Lecture #7

What makes a good eye? Part I. Resolution

Land and Nilsson first half of chapter 3

2/14/13

Two key features of an eye

• Resolution (today)Precision with which eye can resolve

or distinguish a spatial image

• Sensitivity (next time)How much light can be detected by the

eye

Eye design

• Optical devices are subject to laws of physics and optimized by the engineers who design them

• Eyes are subject to laws of physics and optimized by evolution

Camera - transfers object image to “film”

Image resolution depends on pixel density

Vertebrate eye - single lens which transfers image to retina

Spatial resolution depends on cone spacing

Wolfe et al Sensory Perception fig 2.9

Human retinal mosaic in foveaJW temporal nasal AN nasal

Roorda and Williams 1999

Huge variation from person to person in distribution of cones and in M/L cone ratios!

Optimizing resolution

• Receptor size and spacing

• Lens focal length

• Optical imperfections

YouAreHere

v v

v

High Resolution Low Resolution

Why does this even exist?

Resolution - seeing one point of light

receptor

Reasons that points of light might end up as blurred points

of light

• Lens does not focus perfectlyAberration = imperfection

• Cornea, lens or humours scatter some of light

• DiffractionWhen light goes through an aperture,

it gets diffracted

Resolution

Optimal receptor spacing to resolve objects

• Need enough photoreceptors to define boundaries between objects

Optimal receptor spacing

• If receptors are too big (spaced too coarsely) they can’t resolve fine detail

Want optimal receptor spacing• No reason to have more receptors than this as

they won’t resolve the images any better

Refraction: Light bends at interface

More dense

Less dense

Bigger incident angle = more bending

Lens focal length, f

• Focal length is distance at which parallel light is focused

f

Lens focal length

• Determined by curvature of lens surface

Lens radius of curvature

Curvature as if cut lens out of surface of sphere with that radius

Lens focal length

• Determined by curvature of lens surface

Radius of curvature of sphere that matches lens’s curvature

Lens focal length is determined by curvature of lens surfaces

Large radius of curvature =

large focal length

Small radius of curvature =

small focal length

Calculate focal length from lens makers equation

Lens focal length

• Determined by curvature of lens surface

For thin lens

Lens focal length

If lens is symmetric, R1=-R2=R

If lens is flat on one side, R2=∞

How can we make focal length longer?

PhET lens simulator

Physics lens equation

Distance of image and object in relation to lens focal length

dO

Object Image

di

f

Physics lens equation

If dO, object distance is very large dO>>di

dO

O I

di

So di = f

Principal rays and nodal point

dO di

Note: both cornea and lens focus the light so retina is

located at combined focal length of lens and cornea

Retina

Eye resolution

Determine resolution based on how well can resolve a periodic pattern or grating

One period of grating

Quantifying resolution

• Minimum resolvable grating will match receptor spacing

• If receptor spacing is s then grating covers 2s on retina

• Angle = 2s / f

f

Angle

2s

One cycle of grating covers two photoreceptors

• Angle = 2s / f

• So can resolve one cycle if it covers no less than that angle

• Max resolvable spatial frequencyv = 1 / angle of one

grating cycle = f / 2s

s

f

Angle

Angles

retina

grating

2s

f

θTanθ= 2s / fBut for small θ tanθ=θSo θ= 2s/f

θ is in radians!

Interreceptor angle

Angle between two adjacent photoreceptors

θ = s / f

s

f

Angleθ

One cycle of grating covers two photoreceptors

• To cover full cycle of grating the angle is 2θ= 2s / f

• Maximum resolvable spatial frequency, vv = 1/2θ= f / 2s

s

f

Angle2θ

Minimum receptor acceptance angle

• Another angle:Δρ minimum

receptor acceptance angle

f eye focal length = distance to retina

d receptor diameter

Resolution =1/ Δρ = f/d

Δρ

f

d

In general s ≈ d and 2s/f ≈ 2d/f

Two ways to characterize eye resolution

• Maximum resolvable spatial frequency, v = f/2s

sf

Angle2θ=2s/f

Δρ

fd

Inter-receptor spacing, Δρ = d/f

Grating resolution = f / 2s Receptor resol = 1/Δρ= f/d = f /s

Some typical spatial frequencies - Vertebrate eyes

Animal Max resolvable spatial freq

Inter-receptor angle

Eagle 8000 cycles/rad

0.0036 deg

Human 4175 0.007

Cat 573 0.05

Goldfish 409 0.07

Rat 57 0.5

Some typical spatial frequencies - Vertebrate eyes

Animal Max resolvable spatial freq

Inter-receptor angle, Δρ

Eagle 8000 cycles/rad

0.0036 deg

Human 4175 0.007

Cat 573 0.05

Goldfish 409 0.07

Rat 57 0.5

Δρ= 0.0036º * π / 180 º = 6.28 x 10-5 rad

Some typical spatial frequencies - Vertebrate eyes

Animal Max resolvable spatial freq

Inter-receptor angle, Δρ

Eagle 8000 cycles/rad

0.0036 deg

Human 4175 0.007

Cat 573 0.05

Goldfish 409 0.07

Rat 57 0.5

Δρ= 0.0036º * π / 180 º = 6.28 x 10-5 rad

v = 1 / 2Δρ = 1 / (2*6.28 x 10-5) = 7960 cycles / rad

WHAT ABOUT ME?

How can we modify an eye to get highacuity night vision?

Chicken Owl

Image size relative to object size: Similar triangles

• f is focal length of the eye

• O = object size• I = image size

• U=distance to object

• O / U = I / f

f

Image size relative to object size: Similar triangles

• O / U = I / f

• f is focal length of the eye

• O = object size• I = image size

• U=distance to object

O

I

U

f

Magnification = I / O = f / U

PhET lens simulator

Spatial frequency = cycles per degree

Spatial frequency Eye resolution

Resolvable spatial frequency = f / 2s

• How can we make spatial frequency and therefore eye resolution increase?

Increasing eye resolution

• Spatial freq = resolution = f / 2s

• Decrease receptor size, sCan’t decrease below 2 μm Smaller receptors leak light

• Increase lens focal length, f

sf

Angle

What happens as f gets bigger?

• O / U = I / f

• O and U stay same

• What happens to I as f gets bigger?

f• If s stays the same, what happens?

Lens

• To increase resolution increase lens focal length, f

• Retina needs to be at a distance f from the focusing elements (lens / cornea combo)

• To increase f, what does this require?

Some typical spatial frequencies - Vertebrate eyes

Animal Max resolvable spatial freq

Inter-receptor angle

Eagle 8000 cycles/rad

0.0036 deg

Human 4175 0.007

Cat 573 0.05

Goldfish 409 0.07

Rat 57 0.5

Some typical spatial resolutions - Invert eyes

Animal Max resolvable spatial freq

Inter-receptor angle

Octopus 2632 cycles/rad

0.011 deg

Jumping spider 716 0.04

Worker bee 30 0.95

Crab 19 1.5

Drosophila 5.7 5

Nautilus 3.6 8

Flatworm 0.8 35

T

Rule of “thumb”

• If you view your thumb at arms length, it subtends about 2 degrees

• Human resolution is 4000 cycles / radSince π radians = 180 degreesThis is 73 cycles / deg

• So you can resolve 146 cycles across your thumb

Eye chart

Further down chart you can read, the better your visual acuity

Eye chart is based on sine gratings

1700 cycles / radian

Typical human spatial acuity

• Can resolve <0.01 deg

• Place letters at 20 ft

• Visual acuity = Resolution you have at 20

ft Distance at which

“normal” viewer would have same resolution as you

Limitations to eye resolution

• Diffraction• Optical aberrations

SphericalChromatic

Diffraction

• When light passes through an aperture or lens, it will be distorted by diffraction

Diffraction

• When light passes through a lens, it will be distorted by diffraction

Edges do not slow down as much as middleWhen arrive at retina, the edge and middle interfere

Diffraction

• Along center line will constructively interfere

• On sides, will destructively interfere

Diffraction

• Airy disk - width of interference peak is w = λ / D

D w

Calculate w two ways and compare

• λ = 650 nm (red laser) • D = 0.088 mm

Spot widthdistance

w= spot width / distance = 1 inch / 17 ft * 12 inch/ft

= 0.005 rad not too bad compared to expectation

Diffraction through aperture

• Distance from bright to first dark spot

Sin θ = λ / Dλ = wavelengthD = slit width

What does this mean for humans

• Pupil diameter = 2 mm• Visible light λ = 500 nm

Spatial frequency = 1/wSo humans can resolve 1/.00025

radians or about 4000 cycles / radian

Diffraction limit

Resolution of human cones in fovea

Cone diameter 2 μm2 s = 4 μmDistance 20 mm

Angle = 2s/f = 4 μm / 20mm

= 0.0002 rad

= 0.011°

Resolution of human cones in fovea

• Angular resolution based on cone spacing0.0002 rad or 0.011°

• Spatial frequency = 1/angular resolution5000 cycles / rad or 90 / deg

• Diffraction limit4000 cycles /radSo cone spacing is close to diffraction limit

Diffraction Spatial frequency = 1/w = D / λ

• How does an eagle see 8000 cycles / rad

• What has to change so it has better resolution (and overcomes diffraction?)

Imperfections in the visual system

• Diffraction when light goes through aperture

• Several other defects of lens systems

Fig 3.6

Optical defects 1. Lens focal defect

• Myopia

• Near sighted

• Cornea / lens focus too much

Optical defects 1. Lens focal defect

• Hyperopia

• Far sighted

• Cornea / lens focus not enough

2. Lens spherical aberration

• Light at edges is bent more than light in center of lens

2. Lens spherical aberration

• Light at edges is bent more than light in center of lens

3. Lens chromatic aberration

• Lens index of refraction and so focal length varies with wavelength • Light of different wavelengths are

focused to different points

Refraction as function of wavelength

λ n2 f

440 1.4155 12.03 cm486 1.410 12.19590 1.4015 12.45650 1.3983 12.55

Blue light bends moreWill focus in front of red light

Human lens index of refractionVaries between edge and center - correct for aberrations

Sivak and Mandelman 1982

Center inner outer periphery

Spherical aberration of human lens

1.386

1.406

Periphery focuses less than in center to exactly compensate for spherical aberration