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

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Page 1: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium

/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

Page 2: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium

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

Page 3: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium

Eye & Retina ~1.5x108

photoreceptors

~ 106

ganglion cells

amacrine cellbipolar cell

ganglion cell

horizontal cell

cone

rod

pigmented cell

Page 4: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium

Visual Pathway

LGN = Lateral Geniculate Nucleus

Page 5: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium

fovea

visual cortex

Visual Pathway

V1 = Visual Striate Cortex

Page 6: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium

Visual Pathway

Page 7: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium

(B)

(A)

(B) Parasol ganglion cells

(A) Midget ganglion cells

Scale ~ Eccentricity

Page 8: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium

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

Page 9: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium

(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

Page 10: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium

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

Retino-Cortical Magnification

Page 11: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium

Retino-Cortical Magnification

(Rodieck, 1998)

Page 12: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium

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

Page 13: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium

log-polar model (Schwartz 1977):

Retino-Cortical Magnification

Page 14: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium

fovea (cones only) 20o eccentricity

rods

cones

Retino-Cortical Magnification

problem: log-polar model fails to capture physical resolution limitation in fovea

Page 15: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium

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.”

Page 16: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium

Geometric Model I: 2-Form Field

(abuse of notation: , i.e. non-exact 1-forms)

Page 17: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium

Geometric Model II: Metric Field

Ricci tensor

Ricci scalar

metric tensor

Page 18: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium

Quantifying Retino-Cortical Magnification

Page 19: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium

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

Page 20: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium

Quantifying Retino-Cortical Magnification

horizonfovea

Page 21: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium

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

Page 22: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium

Canonical Coordinates

Page 23: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium

Canonical Coordinates

note:

Page 24: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium

Canonical Coordinates

Courtesy of Prof. P. H. Schiller, MIT

Page 25: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium

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…

Page 26: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium

Receptor Responses

Baylor 1987

rod

cone

rod

cone

Page 27: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium

Receptive Fields

x-y

x-t

x-y

Page 28: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium

Receptive Fields

DeAngelis, Ohzawa, Freeman

Page 29: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium

Receptive Fields Schwartz space (Schwartz, 1951):

Gaussian family (Koenderink, 1984):

i.e. retinal irradiance function = tempered distribution:

well-posed & operationally defined differential operators:

Page 30: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium

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!

Page 31: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium

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?

Page 32: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium

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?

Page 33: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium
Page 34: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium

“[…] 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!

Page 35: 1212 /department of biomedical engineering/biomedical imaging 1212 Modeling Foveal Vision Luc Florack TU/e Biomedical Engineering TU/e Cluster Symposium

The End

acknowledgement: NWO, Vernieuwingsimpuls