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Transcript of 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc...
/department of biomedical engineering /biomedical imaging
/department of biomedical engineering /biomedical imaging
Modeling Foveal Vision
Luc FlorackTU/e Biomedical Engineering
TU/e Cluster SymposiumW&I, Eindhoven, 13-12-2006
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 21st Century
Eye & Retina ~1.5x108
photoreceptors
~ 106
ganglion cells
amacrine cellbipolar cell
ganglion cell
horizontal cell
cone
rod
pigmented cell
Visual Pathway
LGN = Lateral Geniculate Nucleus
fovea
visual cortex
Visual Pathway
V1 = Visual Striate Cortex
Visual Pathway
(B)
(A)
(B) Parasol ganglion cells
(A) Midget ganglion cells
Scale ~ Eccentricity
ganglion cell group small bistratified parasol midget
retinal eccentricity (mm)
den
dri
tic fi
eld
d
iam
ete
r (
m)
(Dacey, 1993; Rodieck, 1998)
Scale ~ Eccentricity
(Weymouth, 1958; McKee & Nakayama, 1984; Rodieck, 1998)
spatialmotion
0 10 20 30 40
2
4
6
8
10
0
min
imu
m a
ng
le (
min
)
visual eccentricity (deg)
Scale ~ Eccentricity
L.M.J. Florack, Proc. First IEEE Workshop on Biologically Motivated Computer Vision,Seoul, Korea, Lecture Notes in Comp. Science, 2000.
Retino-Cortical Magnification
Retino-Cortical Magnification
(Rodieck, 1998)
Retino-Cortical Magnification
(J.S. Sunness, T. Liu, S. Yantis, 2004)
expanding annular retinal stimulus
pseudocolor timing representationof retinal stimulus
corresponding fMRI cortical activity pattern
log-polar model (Schwartz 1977):
Retino-Cortical Magnification
fovea (cones only) 20o eccentricity
rods
cones
Retino-Cortical Magnification
problem: log-polar model fails to capture physical resolution limitation in fovea
L. S. Balasuriya & J. P. Siebert (2006)
Retino-Cortical Magnification
“Conventional approaches for creatingretinal tessellations have been based on analytictransforms. However, the authors question thetractability of the problem, from an analyticperspective, that meets the constraints of acontinuous regular (in the fovea) to log-polar (inthe periphery) sampling regimen.”
Geometric Model I: 2-Form Field
(abuse of notation: , i.e. non-exact 1-forms)
Geometric Model II: Metric Field
Ricci tensor
Ricci scalar
metric tensor
Quantifying Retino-Cortical Magnification
Quantifying Retino-Cortical Magnification
vt(t,T): retino-cortical magnificationv(t,T): integrated retino-cortical magnification
t=1 t=T t=T ½ t=T
Quantifying Retino-Cortical Magnification
horizonfovea
Biological SignificanceRodieck 1998: R 21mm, ½ 2.15mm = (½/R) ½
0.22mmRodieck 1998: foveola 0.21mm
2 (Rodieck, 1998)
pedicle-free zone
avascular zone
rod-free zone
500 m
human
cones
rods
cones only
Canonical Coordinates
Canonical Coordinates
note:
Canonical Coordinates
Courtesy of Prof. P. H. Schiller, MIT
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…
Receptor Responses
Baylor 1987
rod
cone
rod
cone
Receptive Fields
x-y
x-t
x-y
Receptive Fields
DeAngelis, Ohzawa, Freeman
Receptive Fields Schwartz space (Schwartz, 1951):
Gaussian family (Koenderink, 1984):
i.e. retinal irradiance function = tempered distribution:
well-posed & operationally defined differential operators:
Jan KoenderinkConjecture: “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 21st Century!
Challenges for 21st Century!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?
Challenges for 21st 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?
“[…] 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 21st Century!
The End
acknowledgement: NWO, Vernieuwingsimpuls