Why Use Sine Waves to Study Vision?

Why Use Sine Waves to Study Vision?

Linear systems analysis provides a compact mathematical description of visual performance.

Varying frequency allows us to measureperformance at all spatial or temporal scales.

Same description can be applied to the eye’s optics and to neural processing.

Can be applied in comparative studies of vision.

Fourier Analysis

FourierSynthesis

FourierAnalysis

Low Pass Filtered High Pass FilteredOriginal

The World Has Structure at Many Spatial Scales

Vernier Acuity5” of arc

1/6 the diameter of a foveal cone

Imax - Imin Imax + Imin

∆II

==Contrast

Imax∆I

IImin

Visual Angle, aa = tan-1 (s/d)

a

Sun and moon each subtend 0.5 degree of visual angle (30 minutes of arc)Index finger nail at arms length subtends 1 degree of visual angle1 deg = 291 microns on the retina

s

d

Demonstration of the shape of the contrast sensitivity function for luminance gratings. Spatialfrequency increases continuously from left to right, and contrast increases from top to bottom.

FIG. 5.1 Photopic luminance CSFs for human and macaque observers. Contrast sensitivity is thereciprocal of the contrast necessary to detect a pattern at threshold. Here it is plotted as a function oftest spatial frequency (from R.L. De Valois et al., 1974, Vision Res., 14, 75-81. Copyright 1974,Pergamon Journals, Inc. Reprinted by permission).

FIG. 5.2 CSFs for several different species. The data are all normalized relative to the point ofmaximum sensitivity for each function. Note the qualitative similarity in shape of the various functionsdespite their displacement along the log spatial frequency axis (data redrawn from Northmore &Dvorak, 1979 [goldfish]; Blake et al., 1974 [cat]; Jacobs, 1977 [owl monkey]; R.L. De Valois et al., 1974[macaque monkey]; Fox et al., 1976, and Fox & Lehmkuhle, personal communication [falcon]. VisionRes., Copyright 1974, 1977, 1979, Pergamon Journals, Inc. Reprinted by permission).

Temporal Contrast Sensitivity Function

Temporal Resolution Limit ~60Hz


Fourier Optics

Point Source

How good is this lens?

Object Plane Image Plane

Blurred Image

Impulse Response = Point Spread Function

A. Perfect Eye

Planar wavefrontSpherical wavefront

B. Aberrated Eye

Aberrated wavefront

Sources of Retinal Image Blur

DiffractionAberrationsLight Scatter

I(r) = [2J1(πr) /πr]2

Radius in radians, ro = 1.22λ/a

3 mm 440 nm 0.62'3 mm 700 nm 0.98'2 mm 555 nm 1.2'8 mm 555 nm 0.29’

FWHM ~= ro

PupilDia., a λ

Visual angle

1 mm 2 mm 3 mm 4 mm

5 mm 6 mm 7 mm

Courtesy A. Roorda

1 mm 2 mm 3 mm 4 mm

5 mm 6 mm 7 mm

Courtesy A. Roorda

Fourier Optics

Sine Wave Grating

How good is this lens?

Object Plane Image Plane

Blurred and Shifted Grating Image

Modulation Transfer Function

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250

Mod

ulat

ion

Tran

sfer

Spatial Frequency (c/deg)

Diffraction8 mm pupil

632.8 nm

Eye's MTF3 mm pupil

Off Limits in Normal Viewing

Mod

ulat

ion

tran

sfer

Spatial frequency (c/deg)

0.01

0.1

1 2 mm 3 mm 4 mm

0.01

0.1

1

0 10 20 30 40 50 60

5 mm

0 10 20 30 40 50 60

6 mm

0 10 20 30 40 50 60

7 mm

Spatial Domain Frequency Domain

Object ObjectSpectrum

Generalized Pupil Function

Image Image Spectrum

Optical Transfer Function

Point Spread Function

Convolved w

ith

Autocorrelation

Equals

Equals

Fourier Transform

Inverse Fourier Transform

Multiplied by

Fourier Transform

Inverse Fourier Transform

Squa

red

mod

ulus

of

Four

ier T

rans

form

Courtesy of David Williams, University of Rochester

PointspreadFunction

RetinalImage

0.5 deg

Perfect eye(diffraction limited) MRB GYY MAK

5.7 mm pupil

ComputingRetinal Images with

ConvolutionObject

Double Pass ProcedureMeasuring the Eye’s MTF:

First PassSecond

Pass

Aerial Image

Point Source

Laser interferometry avoids blur from diffraction and aberrations

0.01

0.1

1

0 50 100 150 200 250

Mod

ulat

ion

Tran

sfer


Eye's MTFsin incoherent light

8 mm pupil632.8 nm

2 3 4 5 6 7.3 mm

Eye's MTF with Interference Fringes

1

10

100

0 10 20 30 40 50 60

Cont

rast

sen

sitivi

ty

Spatial frequency (c/deg)

interferometry

normal viewing

CSFnv = MTF x CSFint

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250

Mod

ulat

ion

Tran

sfer


Diffraction8 mm pupil

632.8 nm

Eye's MTF3 mm pupil

Off Limits in Normal Viewing

High Frequency Cut: Partly Optical Blur

Ganglion Cell BodiesBeth Peterson and Dennis Dacey, UW

ON Center OFF Center

+-+

-

Center-Surround Antagonism in Ganglion Cell Receptive Fields

FIG. 3.16 Cross-sectional diagram of a bipolar cell receptive field profile. In the top drawing (A) thecenter and surround inputs are shown separately. Since they are opposite in sign and sumalgebraically, the resultant combined RF profile is as shown in B. In C is a three-dimensionalrepresentation of the entire RF, showing its radial symmetry (after the receptive field model ofRodieck, 1965).

FIG. 3.17 Superposition of a ganglion cell RF on gratings of different spatial frequencies. In themiddle figure the RF is superimposed on a grating whose light and dark bars are of such a width as tooptimally stimulate both RF center and surround. In the top figure the same RF is shown on a gratingof half optimal spatial frequency, and in the bottom, on one of twice optimal frequency. These latterpatterns would produce little net excitation or inhibition from the cell whose RF is shown.

Low Medium High

FIG. 3.16 Cross-sectional diagram of a bipolar cell receptive field profile. In the top drawing (A) thecenter and surround inputs are shown separately. Since they are opposite in sign and sumalgebraically, the resultant combined RF profile is as shown in B. In C is a three-dimensionalrepresentation of the entire RF, showing its radial symmetry (after the receptive field model ofRodieck, 1965).

Midget Ganglion Cell Receptive Field Center

+

Horizontal Cells

Midget Ganglion Cell Receptive Field

+ -

ON cells in layers 5 and 6

OFF cells in layers 3 and 4

I

I

I

C

C

C

Low Spatial Frequency:Little or No Response

Medium Spatial Frequency:Good Response

High Spatial Frequency:Little or No Response

Low Medium High

Lateral Inhibition? Finite Grating Summation Area Too!

0 20 40 60 80 100 120

104q

FT a

mpl

itude

(spi

kes/

sec)

0

10

20

30

40

50

60

70

20 40 60 80 100 120

94t

0

10

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00

5

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20 40 60 80 100 120 140

93y

0


Monkey LGN Interferometry

Section Through Primary Visual CortexAlso called Striate Cortex or V1

FIG. 4.18 Three-dimensional representations of the RF profiles (in both space and frequency domains)of a fairly narrowly tuned cat simple cell. In the frequency domain plot it can be seen that the cellresponds to a delimited, compact range of spatial frequencies. Note in the space domain plot that thiscell has an oscillatory RF with multiple lobes along the x axis and is elongated in the y direction (fromWebster & R.L. De Valois, 1985, J. Opt. Soc. Am. A, 2, 1124-1132. Reprinted by permission).

FIG. 4.14 Quantitative simple cell receptive field profile (space domain) and corresponding spatialfrequency tuning function. The RF profile was measured by recording the responses (y axis) to anarrow, flickering black-white bar in different spatial positions (x axis). The solid lines in the columnon the left represent the RF profiles predicated by measuring the response to gratings of differentspatial frequencies (right). The data points are actual responses (from Albrecht, 1978. Reprinted bypermission).

FIG. 6.10 Macaque behavioral CSF (crosses) and the CSF of a single macaque striate cortex cell(circles). Note that the cell’s CSF has a very much narrower bandwidth than that of the overallbehavioral CSF. The latter presumably reflects the activity of many more narrowly tuned individualcells, as depicted in figure 6.1.

FIG 6.4 (A) CSF before (solid line) and after (data points) adaptation to a grating of a single spatialfrequency. (B) Threshold elevation at different spatial frequencies produced by the adaptation. Notethe band-pass loss in contrast sensitivity around the adaptation frequency (indicated by arrows) (fromBlakemore & Campbell, 1969. Reprinted b permission).

FIG. 6.7 Demonstration of the Blakemore-Sutton effect. First note that the two test gratings on theright are identical. Now adapt for about 1 min while scanning back and forth along the fixation barbetween the two gratings on the left, then quickly shift your gaze to the fixation line between the testgratings on the right. They should now appear different in spatial frequency, each one being shiftedaway from the frequency of the adaptation grating that occupied the same retinal area.

FIG. 6.8 A model of spatial frequency channels and the effects of adaptation, to account for theBlakemore-Sutton apparent spatial frequency shift. Three spatial frequency channels (A, B, and C)are presumed to signal the apparent spatial frequency in this region by their relative activity rates. Forinstance, x will have a certain apparent spatial frequency because it equally stimulates channels A andB. Following adaptation to a grating of the middle frequency, the sensitivity of channel B is reduced toB’. Now, a grating of the frequency y will equally stimulate channels A and B (now B’), and thus willlook the same as x had looked before adaptation.

FIG. 6.1 Spatial contrast sensitivity function as the envelope of many more narrowly tuned spatialfrequency selective channels.


Demonstration of the shape of the contrast sensitivity function for luminance gratings. Spatialfrequency increases continuously from left to right, and contrast increases from top to bottom.

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Why Use Sine Waves to Study Vision?