Lecture #8
SensitivityLand + Nilsson ch3 end
2/19/13
Topics for today• Challenges for high resolution
1) 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
DiffractionResolution 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 frequencyw 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 uncertainty
n √n 100 1010 3.31 1
Photon counting• Uncertainty in photons arriving
√n is 1 standard deviation = 66% of variation
2 √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 twoAverage 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=0.3S = 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=0.3S = 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 2Bees 2Fish / 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 powerSpecies 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