lecture #8
DESCRIPTION
Lecture #8. Sensitivity Land + Nilsson ch3 end 2 /19/13. Topics for today. Challenges for high resolution Contrast Diffraction Low light levels Sensitivity. Vertebrate spatial frequencies: best case scenarios. Resolution problem #1) What if there is less contrast?. Contrast - PowerPoint PPT PresentationTRANSCRIPT
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Lecture #8
SensitivityLand + Nilsson ch3 end
2/19/13
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Topics for today• Challenges for high resolution
1) Contrast2) Diffraction3) Low light levels
• Sensitivity
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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
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Resolution problem #1) What if there is less contrast?
• Contrast
If Imin= 0 then contrast is maximum = 100%
White vs black
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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%
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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
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Problem #2) What if there is diffraction
• Diffraction causes angular spreadingWidth of central interference peak is w = λ / D
D w
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DiffractionResolution is limited - can’t resolve
anything smaller than this angle
D w
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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
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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
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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
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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
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Contrast sensitivity function
Contrast sensitivity
Frequency
Fall off due to blurring by lens and diffraction from pupil
Diffraction limit, vCO
Hi contrast
Lo contrast
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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
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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
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Contrast sensitivity decreases with age
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Contrast sensitivit
y test
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Contrast sensitivity test
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Problem #3) Low light levels limit detection
• Random arrival of photons at each receptor
• Very low light levels cause image to be less certain
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Seeing object - high light levels
Land & Nilsson fig 3.8
Black object on bright background
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Seeing object - low light levels
Land & Nilsson fig 3.8
Black object on dim background
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Seeing object at low light level
Land & Nilsson fig 3.8
Very few photons
At light detection threshold
Photoreceptor detecting light
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Seeing object at low light level
Land & Nilsson fig 3.8
10x more light - more receptors detect photons
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Seeing object at low light level
Land & Nilsson fig 3.8
10x
100x 1000x
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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
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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
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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)
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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
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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
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How many photons are needed?
• To detect contrast, C
Contrast is between 0 and 1.
n will be greater than 1
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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
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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
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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
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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
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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
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Light arriving at one photoreceptor - Bright sun
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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
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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
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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
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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
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Eye sensitivity• Sensitivity tells how well
photoreceptors detect light• Sensitivity = # photons (n) caught
per receptor for standard radiance
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What impacts eye sensitivity?D
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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
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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
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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
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Example sensitivities
cones
rods
S in μm2sr
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Sensitivity correlates with light regime
• Diurnal or surface dwelling S < 1
• Crepuscular or mid water S = 1-100
• Nocturnal or deep sea 100-10000
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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
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Pupil aperture• Pupil aperture changes
• Sensitivity goes as D2
Change in D x4 gives change in S x 16
Day Night2 mm 8 mm
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Nocturnal animals• Pupil opens
almost to full eye size
• After this, must increase eye size to get bigger aperture
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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
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Large eyes = good eye sight
• Good resolution
Humans hawks dragonflies
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Large eyes = good eye sight
• Good sensitivity
Cats owls moths
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Large eyes = good eye sight
• Both resolution and sensitivity
Blue whale : 12-15 cm eyeGiant squid : 40 cm eye (16 inches)
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Blue whale• Blue whale : softball sized eye 12-
15 cm
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Giant squid eyes
http://www.youtube.com/watch?v=JSBDoCoJTZg
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Another way to think about sensitivity F# = f /D
F/#=focal length / aperture
D
f
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F# = focal length / aperture
Short focal length
Long focal length
For constant aperture
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F# = focal length / aperture
Short focal length Small f/#
Long focal length Big f/#
For constant aperture
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F# = focal length / aperture
Big aperture
Small aperture
For constant focal length
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F# = focal length / aperture
Big aperture Small f/#
Small aperture Big f/#
For constant focal length
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F# = focal length / aperture
If focal length = aperture
F/# is 1
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F # of eye• F # =
Eye focal lengthPupil diameter
= f/D
Humans (daytime)F# = 16 mm / 2 mm = 8
D
f
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F number, F# = f / D
Species F#Humans - day 8 Humans - night 2Bees 2Fish / nocturnal verts
1
Arthropods 0.5
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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
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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
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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
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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
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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
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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