# Lecture #8 Sensitivity Land + Nilsson ch3 end 2/19/13.

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Lecture #8 Sensitivity Land + Nilsson ch3 end 2/19/13

Topics for today Challenges for high resolution 1)Contrast 2)Diffraction 3)Low light levels Sensitivity

Vertebrate spatial frequencies: best case scenarios AnimalMax resolvable spatial freq Inter-receptor angle Eagle8000 cycles/rad0.0036 deg Human41750.007 Cat5730.05 Goldfish4090.07 Rat570.5

Resolution problem #1) What if there is less contrast? Contrast If I min = 0 then contrast is maximum = 100% White vs black

Contrast I maxI minC White/Black100%0% White/gray100%20% Lt gray / gray70%30% Med gray / med gray 50%

Contrast I maxI minC White/Black100%0%1.0 White/gray100%20%0.66 Lt gray / gray70%30%0.4 Med gray / med gray 50% 0.0

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Problem #2) What if there is diffraction Diffraction causes angular spreading Width 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 v co is max cut-off frequency w is width of diffraction peak (radians) λ is wavelength D is aperture

Detectable grating frequency - Humans Max frequency that can be detected depends on diffraction λ is wavelength500 nm D is pupil aperture2 mm w = 500 x 10 -9 m / 2 x 10 -3 m = 0.00025 rad v co = 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 I max I min

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, v CO Land and Nilsson fig 3.3

Contrast sensitivity function Contrast sensitivity Frequency Fall off due to blurring by lens and diffraction from pupil Diffraction limit, v CO 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/λ Land and Nilsson fig 3.3

Contrast sensitivity function Contrast sensitivity Frequency Fall off due to blurring by lens and diffraction from pupil Diffraction limit, v CO Hi contrast On low frequency side size of neurons matter

Contrast sensitivity decreases with age

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 10 10 3.3 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 I max is intensity of brighter stripe I min is intensity of darker stripe ΔI is difference between these two Average intensity, I = 1/2 (I max + I min )

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/C 2 Contrast# photons# detected photons/s #photons needed/s 100%11030 50%440120 10%10010003000 1%10000100,000300,000

How many photons are needed to detect contrast? # photons needed n >1/C 2 Contrast# photons# photons detected/s #photons needed/s 100%11030 50%440120 10%10010003000 1%10000100,000300,000 Takes rod 0.1s to detect light so rate = # photons / 0.1s

How many photons are needed to detect contrast? # photons needed n >1/C 2 Contrast# photons# photons detected/s #photons needed/s 100%11030 50%440120 10%10010003000 1%10000100,000300,000 Only detect 30% of photons that arrive at eye so need 3x more

How many photons are out there? Bright sun is 10 20 photons / m 2 sr s But a photoreceptor is only 5 μm 2 Collection angle is 0.0003 sr Land&Nilsson Table 2.1

Measuring incident light (lecture 3) Irradiance Light flux on a surface - from all directions Photons /s m 2 Radiance Irradiance Radiance Light flux on a surface: from a particular direction and angle Photons /s m 2 sr

Light arriving at one photoreceptor - Bright sun

How many photons arrive at one photoreceptor Light levelPhoton flux photons / m 2 /sr/s Photon rate Photons/s Bright sun10 20 1.5 x 10 5 Room light10 17 150 Moon light10 14 0.15 Star light10 12 0.0015

How many photons are needed to detect contrast? Contrast# photons needed/s Light# photons arriving/s 100%30Moon light0.15 50%120Room light150 10%3000 1%300,000Bright sun150,000

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%30Moon light0.15 50%120Room light150 10%3000 1%300,000Bright sun150,000

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 levels Gets 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 S = n/R = # photons / radiance (W/m 2 sr s) (photons m 2 sr ) Fig 3.11 D = diameter of pupil Δρ = receptor acceptance angle P abs = probability photon is absorbed

Human sensitivities Human S=0.62 D 2 Δρ  P abs Daytime: D=2 mm = 2000μm Δρ=1.2x10 -4 rad P abs =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 D 2 Δρ  P abs Daytime: D=2 mm = 2000μm Δρ=1.2x10 -4 rad P abs =0.3 S = 0.01 μm 2 sr

Example sensitivities cones rods S in μm 2 sr

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 D 2 Δρ 2 P abs Resolution, 1/Δρ = f/d focal length / receptor diam

Pupil aperture Pupil aperture changes Sensitivity goes as D 2 Change in D x4 gives change in S x 16 Day Night 2 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 P abs (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 eye Giant squid : 40 cm eye (16 inches)

Blue whale Blue whale : softball sized eye 12-15 cm

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 length Pupil diameter = f/D Humans (daytime) F# = 16 mm / 2 mm = 8 D f

F number, F# = f / D SpeciesF# Humans - day 8 Humans - night 2 Bees 2 Fish / nocturnal verts 1 Arthropods0.5

Sensitivity in terms of F/# Sensitivity, S=0.62 D 2 Δρ  P abs So how should an eye’s sensitivity be increased? Δ ρ =d/f F# = f / D

F number As F# goes down, sensitivity increases to second power SpeciesF#Sensitivity = Relative brightness Humans - day 8 1 Humans - night 2 16 Bees 2 16 Fish / nocturnal verts 1 64 Arthropods0.5 256

To optimize resolution and sensitivity, eyes get large CharacterOptimizesEquation Long focal length, fMinimum resolvable angle Maximum sampling frequency Δρ=d/f ν s =f/2s

A good eye is large - resolution and sensitivity CharacterOptimizesEquation Long focal length, fMinimum resolvable angle Maximum sampling frequency Δρ=d/f ν s =f/2s Wide aperture, DMinimize diffraction High optical cut-off frequency w=λ/D ν co =1/w=D/λ R e s o lu t i o n

A good eye is large - resolution and sensitivity CharacterOptimizesEquation Long focal length, fMinimum resolvable angle Maximum sampling frequency Δρ=d/f ν s =f/2s Wide aperture, DMinimize diffraction High optical cut-off frequency w=λ/D ν co =1/w=D/λ Wide aperture, DIncrease light to eye Good contrast detection S=0.62D 2 Δρ 2 P abs C>1/√n SensiSensi tivity

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