light and the retinal image: key points
DESCRIPTION
LIGHT AND THE RETINAL IMAGE: KEY POINTS Light travels in (more or less) straight lines: the pinhole camera’s inverted image Enlarging the pinhole leads to BLUR How a lens prevents blur: refraction reunites light rays by bending them Point-to-point projection from object to inverted image - PowerPoint PPT PresentationTRANSCRIPT
LIGHT AND THE RETINAL IMAGE: KEY POINTSLight travels in (more or less) straight lines: the pinhole camera’s inverted
imageEnlarging the pinhole leads to BLURHow a lens prevents blur: refraction reunites light rays by bending themPoint-to-point projection from object to inverted imageRefraction: which way is light bent? Slowing in glass: lifeguard analogy.The eye: retina, lens and cornea; fovea, periphery and blind spotFocus errors; distant vision and near visionMyopia, hypermetropria, emmetropia, accommodation; emmetropizationVisual angle and image size: (in radians) = size/distance(in degrees) = (180/ ) * size/distance
(minutes of arc) = 60 * (180/ ) * size/distancePoint spread function: width is 1 minute in visual angle, or 5 microns (.005 mm) Sources of light spread making the image imperfect:
focus error; chromatic aberration; other aberrations; diffractionDirect observation of the image: Helmholtz’s ophthalmoscopeQuality of the image: spread is about 5 microns (1 minute of arc)Visual resolution limit: about 1 minute of arc or 30 cpd (for 20/20 vision)Can vision be perfected?? William’s magic mirror and laser surgeryAliasing through sampling by the photoreceptor mosaic: Nyquist limit (60cpd)
A Review of OpticsA Review of Optics
Austin Roorda, Ph.D.University of Houston College of Optometry
(Most of) these slides were prepared by Austin Roorda, (UC Berkeley Optometry School) and used by permission.
Geometrical Optics
Relationships between pupil size, refractive error
and blur
Optics of the eye: Depth of Focus Optics of the eye: Depth of Focus
2 mm 4 mm 6 mm
2 mm 4 mm 6 mm
In focus
Focused in front of retina
Focused behind retina
Optics of the eye: Depth of Focus Optics of the eye: Depth of Focus
7 mm pupil
Bigger blurcircle
Courtesy of RA ApplegateCourtesy of RA Applegate
Smaller blurcircle
2 mm pupil
Courtesy of RA ApplegateCourtesy of RA Applegate
DemonstrationRole of Pupil Size and Defocus on Retinal Blur
Draw a cross like this one on a page, hold it so close that is it completely out of focus, then squint. You should see the horizontal line become clear. The line becomes clear because you have made you have used your eyelids to make your effective pupil size smaller, thereby reducing the blur due to defocus on the retina image. Only the horizontal line appears clear because you have only reduced the blur in the horizontal direction.
Physical Optics
The Wavefront
What is the Wavefront?What is the Wavefront?converging beam
=spherical wavefront
parallel beam=
plane wavefront
What is the Wavefront?What is the Wavefront?ideal wavefrontparallel beam
=plane wavefront
defocused wavefront
What is the Wavefront?What is the Wavefront?parallel beam
=plane wavefront aberrated beam
=irregular wavefront
ideal wavefront
What is the Wavefront?What is the Wavefront?aberrated beam
=irregular wavefront
diverging beam=
spherical wavefront
ideal wavefront
The Wave Aberration
What is the What is the Wave AberrationWave Aberration??diverging beam
=spherical wavefront wave aberration
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
Wavefront Aberration
mm (right-left)m
m (s
uper
ior-
infe
rior)
Wave Aberration of a SurfaceWave Aberration of a Surface
Diffraction
DiffractionDiffraction
“Any deviation of light rays from a rectilinear path which cannot be interpreted as reflection or refraction”
Sommerfeld, ~ 1894
Diffraction and InterferenceDiffraction and Interference
• diffraction causes light to bend perpendicular to the direction of the diffracting edge
• interference due to the size of the aperture causes the diffracted light to have peaks and valleys
rectangular aperture
square aperture
Airy Disc
circular aperture
The Point Spread Function
The Point Spread Function, or PSF, is the image that an optical system
forms of a point source.
The point source is the most fundamental object, and forms the
basis for any complex object.
The PSF is analogous to the Impulse Response Function in electronics.
Airy Disc
The Point Spread FunctionThe Point Spread Function
The PSF for a perfect optical system is the Airy disc, which is the Fraunhofer diffraction pattern for a circular pupil.
Airy DiskAiry Disk
1.22a
angle subtended at the nodal point
wavelength of the light
pupil diameter
1.22
a
a
0
0.5
1
1.5
2
2.5
1 2 3 4 5 6 7 8
pupil diameter (mm)sepa
ratri
on b
etw
een
Airy
dis
k pe
ak a
nd 1
st m
in
(min
utes
of a
rc 5
00 n
m li
ght)
As the pupil size gets larger, the Airy disc gets smaller.
Point Spread Function vs. Pupil SizePoint Spread Function vs. Pupil Size
1 mm 2 mm 3 mm 4 mm
5 mm 6 mm 7 mm
Small PupilSmall PupilLittle spreading due to defocus or aberrationsLittle spreading due to defocus or aberrations
So diffraction is limitingSo diffraction is limiting
Larger pupil: Larger pupil: Less diffraction (not shown)Less diffraction (not shown)
But more blur and more aberrationsBut more blur and more aberrations
Aberrations
1 mm 2 mm 3 mm 4 mm
5 mm 6 mm 7 mm
Point Spread Function vs. Pupil Size
Perfect Eye (Diffraction Limited)
pupil images
followed by
psfs for changing pupil size
1 mm 2 mm 3 mm 4 mm
5 mm 6 mm 7 mm
Point Spread Function vs. Pupil Size
Typical Eye with aberrations
DemonstrationObserve Your Own Point Spread Function
Resolution
Rayleigh resolution
limit
Unresolved point sources
Resolved
Keck telescope: (10 m reflector) About 4500 times better than the eye!“Pupil” is 10M: almost no diffraction
Wainscott
• Compound eye:• Each facet must
be large to fight diffraction
• Many facets (pixels) needed to capture details
Convolution with the PSF
( , ) ( , ) ( , )PSF x y O x y I x y
Convolution
Simulated Images
20/40 letters
20/20 letters
MTFModulation Transfer
Function
low medium high
object:100% contrast
image
spatial frequency
cont
rast
1
0
• The modulation transfer function (MTF) indicates the ability of an optical system to reproduce (transfer) various levels of detail (spatial frequencies) from the object to the image.
• Its units are the ratio of image contrast over the object contrast as a function of spatial frequency.
• It is the optical contribution to the contrast sensitivity function (CSF).
MTF: Cutoff FrequencyMTF: Cutoff Frequency
0
0.5
1
0 50 100 150 200 250 300
1 mm2 mm4 mm6 mm8 mm
mod
ulat
ion
tran
sfer
spatial frequency (c/deg)
cut-off frequency
57.3cutoffaf
Rule of thumb: cutoff frequency increases by ~30 c/d for each mm increase in pupil size
Effect of Defocus on the MTF
Charman and Jennings, 1976
450 nm
650 nm
Relationships Between Wave Aberration,
PSF and MTF
Retinal Sampling
Projected Image Sampled Image
5 arc minutes20/20 letter
Sampling by Foveal ConesSampling by Foveal Cones
5 arc minutes20/5 letter
Projected Image Sampled Image
Sampling by Foveal ConesSampling by Foveal Cones
Nyquist Sampling TheoremNyquist Sampling Theorem
Photoreceptor Sampling >> Spatial Frequency
I
0
1
I
0
1
nearly 100% transmitted
I
0
1
I
0
1
nearly 100% transmitted
Photoreceptor Sampling = 2 x Spatial Frequency
I
0
1
I
0
1
nothing transmitted
Photoreceptor Sampling = Spatial Frequency
Nyquist theorem:The maximum spatial frequency that can be detected is equal to ½ of the sampling frequency.
foveal cone spacing ~ 120 samples/deg
maximum spatial frequency: 60 cycles/deg (20/10 or 6/3 acuity)