RADIOMETRY OF LAMBERTIAN SOURCES - engr.arizona.edu

1 ECE 425 CLASS NOTES 2000 DR. ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax) 157 RADIOMETRY OF LAMBERTIAN SOURCES Relate M and L Valuable exercise in spherical coordinate integration Area element of sphere, radius r Start with radiance to get flux within area d A 2 LdA2dA1rdA2rd r d sin=r2 d d sin= ECE 425 CLASS NOTES 2000 DR. ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax) 158 Total flux in hemisphere above source Now, in general L = L ( , ), but for a LAMBERTIAN source L ,()d2 dA1 d cos-------------------------------=d2 L ,() dA1d cos=L ,() dA1dA2r2----------cos=L ,() dA1 d d sincos= dA1d L ,() d sincos0 2 02 = ECE 425 CLASS NOTES 2000 DR.

2 ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax) 159 L = constant Therefore Radiant exitance definition dA12 L sin()22------------------0 2 = Ld A1=M dA1----------= L= ECE 425 CLASS NOTES 2000 DR. ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax) 160 Include spectral variation Can think of as having units of sr , which cancels the sr -1 units of L Note: M 2 L , as one might guess since hemisphere is 2 sr Why?

3 M L =or L M = ECE 425 CLASS NOTES 2000 DR. ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax) 161 TRANSMITTANCE Defined as ratio of transmitted output flux to input fluxfilter t( ) in out = in ( )t () out in =ECE 425 CLASS NOTES 2000DR. ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax)162 REFLECTANCET hree types:Most natural surfaces are approximately LAMBERTIAN ELspecular (mirror)ELdirectionalELdiffuse ( LAMBERTIAN )ECE 425 CLASS NOTES 2000DR.

4 ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax)163for < 40 , snow and sand for < 60 At larger , natural surfaces tend to become directionalReflectance of LAMBERTIAN surface Reflectance defined similarly to transmittance From earlier derivation LdA2dA1Er () out in =ECE 425 CLASS NOTES 2000DR. ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax)164 and Therefore,, Given reflectance of surface and incident irrradiance, can get radiance out L dA1= inE dA1= () L E =0 ()1 L ()E =ECE 425 CLASS NOTES 2000DR.

5 ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax)165 BAND-AVERAGED IRRADIANCEC ascade spectral quantitites from source-to-surface-to-surface .. until arriving at a spectrally-integrating element, a detector, where S( ) is the detector s spectral sensitivityExample of a specific detector S( ) is human vision system sensitivity V( )Etotal is the effective irradiance, because it is what is E (geometric factors)L t1 ()t2 ().. 1 ().. =EtotalE S () d =ECE 425 CLASS NOTES 2000DR. ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax)166detected and measuredBand-averaged spectral irradianceEbEtotalS () d0 ----------------------=ECE 425 CLASS NOTES 2000DR.

6 ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax)167 RADIOMETRY OF OPTICAL SYSTEMS telescope collects light from point source light bucket assume: source and detector on optical axis object-to-sensor distance much greater than aperture lens (or mirror)detectorpoint sourceaperture (diameter d)z0zi(radiant intensity I)ECE 425 CLASS NOTES 2000DR. ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax)168diameter no transmission losses irradiance at lens aperture (inverse square law) flux collected by aperture flux at detector (no losses) proportional to radiant intensity of source proportional to square of aperture diameterz0d EIz02 = cEAaperture =E d24--------- =Iz02----- d24--------- = i c=ECE 425 CLASS NOTES 2000DR.

7 ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax)169 inversely proportional to square of object-to-sensor distanceWhat is the only way to increase the amount of light collected by a telescope?Camera assume: LAMBERTIAN source and detector plane normal to optical axis magnification mhiho zizo ==ECE 425 CLASS NOTES 2000DR. ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax)170 flux collected by aperture NOTE: similarity to light bucket equation flux at detector (no losses) lens (or mirror)imageLambertian sourceaperture (diameter d)z0zi(radiance Lo, height ho, area Ao)(height hi, area Ai) cLoAo =LoAoAzo2-----=LoAozo2------------- d24--------- = i c=ECE 425 CLASS NOTES 2000DR.

8 ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax)171 irradiance at detector by definition of magnification (see Geometrical Optics) define effective f-number N of sensor, then, Camera Equation NOTE: proportional to radiance of sourceEi cAi = cm2Ao--------------=Lo d2m24zo2----------------=EiLo d24zi2---------=Nzid =EiLo 4N2----------=ECE 425 CLASS NOTES 2000DR. ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax)172 inversely proportional to square of sensor f-number does not depend on zo, the source-to-sensor distance f-number N typically preset on camera to , 2, , 4, , 8, 11, 16, 22 What is rationale for the above preset values of N?

9 ECE 425 CLASS NOTES 2000DR. ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax)173 Camera imaging reflecting object Irradiance on object E Reflectance of object From earlier derivation:lens (or mirror)imageLambertian reflectoraperture (diameter d)(reflectance o, height ho, area Ao)(height hi, area Ai)sourceirradiance EEiEo 4N2----------=ECE 425 CLASS NOTES 2000DR. ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax)174 Camera imaging extended reflecting object Cos4 Law applies:With object irradiance and reflectance fixed, what is the only way to increase the lens (or mirror)imageLambertian reflectoraperture (diameter d)(reflectance o, height ho, area Ao)(height hi, area Ai)sourceirradiance E Ei ()Eo 4N2---------- cos()4=ECE 425 CLASS NOTES 2000DR.

10 ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax)175amount of light collected by a camera?