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Photon Exposure

September 26 by John Dolese 35

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This application calculates the number of photons reaching a camera sensor for a given exposure. A blackbody model of the sun is generated. The "Sunny 16" rule for exposure is demonstrated. Calculations are done using units.Photon_Exposure_Array.mw

Photon ExposureNULLNULL

Blackbody Model of the Sun

    h := Units:-Standard:-`*`(Units:-Standard:-`*`(0.6626069e-33, Units:-Standard:-`^`(Unit('m'), 2)), Units:-Standard:-`*`(Unit('kg'), Units:-Standard:-`/`(Unit('s')))): 

Plank Constant       

  kb := Units:-Standard:-`*`(Units:-Standard:-`*`(0.1380650e-22, Units:-Standard:-`*`(Units:-Standard:-`^`(Unit('m'), 2), Units:-Standard:-`/`(Units:-Standard:-`^`(Unit('s'), 2)))), Units:-Standard:-`*`(Unit('kg'), Units:-Standard:-`/`(Unit('K')))): 

Boltzman Constant  

c := Units:-Standard:-`*`(0.2997925e9, Units:-Standard:-`*`(Unit('m'), Units:-Standard:-`/`(Unit('s')))):  ``

Light Speed

Rsun := Units:-Standard:-`*`(Units:-Standard:-`*`(6.955, Units:-Standard:-`^`(10, 8)), Unit('m')): ``

Sun Radius  

Re_orb := Units:-Standard:-`*`(Units:-Standard:-`*`(1.496, Units:-Standard:-`^`(10, 11)), Unit('m')): ``

Earth Orbit

Tsun := Units:-Standard:-`*`(5800, Unit('K')): ``

Sun Color Temperature     

 tf_atm := .718: 

Transmission Factor  

 

Sun: Spectral Radiant Exitance to Earth: Spectral Irradiance                   

  "M(lambda):=(2*Pi*h*c^(2))/((lambda)^(5))*1/((e)^((h*c)/(lambda*kb*Tsun))-1)*(Rsun/(Re_orb))^(2)*tf_atm:" NULL

evalf(M(Units:-Standard:-`*`(555, Unit('nm')))) = 1277414308.*Units:-Unit(('kg')/(('m')*('s')^3))"(->)"1.277414308*Units:-Unit(('W')/(('nm')*('m')^2))NULL

Photopic Relative Response VP vs λ

 

csvFile := FileTools[Filename]("/VPhotopic.csv")NULL = "VPhotopic.csv"NULL

VPdata := ImportMatrix(csvFile) = Vector(4, {(1) = ` 471 x 2 `*Matrix, (2) = `Data Type: `*float[8], (3) = `Storage: `*rectangular, (4) = `Order: `*Fortran_order})NULLNULL

 

`λP` := [seq(1 .. 4000)]:

VP := ArrayInterpolation(VPdata, `λP`):             (ArrayInterpolation for x,y data VPdata returns y' for new x data lambdaP)

NULLVParray := [`$`([`λP`[n], VP[n]], n = 1 .. 4000)]:                     

Mearth := [`$`([n, Units:-Standard:-`*`(Units:-Standard:-`*`(M(Units:-Standard:-`*`(n, Unit('nm'))), Unit('nm')), Units:-Standard:-`*`(Units:-Standard:-`^`(Unit('s'), 3), Units:-Standard:-`/`(Unit('kg'))))], n = 1 .. 4000)]:````

``

dualaxisplot(plot([Mearth], lambda = 300 .. 900, style = line, color = [blue], labels = ["λ (nm)", "M (W/nm m^2)"], title = "Spectral Radiant Exitance of the Sun", titlefont = ["ARIAL", 15], legend = [Exitance], size = [800, 300]), plot([VParray], style = line, color = [green], labels = ["λ (nm)", "Relative Response"], legend = [Units:-Standard:-`*`(Units:-Standard:-`*`(Photopic, Relative), Response)]))

 

``

 

 

 

Illuminance in Radiometric and Photometric Units:

E__r := sum(Units:-Standard:-`*`(M(Units:-Standard:-`*`(lambda, Unit('nm'))), Unit('nm')), lambda = 200 .. 4000) = 984.7275549*Units:-Unit(('kg')/('s')^3)"(->)"984.7275549*Units:-Unit(('W')/('m')^2)NULL

NULL

E__po := Units:-Standard:-`*`(Units:-Standard:-`*`(683.002, Units:-Standard:-`*`(Unit('lm'), Units:-Standard:-`/`(Unit('W')))), sum(Units:-Standard:-`*`(Units:-Standard:-`*`(VP[lambda], M(Units:-Standard:-`*`(lambda, Unit('nm')))), Unit('nm')), lambda = 200 .. 4000)) = HFloat(91873.47376063903)*Units:-Unit('lx')NULL

Translation from Illuminance to Luminance for Reflected Light;

 

Object Reflectance          R__o:      

Object Luminance           L__po := proc (R__o) options operator, arrow; R__o*E__po/(Pi*Unit('sr')) end proc:                evalf(L__po(1)) = HFloat(29244.234968360346)*Units:-Unit(('cd')/('m')^2) 

 

Illuminance of a Camera Sensor  Eps applied for time texp determines Luminous Exposure Hp;

Ideal Illuminance is determined by the exposure time texp, effective f-number N and to a less extent the angle to the optical axis θ;

 

• 

H       Luminous Exposure

• 

Eps     Illuminance to the Camera

• 

N                                               Effective F-Number

• 

texp             Exposure Time

• 

θ        Angle to the Optical Axis    

 

E__ps_ideal = Units:-Standard:-`*`(Units:-Standard:-`*`(Units:-Standard:-`*`(Pi, Units:-Standard:-`/`(4)), L__po), Units:-Standard:-`*`(Units:-Standard:-`^`(cos(theta), 4), Units:-Standard:-`/`(Units:-Standard:-`^`(N, 2)))):

H__p_ideal = Units:-Standard:-`*`(E__ps_ideal, t__exp):

 

The camera meter determines the exposure time texp to balance the object luminance, reflectance and effective f-number. It does this based on an internal constant k and the camera ISO s.

• 

s        ISO Gain (Based on saturation at 3 stops above the average scene luminance)

• 

k       Reflected Light Meter Calibration Constant      k__m := Units:-Standard:-`*`(Units:-Standard:-`*`(12.5, Unit('lx')), Unit('s')):  

                                                                                                  for Nikon, Canon and Sekonic

• 

c        Incident Light Meter Calibration Constant       c__m := Units:-Standard:-`*`(Units:-Standard:-`*`(250, Unit('lx')), Unit('s')):        

                                                                                                  for Sekonic with flat domeNULL

N^2/t__exp = `#mrow(mi("\`E__po\`"),mo("⋅"),mi("s"))`/c__m                        (Incident Light Meter)  NULL 

Units:-Standard:-`*`(Units:-Standard:-`^`(N, 2), Units:-Standard:-`/`(t__exp)) = Units:-Standard:-`*`(`#mrow(mi("\`L__po\`"),mo("⋅"),mi("s"))`, Units:-Standard:-`/`(k__m)):                        (Reflected Light Meter)

NULL

Solve for H in terms of the Camera Meter Constant k and s

 

Es = Units:-Standard:-`*`(Units:-Standard:-`*`(Units:-Standard:-`*`(Pi, Units:-Standard:-`/`(4)), Lo), Units:-Standard:-`*`(Units:-Standard:-`^`(cos(theta), 4), Units:-Standard:-`/`(Units:-Standard:-`^`(N, 2)))): NULL

t = Units:-Standard:-`*`(Units:-Standard:-`*`(km, Units:-Standard:-`^`(N, 2)), Units:-Standard:-`/`(Units:-Standard:-`*`(Lo, s))):NULL

NULL

NULL

H = Es*t

H = Units:-Standard:-`*`(Units:-Standard:-`*`(Units:-Standard:-`*`(Units:-Standard:-`*`(Pi, Units:-Standard:-`/`(4)), Lo), Units:-Standard:-`*`(Units:-Standard:-`^`(cos(theta), 4), Units:-Standard:-`/`(Units:-Standard:-`^`(N, 2)))), Units:-Standard:-`*`(Units:-Standard:-`*`(km, Units:-Standard:-`^`(N, 2)), Units:-Standard:-`/`(Units:-Standard:-`*`(Lo, s))))"(=)"H = (1/4)*Pi*cos(theta)^4*km/sNULLNULL

 t = H/Es

t = Units:-Standard:-`*`(Units:-Standard:-`*`(Units:-Standard:-`*`(Pi, Units:-Standard:-`/`(4)), Units:-Standard:-`*`(Units:-Standard:-`*`(Units:-Standard:-`^`(cos(theta), 4), km), Units:-Standard:-`/`(s))), Units:-Standard:-`/`(Units:-Standard:-`*`(Units:-Standard:-`*`(Units:-Standard:-`*`(Pi, Units:-Standard:-`/`(4)), Lo), Units:-Standard:-`*`(Units:-Standard:-`^`(cos(theta), 4), Units:-Standard:-`/`(Units:-Standard:-`^`(N, 2))))))"(=)"t = km*N^2/(Lo*s)NULLNULL

H__p := proc (s, theta) options operator, arrow; (1/4)*Pi*k__m*cos(theta)^4/s end proc:                                              

  evalf(H__p(100, 0)) = 0.9817477044e-1*Units:-Unit(('cd')*('s')/('m')('radius')^2)"(->)"0.9817477044e-1*Units:-Unit(('lx')*('s'))NULL

 

Note:  Meters are typically set for a scene reflectance 3 stops below 100% or 12.5%.

           

  E__ps := proc (N, R__o, theta) options operator, arrow; (1/4)*Pi*Unit('sr')*R__o*E__po*cos(theta)^4/(Pi*Unit('sr')*N^2) end proc:               

 evalf(E__ps(16, Units:-Standard:-`/`(Units:-Standard:-`^`(2, 3)), 0)) = HFloat(11.215023652421756)*Units:-Unit('lx')                                                                                                   

t__exp_ideal := proc (N, s, R__o) options operator, arrow; H__p(s, theta)/E__ps(N, R__o, theta) end proc:                                     

  evalf(t__exp_ideal(16, 100, Units:-Standard:-`/`(Units:-Standard:-`^`(2, 3)))) = HFloat(0.008753862094289947)*Units:-Unit('s') NULL NULL

 

 

Actual exposure time includes typical lens losses;

 m := Units:-Standard:-`/`(80):``

Magnification  

  T := .9:``

Lens Transmittance

 F := 1.03:``

Lens Flare

V := 1: ``

Vignetting

 

                                                  ``

Total Lens Efficiency

q := Units:-Standard:-`*`(Units:-Standard:-`*`(Units:-Standard:-`*`(T, F), V), Units:-Standard:-`^`(Units:-Standard:-`+`(1, Units:-Standard:-`-`(m)), 2)):                                      evalf(q) = .9039698438NULL

 

Replacing Eps with q*Eps we get the "Sunny 16" relation between exposure time and ISO;  NULL

t__exp := proc (N, s, R__o) options operator, arrow; H__p(s, theta)/(q*E__ps(N, R__o, theta)) end proc:NULL               evalf(t__exp(16, 100, Units:-Standard:-`/`(Units:-Standard:-`^`(2, 3)))) = HFloat(0.009683798806264942)*Units:-Unit('s')NULL

t__exp_alt := proc (N, s, R__o) options operator, arrow; k__m*N^2*Pi/(s*q*R__o*E__po) end proc:                  evalf(t__exp_alt(16, 100, Units:-Standard:-`/`(Units:-Standard:-`^`(2, 3)))) = HFloat(0.00968379880412244)*Units:-Unit('s') 

• 

The Number of Photons NP Reaching the Sensor Area A;

• 

Circle of confusion for 24x36mm "Full Frame" for 1 arcminute view at twice the diagonal:

                          A__cc := Units:-Standard:-`*`(Units:-Standard:-`*`(Pi, Units:-Standard:-`^`(Units:-Standard:-`*`(12.6, Unit('`μm`')), 2)), Units:-Standard:-`/`(4)):    

     

• 

  Sensor Bandwidth                                          Photopic Response VP

• 

  Exposure Time for Zone 5: Rscene=12.5% , Saturation in Zone 8 Rscene=100%

• 

  Camera ISO differs from Saturation ISO. Typical Saturation ISO is 2300 when the camera is set to 3200. See DxoMark.

 

NULL

The average number of photons for exposure time based on Reflectance of the scene  relative to the metered value:    

Zone 5;   R__meter := R__scene: 

NP := proc (s, R__o, theta) options operator, arrow; (1/4)*t__exp(N, s, R__meter)*A__cc*q*R__scene*cos(theta)^4*(sum(VP[lambda]*M(lambda*Unit('nm'))*Unit('nm')*lambda*Unit('nm')/(h*c), lambda = 200 .. 4000))/N^2 end proc: 

                                                                               evalf(NP(2300, 1, Units:-Standard:-`*`(0, Unit('deg')))) = HFloat(2191.5645712603696)  NULL

Zone 8;       R__meter := Units:-Standard:-`*`(R__scene, Units:-Standard:-`/`(Units:-Standard:-`^`(2, 3))):   NULL

NP__sat := proc (s, theta) options operator, arrow; (1/4)*t__exp(N, s, R__meter)*A__cc*q*R__scene*cos(theta)^4*(sum(VP[lambda]*M(lambda*Unit('nm'))*Unit('nm')*lambda*Unit('nm')/(h*c), lambda = 200 .. 4000))/N^2 end proc:  NULL

                                                                              evalf(NP__sat(2300, Units:-Standard:-`*`(0, Unit('deg')))) = HFloat(17532.516570082957)NULL

NULL

 

Approximate Formula

 

H__sat := proc (s__sat) options operator, arrow; H__p(s__sat, 0)*E__ps(N, 1, 0)/E__ps(N, 1/8, 0) end proc:      

                                                                                       evalf(H__sat(s__sat)) = HFloat(78.53981635)*Units:-Unit(('cd')*('s')/('m')('radius')^2)/s__satNULLNULL

Average Visible Photon Energy

P__e_ave := Units:-Standard:-`*`(Units:-Standard:-`/`(Units:-Standard:-`+`(850, -350)), sum(Units:-Standard:-`*`(Units:-Standard:-`*`(h, c), Units:-Standard:-`/`(Units:-Standard:-`*`(lambda, Unit('nm')))), lambda = 350 .. 850)):                    evalf(P__e_ave) = 0.3533174192e-18*Units:-Unit('J') 

NPtyp := proc (s__sat) options operator, arrow; H__sat(s__sat)*A__cc/(683.002*(Unit('lm')/Unit('W'))*P__e_ave) end proc: 

                               evalf(NPtyp(2300)) = HFloat(17644.363333654386)"(->)"HFloat(17644.363333654386)NULL

NULL

 

Download Photon_Exposure_Array.mw

This is an application of vector position to better understand the vector speed and acceleration is a well defined vector space. Fully developed with embedded components for proper use.

 

    Vector_Posición.mw                   (in spanish)

 

L. Araujo C.

Physics Pure

Computer Science


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