Lecture #8

SensitivityLand + Nilsson ch3 end

2/19/13

Topics for today

• Challenges for high resolution1) Contrast2) Diffraction3) Low light levels

• Sensitivity

Vertebrate spatial frequencies: best case scenarios

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

Resolution problem #1) What if there is less contrast?

• Contrast

If Imin= 0 then contrast is maximum = 100%

White vs black

Contrast

I max I min C

White/Black

100% 0%

White/gray 100% 20%

Lt gray / gray

70% 30%

Med gray / med gray

50% 50%

Contrast

I max I min C

White/Black

100% 0% 1.0

White/gray 100% 20% 0.66

Lt gray / gray

70% 30% 0.4

Med gray / med gray

50% 50% 0.0

T

T

T

Problem #2) What if there is diffraction

• Diffraction causes angular spreadingWidth of central interference peak is w = λ / D

D w

Diffraction

Resolution is limited - can’t resolve anything smaller than this angle

D w

Detectable grating frequency

• Max frequency that can be detected depends on diffraction

vco is max cut-off frequency

w is width of diffraction peak (radians)λ is wavelengthD is aperture

Detectable grating frequency - Humans

• Max frequency that can be detected depends on diffraction

•λ is wavelength 500 nmD is pupil aperture 2 mmw = 500 x 10-9 m / 2 x 10-3 m = 0.00025 radvco = 4000 cycles / rad

Diffraction in optical systems blurs images

• This decreases contrast• This makes gratings even harder to

detect

http://www.microscopyu.com/tutorials/java/mtf/spatialvariation/index.html

Lp/mm = line pairs/mm

Contrast

Imax

Imin

Diffraction decreases contrast and contrast ratio

• Contrast of image decreases compared to contrast of object = contrast ratio

• More loss of contrast with higher frequency grating

• Spatial freq is normalized to diffraction limited cutoff, vCOLand and Nilsson fig 3.3

Contrast sensitivity function

Contrast sensitivity

Frequency

Fall off due to blurring by lens and diffraction from pupil

Diffraction limit, vCO

Hi contrast

Lo contrast

Diffraction decreases contrast and contrast ratio

• Contrast of image decreases compared to contrast of object = contrast ratio

• More loss of contrast with higher frequency grating

• Spatial freq is normalized to diffraction limited cutoff, w=D/λLand and Nilsson fig 3.3

Contrast sensitivity function

Contrast sensitivity

Frequency

Fall off due to blurring by lens and diffraction from pupil

Diffraction limit, vCO

Hi contrast

On low frequency side size of neurons matter

Contrast sensitivity decreases with age

Contrast sensitivit

y test

Contrast sensitivity test

Problem #3) Low light levels limit detection

• Random arrival of photons at each receptor

• Very low light levels cause image to be less certain

Seeing object - high light levels

Land & Nilsson fig 3.8

Black object on bright background

Seeing object - low light levels

Land & Nilsson fig 3.8

Black object on dim background

Seeing object at low light level

Land & Nilsson fig 3.8

Very few photons

At light detection threshold

Photoreceptor detecting light

Seeing object at low light level

Land & Nilsson fig 3.8

10x more light - more receptors detect photons

Seeing object at low light level

Land & Nilsson fig 3.8

10x

100x 1000x

Photon counting

• At low light levels, rod will “count” the number of photons, n

• Photon arrival is a poisson process Uncertainty in photon arriving goes as √n

• Fewer photons means more uncertaintyn √n 100 1010 3.31 1

Photon counting

• Uncertainty in photons arriving √n is 1 standard deviation

= 66% of variation2 √n is 2 standard deviations

= 95% of variation

• So if 9 photons arrive on average in 1 s, for any particular second 9 ± 6 photons will arrive with 95% confidence

Contrast detection• The bright / dark stripe of a

grating falls across two receptors

• Contrast

Imax is intensity of brighter stripe

Imin is intensity of darker stripe

ΔI is difference between these two

Average intensity, I = 1/2 (Imax + Imin)

Contrast detection• To detect stripes as being

different, average number of photons must be greater than uncertainty in photon number

95% confidence

• So contrast in terms of photon number is

Contrast detection• To detect stripes as being

different, average number of photons must be greater than uncertainty in photon number

95% confidence

• So contrast in terms of photon number is

Detectable contrast

How many photons are needed?

• To detect contrast, C

Contrast is between 0 and 1.

n will be greater than 1

How many photons are needed to detect contrast?

• # photons needed n >1/C2

Contrast # photons # detected photons/s

#photons needed/s

100% 1 10 30

50% 4 40 120

10% 100 1000 3000

1% 10000 100,000 300,000

How many photons are needed to detect contrast?

• # photons needed n >1/C2

Contrast # photons # photons detected/s

#photons needed/s

100% 1 10 30

50% 4 40 120

10% 100 1000 3000

1% 10000 100,000 300,000

Takes rod 0.1s to detect light so rate = # photons / 0.1s

How many photons are needed to detect contrast?

• # photons needed n >1/C2

Contrast # photons # photons detected/s

#photons needed/s

100% 1 10 30

50% 4 40 120

10% 100 1000 3000

1% 10000 100,000 300,000

Only detect 30% of photons that arrive at eye so need 3x more

How many photons are out there?

Bright sun is 1020 photons / m2 sr s

But a photoreceptor is only 5 μm2

Collection angle is 0.0003 sr

Land&Nilsson Table 2.1

Measuring incident light (lecture 3)

• IrradianceLight flux on a surface - from all directions

Photons /s m2

RadianceIrradiance

• RadianceLight flux on a surface: from a particular direction and angle

Photons /s m2 sr

Light arriving at one photoreceptor - Bright sun

How many photons arrive at one photoreceptor

Light level Photon flux photons / m2/sr/s

Photon ratePhotons/s

Bright sun 1020 1.5 x 105

Room light 1017 150

Moon light 1014 0.15

Star light 1012 0.0015

How many photons are needed to detect contrast?

Contrast # photons needed/s

Light # photons arriving/s

100% 30 Moon light

0.15

50% 120 Room light

150

10% 3000

1% 300,000 Bright sun

150,000

How many photons are needed to detect contrast?

• Can only detect high contrast in bright sun

Contrast # photons needed/s

Light # photons arriving/s

100% 30 Moon light

0.15

50% 120 Room light

150

10% 3000

1% 300,000 Bright sun

150,000

Some caveats

• In dark, rods gang together so you get a larger area of light collection to increase photon #s and so ability to detect contrast

• To maximize ability to resolve fine detail requires high light levelsGets worse with age

Eye sensitivity

• Sensitivity tells how well photoreceptors detect light

• Sensitivity = # photons (n) caught per receptor for standard radiance

What impacts eye sensitivity?

D

Eye sensitivity

• Eye sensitivityS = n/R = # photons / radiance (W/m2 sr s)

(photons m2 sr )

Fig 3.11

D = diameter of pupilΔρ = receptor acceptance anglePabs = probability photon is absorbed

Human sensitivities• Human

S=0.62 D2 Δρ2 Pabs Daytime:

D=2 mm = 2000μm

Δρ=1.2x10-4 rad

Pabs=0.3

S = 0.62 (2000 μm)2 (1.2x10-4 rad)2 (0.3)

Note: D must be in μm and Δρ in radians

Human sensitivities• Human

S=0.62 D2 Δρ2 Pabs Daytime:

D=2 mm = 2000μm

Δρ=1.2x10-4 rad

Pabs=0.3

S = 0.01 μm2 sr

Example sensitivities

cones

rods

S in μm2sr

Sensitivity correlates with light regime

• Diurnal or surface dwelling S < 1

• Crepuscular or mid water S = 1-100

• Nocturnal or deep sea 100-10000

How do you increase

sensitivity and not change resolution?

• Sensitivity S = 0.62 D2 Δρ2 Pabs

• Resolution, 1/Δρ = f/d focal length / receptor diam

Pupil aperture

• Pupil aperture changes

• Sensitivity goes as D2

Change in D x4 gives change in S x 16

Day Night2 mm 8 mm

Nocturnal animals

• Pupil opens almost to full eye size

• After this, must increase eye size to get bigger aperture

How can you increase Pabs

(probability absorb photon)?

• A=1-T=1-e-αl

• Pack in more pigment

• Make photoreceptors longer

• Have light do a double pass through the retina by adding reflector at back

Large eyes = good eye sight

• Good resolution

Humans hawks dragonflies

Large eyes = good eye sight

• Good sensitivity

Cats owls moths

Large eyes = good eye sight

• Both resolution and sensitivity

Blue whale : 12-15 cm eyeGiant squid : 40 cm eye (16 inches)

Blue whale• Blue whale : softball sized eye 12-

15 cm

Giant squid eyes

http://www.youtube.com/watch?v=JSBDoCoJTZg

Another way to think about sensitivity F# = f /D

F/#=focal length / aperture

D

f

F# = focal length / aperture

Short focal length

Long focal length

For constant aperture

F# = focal length / aperture

Short focal length Small f/#

Long focal length Big f/#

For constant aperture

F# = focal length / aperture

Big aperture

Small aperture

For constant focal length

F# = focal length / aperture

Big aperture Small f/#

Small aperture Big f/#

For constant focal length

F# = focal length / aperture

If focal length = aperture

F/# is 1

F # of eye• F # =

Eye focal lengthPupil diameter

= f/D

Humans (daytime)F# = 16 mm / 2 mm = 8

D

f

F number, F# = f / D

Species F#

Humans - day 8

Humans - night 2

Bees 2

Fish / nocturnal verts

1

Arthropods 0.5

Sensitivity in terms of F/#

• Sensitivity, S=0.62 D2 Δρ2Pabs

So how should an eye’s sensitivity be increased?

Δρ=d/f

F# = f / D

F number

• As F# goes down, sensitivity increases to second power

Species F# Sensitivity = Relative brightness

Humans - day 8 1

Humans - night 2 16

Bees 2 16

Fish / nocturnal verts 1 64

Arthropods 0.5 256

To optimize resolution and sensitivity, eyes get large

Character Optimizes Equation

Long focal length, f

Minimum resolvable angleMaximum sampling frequency

Δρ=d/f

νs=f/2s

A good eye is large - resolution and sensitivity

Character Optimizes Equation

Long focal length, f

Minimum resolvable angleMaximum sampling frequency

Δρ=d/f

νs=f/2s

Wide aperture, D Minimize diffractionHigh optical cut-off frequency

w=λ/Dνco=1/w=D/λ

Resolution

A good eye is large - resolution and sensitivity

Character Optimizes Equation

Long focal length, f

Minimum resolvable angleMaximum sampling frequency

Δρ=d/f

νs=f/2s

Wide aperture, D Minimize diffractionHigh optical cut-off frequency

w=λ/Dνco=1/w=D/λ

Wide aperture, D Increase light to eyeGood contrast detection

S=0.62D2Δρ2Pabs

C>1/√n

Sensitivity

Conclusions

• Resolution is best for high contrast, minimal diffraction, and high light intensities

• Sensitivity and resolution are inversely correlated

• Next few lectures - aquatic and terrestrial examples