1. 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium W&I, Eindhoven, 13-12-2006

2. Overview Facts on Human Vision: Foveal Vision Geometric Model for Foveal Vision Biological Plausibility of Foveal Vision Model Summary of Foveal Vision Model Challenges for the 21 st Century

3. Eye & Retina ~1.5x10 8 photoreceptors ~ 10 6 ganglion cells amacrine cell bipolar cell ganglion cell horizontal cell cone rod pigmented cell

4. Visual Pathway LGN = Lateral Geniculate Nucleus

5. fovea visual cortex Visual Pathway V1 = Visual Striate Cortex

6. Visual Pathway

7. (B) (A) (B) Parasol ganglion cells (A) Midget ganglion cells Scale ~ Eccentricity

8. ganglion cell group small bistratified parasol midget retinal eccentricity (mm) dendritic field diameter (  m) (Dacey, 1993; Rodieck, 1998) Scale ~ Eccentricity

9. (Weymouth, 1958; McKee & Nakayama, 1984; Rodieck, 1998) spatial motion 010203040 2 4 6 8 10 0 minimum angle (min) visual eccentricity (deg) Scale ~ Eccentricity

10. L.M.J. Florack, Proc. First IEEE Workshop on Biologically Motivated Computer Vision, Seoul, Korea, Lecture Notes in Comp. Science, 2000. Retino-Cortical Magnification

11. (Rodieck, 1998)

12. Retino-Cortical Magnification (J.S. Sunness, T. Liu, S. Yantis, 2004) expanding annular retinal stimulus pseudocolor timing representation of retinal stimulus corresponding fMRI cortical activity pattern

13. log-polar model (Schwartz 1977): Retino-Cortical Magnification

14. fovea (cones only)20 o eccentricity rods cones Retino-Cortical Magnification problem: log-polar model fails to capture physical resolution limitation in fovea

15. L. S. Balasuriya & J. P. Siebert (2006) Retino-Cortical Magnification “Conventional approaches for creating retinal tessellations have been based on analytic transforms. However, the authors question the tractability of the problem, from an analytic perspective, that meets the constraints of a continuous regular (in the fovea) to log-polar (in the periphery) sampling regimen.”

16. Geometric Model I: 2-Form Field (abuse of notation:, i.e. non-exact 1-forms)

17. Geometric Model II: Metric Field Ricci tensor Ricci scalar metric tensor

18. Quantifying Retino-Cortical Magnification   

19. Quantifying Retino-Cortical Magnification v t (t,T): retino-cortical magnificationv(t,T): integrated retino-cortical magnification t=1t=T t=T ½ t=T

20. Quantifying Retino-Cortical Magnification horizon fovea

21. Biological Significance Rodieck 1998: R  21mm,  ½  2.15mm   = (  ½/R)  ½  0.22mm Rodieck 1998: foveola  0.21mm 2  (Rodieck, 1998) pedicle-free zone avascular zone rod-free zone 500  m human cones rods cones only

22. Canonical Coordinates

23. note:

24. Canonical Coordinates Courtesy of Prof. P. H. Schiller, MIT

25. Summary of Foveal Vision Model New paradigm for modeling foveal vision: –exterior differential calculus –Riemannian geometry Model: –is based on axioms expressing natural invariances –accounts for transient structure connecting fovea to periphery –gracefully removes singularity in classical log-polar paradigm –suggests canonical coordinates with biological significance –provides quantitative explanation of retino-cortical magnification –relates seemingly unrelated biological scale parameters –may have (hitherto unexplored) predictive power –is falsifiable…

26. Receptor Responses Baylor 1987 rod cone rod cone

27. Receptive Fields x-y x-t x-y

28. Receptive Fields DeAngelis, Ohzawa, Freeman

29. Receptive Fields Schwartz space (Schwartz, 1951): Gaussian family (Koenderink, 1984): i.e. retinal irradiance function = tempered distribution: well-posed & operationally defined differential operators:

30. Jan Koenderink Conjecture: “The brain can organize itself through information obtained via interactions with the physical world into an embodiment of geometry, it becomes a veritable geometry engine ” Challenges for 21 st Century!

31. many open problems, e.g. “local sign” (Localzeichen, Herman Lotze, 1884): how is spatial topology embodied in the visual system? key: correlation structure of receptive fields?

32. Challenges for 21 st Century! many open problems, e.g. “Gestalt laws” and visual illusions: how is retinal irradiance represented in the visual system? how does the visual system establish neighbourhood relations? key: fibre bundles, sections, connections?

34. “[…] while geometry is supposed to deal with properties of space or of space- time itself, the evidence for a geometry must always be provided by what is material” Clark Glymour Challenges for 21 st Century!

35. The End acknowledgement: NWO, Vernieuwingsimpuls