movements of the eyes and the movements of the head. Lancet. 1895;145(3743):1293–8. https://doi.org/10.1016/S0140-6736(01)94423-X.

    Article  Google Scholar 

  8. Bruce NDB, Tsotsos JK. A statistical basis for visual field anisotropies. Neurocomputing. 2006; https://doi.org/10.1016/j.neucom.2005.12.096.

  9. Burch M, Kumar A, Mueller K, Kervezee T, Nuijten W, Oostenbach R, Peeters L, Smit G. Finding the outliers in scanpath data. In: Eye tracking research and applications symposium (ETRA); 2019. https://doi.org/10.1145/3317958.3318225.

    Chapter  Google Scholar 

  10. Burn RP, Mandelbrot BB. The fractal geometry of nature. Math Gaz. 1984;68(443):71. https://doi.org/10.2307/3615422.

    Article  Google Scholar 

  11. Chang R-C, Tsai M-J. Visual behavior patterns of successful decision makers in crime scene photo investigation: an eye tracking analysis. J Forensic Sci. 2022;67(3):1072–83. https://doi.org/10.1111/1556-4029.14970.

    Article  PubMed  Google Scholar 

  12. Cover TM, Hart PE. Nearest neighbor pattern classification. IEEE Trans Inf Theory. 1967; https://doi.org/10.1109/TIT.1967.1053964.

  13. Cristino F, MathĂ´t S, Theeuwes J, Gilchrist ID. ScanMatch: a novel method for comparing fixation sequences. Behav Res Methods. 2010; https://doi.org/10.3758/BRM.42.3.692.

  14. Crowe EM, Gilchrist ID, Kent C. New approaches to the analysis of eye movement behaviour across expertise while viewing brain MRIs. Cogn Res Princ Implic. 2018; https://doi.org/10.1186/s41235-018-0097-4.

  15. Crutcher MD, Calhoun-Haney R, Manzanares CM, Lah JJ, Levey AI, Zola SM. Eye tracking during a visual paired comparison task as a predictor of early dementia. Am J Alzheimers Dis Other Dement. 2009;24(3):258–66. https://doi.org/10.1177/1533317509332093.

    Article  Google Scholar 

  16. Davies A, Vigo M, Harper S, Jay C. The visualisation of eye-tracking scanpaths: what can they tell us about how clinicians view electrocardiograms? In: Proceedings of the 2nd workshop on eye tracking and visualization, ETVIS 2016; 2017. https://doi.org/10.1109/ETVIS.2016.7851172.

    Chapter  Google Scholar 

  17. Dewhurst R, Nyström M, Jarodzka H, Foulsham T, Johansson R, Holmqvist K. It depends on how you look at it: scanpath comparison in multiple dimensions with MultiMatch, a vector-based approach. Behav Res Methods. 2012; https://doi.org/10.3758/s13428-012-0212-2.

  18. Dewhurst R, Foulsham T, Jarodzka H, Johansson R, Holmqvist K, Nyström M. How task demands influence scanpath similarity in a sequential number-search task. Vis Res. 2018; https://doi.org/10.1016/j.visres.2018.05.006.

  19. Di Ieva A. Fractal geometry of the brain/Antonio Di Ieva, editor. Edited by SpringerLINK ebooks – Biomedical and Life Sciences (2016) and Antonio Di Ieva. Springer Series in Computational Neuroscience. 2016.

    Google Scholar 

  20. Engbert R, Mergenthaler K, Sinn P, Pikovsky A. An integrated model of fixational eye movements and microsaccades. Proc Natl Acad Sci U S A. 2011;108(39) https://doi.org/10.1073/pnas.1102730108.

  21. Essen V, David C, Newsome WT, Maunsell JHR. The visual field representation in striate cortex of the macaque monkey: asymmetries, anisotropies, and individual variability. Vis Res. 1984; https://doi.org/10.1016/0042-6989(84)90041-5.

  22. Eviatar A, Goodhill V. XXVII vector electronystagmography: a method of localizing nystagmic impulses within the vestibular system: a preliminary report. Ann Otol Rhinol Laryngol. 1968;77(2):264–74. https://doi.org/10.1177/000348946807700211.

    Article  CAS  PubMed  Google Scholar 

  23. Ieva D, Antonio FJ, Esteban FG, Klonowski W, Martín-Landrove M. Fractals in the neurosciences, part II: clinical applications and future perspectives. Neuroscientist. 2015;21(1):30–43. https://doi.org/10.1177/1073858413513928.

    Article  Google Scholar 

  24. Gandomkar Z, Tay K, Brennan P, Mello-Thoms C. Recurrence quantification analysis of radiologists’ Scanpaths when interpreting mammograms. Med Phys. 2018;45(April) https://doi.org/10.1002/mp.12935.

  25. Hausdorff F. Dimension und äußeres Maß. Math Ann. 1918; https://doi.org/10.1007/BF01457179.

  26. Hering E. Uber Muskelgeräusche. Sitzberichte Der Kaiserlichen Akademie Der Wissenschaften in Wien. 1879;79:137–54.

    Google Scholar 

  27. Higuchi T. Approach to an irregular time series on the basis of the fractal theory. Physica D: Nonlinear Phenomena. 1988; https://doi.org/10.1016/0167-2789(88)90081-4.

  28. Hutton JT, Nagel JA, Loewenson RB. Eye tracking dysfunction in Alzheimer-type dementia. Neurology. 1984;34(1):99. https://doi.org/10.1212/WNL.34.1.99.

    Article  CAS  PubMed  Google Scholar 

  29. Jacob JE, Nair GK, Cherian A, Iype T. Application of fractal dimension for EEG based diagnosis of encephalopathy. Analog Integr Circ Sig Process. 2019; https://doi.org/10.1007/s10470-019-01388-z.

  30. Javal E. Essai Sur La Physiologie de La Lecture. Annales d’Ocilistique. 1878;80:97–117.

    Google Scholar 

  31. Jolliffe IT. Principal component analysis, second edition. Encyclopedia Stat Behav Sci. 2002; https://doi.org/10.2307/1270093.

  32. Jones MC, Sibson R. What is projection pursuit? J R Stat Soc Ser A. 1987;150(1):1–18. https://doi.org/10.2307/2981662.

    Article  Google Scholar 

  33. Jossberger H. Eye tracking in professional learning and development: uncovering expertise development among residents in radiology. In: Goller M, Kyndt E, Paloniemi S, Damşa C, editors. Methods for researching professional learning and development: challenges, applications and empirical illustrations. Cham: Springer International Publishing; 2022. p. 467–89. https://doi.org/10.1007/978-3-031-08518-5_21.

    Chapter  Google Scholar 

  34. Just MA, Carpenter PA. The psychology of reading and language comprehension. Boston: Allyn & Bacon; 1987.

    Google Scholar 

  35. Klonowski W. Fractal analysis of electroencephalographic time series (EEG Signals). 2016. https://doi.org/10.1007/978-1-4939-3995-4_25.

  36. Krejtz K, Duchowski AT, Niedzielska A, Biele C, Krejtz I. Eye tracking cognitive load using pupil diameter and microsaccades with fixed gaze. PLoS One. 2018;13(9):1–23. https://doi.org/10.1371/journal.pone.0203629.

    Article  CAS  Google Scholar 

  37. KrĂłl ME, KrĂłl M. Scanpath similarity measure reveals not only a decreased social preference, but also an increased nonsocial preference in individuals with autism. Autism. 2020; https://doi.org/10.1177/1362361319865809.

  38. Kundel HL. How to minimize perceptual error and maximize expertise in medical imaging. In: Jiang Y, Sahiner B, editors. Medical imaging 2007: image perception, observer performance, and technology assessment, vol. 6515. International Society for Optics/Photonics/SPIE; 2007. p. 651508. https://doi.org/10.1117/12.718061.

    Chapter  Google Scholar 

  39. Kundel HL, Nodine CF, Conant EF, Weinstein SP. Holistic component of image perception in mammogram interpretation: gaze-tracking study. Radiology. 2007;242(2):396–402. https://doi.org/10.1148/radiol.2422051997.

    Article  PubMed  Google Scholar 

  40. Le Meur O, Liu Z. Saccadic model of eye movements for free-viewing condition. Vis Res. 2015;116:152–64. https://doi.org/10.1016/j.visres.2014.12.026.

    Article  PubMed  Google Scholar 

  41. Levenshtein V. Binary codes capable of correcting deletions, insertions, and reversals. Soviet Physics Doklady; 1966.

    Google Scholar 

  42. Liu Z, Gu Z, Yang. The effectiveness of eye tracking in the diagnosis of cognitive disorders: a systematic review and meta-analysis. PLoS One. 2021;16(7):1–16. https://doi.org/10.1371/journal.pone.0254059.

    Article  CAS  Google Scholar 

  43. Lloyd EH, Hurst HE, Black RP, Simaika YM. Long-term storage: an experimental study. J R Stat Soc Ser A. 1966; https://doi.org/10.2307/2982267.

  44. Lopes R, Betrouni N. Fractal and multifractal analysis: a review. Med Image Anal. 2009;13(4):634–49. https://doi.org/10.1016/j.media.2009.05.003.

    Article  CAS  PubMed  Google Scholar 

  45. Losa GA. The fractal geometry of life. Riv Biol. 2009;102(1):29–59.

    PubMed  Google Scholar 

  46. Losa G, Nonnenmacher T. Self-similarity and fractal irregularity in pathologic tissues. Mod Pathol. 1996;9(April):174–82.

    CAS  PubMed  Google Scholar 

  47. Lyons RG. Prentice hall – understanding digital signal processing. Upper Saddle River: Prentice Hall; 2001.

    Google Scholar 

  48. Mandelbrot B. How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science. 1967;156(3775):636–8.

    Article  CAS  PubMed  Google Scholar 

  49. Mandelbrot BB. The fractal geometry of nature. New York: W.H. Freeman; 1983.

    Book  Google Scholar 

  50. Mengoudi K, Ravi D, Yong KXX, Primativo S, Pavisic IM, Brotherhood E, Kirsty L, Schott JM, Crutch SJ, Alexander DC. Augmenting dementia cognitive assessment with instruction-less eye-tracking tests. IEEE J Biomed Health Inform. 2020;24(11):3066–75. https://doi.org/10.1109/JBHI.2020.3004686.

    Article  PubMed  Google Scholar 

  51. Needleman SB, Wunsch CD. A general method applicable to the search for similarities in the amino acid sequence of two proteins. J Mol Biol. 1970; https://doi.org/10.1016/0022-2836(70)90057-4.

  52. Neumann JV. Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren. Math Ann. 1930; https://doi.org/10.1007/BF01782338.

  53. Newport RA, Liu S, Di Ieva A. Integrating eye gaze into machine learning using fractal curves. In: NeuRIPS 2022 workshop on gaze meets ML; 2022a. https://openreview.net/forum?id=-tFBD0sLQJL.

    Google Scholar 

  54. Newport RA, Russo C, Al Suman A, Di Ieva A. Assessment of eye-tracking Scanpath outliers using fractal geometry. Heliyon. 2021;7(7):e07616. https://doi.org/10.1016/j.heliyon.2021.e07616.

    Article  PubMed  PubMed Central  Google Scholar 

  55. Newport RA, Russo C, Liu S, Al Suman A, Di Ieva A. SoftMatch: comparing scanpaths using combinatorial spatio-temporal sequences with fractal curves. Sensors. 2022b;22(19) https://doi.org/10.3390/s22197438.

  56. Noton D, Stark L. Scanpaths in eye movements during pattern perception. Science. 1971;171(3968):308–11.

    Article  CAS  PubMed  Google Scholar 

  57. Peano G. Sur une courbe, qui remplit toute une aire plane. Math Ann. 1890; https://doi.org/10.1007/BF01199438.

  58. Rosenfeld A. Picture processing by computer. ACM Comp Surv (CSUR). 1969; https://doi.org/10.1145/356551.356554.

  59. Rozenshtein A, Pearson GDN, Yan SX, Liu AZ, Toy D. Effect of massed versus interleaved teaching method on performance of students in radiology. J Am Coll Radiol. 2016;13(8):979–84. https://doi.org/10.1016/j.jacr.2016.03.031.

    Article  PubMed  Google Scholar 

  60. Suman C, Al Carrigan A, Russo C. Spatial and time domain analysis of eye-tracking data during screening of brain magnetic resonance images. PLoS One. 2021;16(12):1–19. https://doi.org/10.1371/journal.pone.0260717.

    Article  CAS  Google Scholar 

  61. Tolle CR, McJunkin TR, Rohrbaugh DT, LaViolette RA. Lacunarity definition for ramified data sets based on optimal cover. Physica D: Nonlinear Phenom. 2003; https://doi.org/10.1016/S0167-2789(03)00029-0.

  62. Tseng PH, Carmi R, Cameron IGM, Munoz DP, Itti L. Quantifying center bias of observers in free viewing of dynamic natural scenes. J Vis. 2009; https://doi.org/10.1167/9.7.4.

  63. Walter A, Mannheim JG, Caruana CJ, editors. Imaging modalities for biological and preclinical research: a compendium, vol. 1. IOP Publishing; 2021. p. 2053–563. https://doi.org/10.1088/978-0-7503-3059-6.

    Book  Google Scholar 

  64. Wishart J. Computer development (SEAC and DYSEAC) at the National Bureau of standards, Washington, D.C. J R Stat Soc Ser A. 1955; https://doi.org/10.2307/2342503.

  65. Wolf A, Ueda K. Contribution of eye-tracking to study cognitive impairments among clinical populations. Front Psychol. 2021;12 https://doi.org/10.3389/fpsyg.2021.590986.

  66. Yarbus AL. Eye movements during perception of complex objects. In: Eye movements and vision. Boston: Springer; 1967. p. 171–211.

    Chapter  Google Scholar 

Download references

Acknowledgments

This work was supported by an Australian Research Council (ARC) Future Fellowship granted to A. Di Ieva in 2019.

Author information

Authors and Affiliations

  1. Computational NeuroSurgery (CNS) Lab, Macquarie Medical School, Faculty of Medicine, Human and Health Sciences, Macquarie University, Sydney, NSW, Australia

    Robert Ahadizad Newport, Sidong Liu & Antonio Di Ieva

Authors
  1. Robert Ahadizad Newport
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sidong Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Antonio Di Ieva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Robert Ahadizad Newport .

Editor information

Editors and Affiliations

  1. Computational NeuroSurgery (CNS) Lab & Macquarie Neurosurgery Macquarie Medical School, Faculty of Medicine, Human and Health Sciences, Macquarie University, Sydney, NSW, Australia

    Antonio Di Ieva

Rights and permissions

Reprints and permissions

Copyright information

© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Newport, R.A., Liu, S., Di Ieva, A. (2024). Analyzing Eye Paths Using Fractals. In: Di Ieva, A. (eds) The Fractal Geometry of the Brain. Advances in Neurobiology, vol 36. Springer, Cham. https://doi.org/10.1007/978-3-031-47606-8_42

Download citation

Publish with us

Policies and ethics

Search

Navigation