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ECE 425 CLASS NOTES – 2000 DR. ROBERT A. SCHOWENGERDT 520 621-2706 (voice), 520 621-8076 (fax) 157 RADIOMETRY OF LAMBERTIAN SOURCES Relate M λ and L λ Valuable exercise in spherical coordinate integration Area element of sphere, radius r Start with radiance to get flux within area dA 2 φ θ L dA 2 dA 1 r dA 2 rd φ r θ d θ sin = r 2 θ d θ d φ sin =
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Transcript of Lambertian Sources

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RADIOMETRY OF LAMBERTIAN SOURCESRelate M and L Valuable exercise in spherical coordinate integrationL dA2

rdA1

Area element of sphere, radius rd A 2 = rdr sin d = r sin dd2

Start with radiance to get flux within area dA2157DR. ROBERT A. SCHOWENGERDT schow[email protected] 520 621-2706 (voice), 520 621-8076 (fax)

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d L ( , ) = -----------------------------d A 1 cos d d = L ( , ) cos d A 1 d d A2 = L ( , ) cos d A 1 --------2 r = L ( , ) cos d A 1 sin dd2

2

Total flux in hemisphere above source2 2

= d A1

d 0 0

L ( , ) cos sin d

Now, in general L = L(,), but for a Lambertian source

158DR. ROBERT A. SCHOWENGERDT schow[email protected] 520 621-2706 (voice), 520 621-8076 (fax)

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L = constant Therefore( sin ) = d A 1 2L ----------------2 = Ld A 12 2 0

Radiant exitance definition M = --------d A1 = L

159DR. ROBERT A. SCHOWENGERDT schow[email protected] 520 621-2706 (voice), 520 621-8076 (fax)

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Include spectral variationM = L or L = M

Can think of as having units of sr, which cancels the sr-1 units of L Note: M 2L, as one might guess since hemisphere is 2 srWhy?

160DR. ROBERT A. SCHOWENGERDT schow[email protected] 520 621-2706 (voice), 520 621-8076 (fax)

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TRANSMITTANCEDefined as ratio of transmitted output flux to input fluxfilter t() in out = in()

t ( ) =

out

in

161DR. ROBERT A. SCHOWENGERDT schow[email protected] 520 621-2706 (voice), 520 621-8076 (fax)

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REFLECTANCEThree types:L L E E specular (mirror) E directional diffuse (Lambertian) L

Most natural surfaces are approximately Lambertian

162DR. ROBERT A. SCHOWENGERDT schow[email protected] 520 621-2706 (voice), 520 621-8076 (fax)

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for < 40, snow and sand for < 60 At larger , natural surfaces tend to become directional

Reflectance of Lambertian surfaceL dA2

E

rdA1

Reflectance defined similarly to transmittance ( ) = out in

From earlier derivation163DR. ROBERT A. SCHOWENGERDT schow[email protected] 520 621-2706 (voice), 520 621-8076 (fax)

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out

= L d A 1

and = E d A1in

Therefore, ( ) = L E , 0 ( ) 1

Given reflectance of surface and incident irrradiance, can get radianceL = ( )E

164DR. ROBERT A. SCHOWENGERDT schow[email protected] 520 621-2706 (voice), 520 621-8076 (fax)

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BAND-AVERAGED IRRADIANCECascade spectral quantitites from source-to-surface-tosurface . . .E = (geometric factors) L t 1 ( ) t 2 ( ) 1 ( )

until arriving at a spectrally-integrating element, i.e. a detector,E total =

E S ( ) d

where S() is the detectors spectral sensitivity

Example of a specific detector S() is human vision system sensitivity V()

Etotal is the effective irradiance, because it is what is

165DR. ROBERT A. SCHOWENGERDT schow[email protected] 520 621-2706 (voice), 520 621-8076 (fax)

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detected and measured Band-averaged spectral irradianceE total E b = --------------------

S ( ) d0

166DR. ROBERT A. SCHOWENGERDT schow[email protected] 520 621-2706 (voice), 520 621-8076 (fax)

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RADIOMETRY OF OPTICAL SYSTEMStelescope collects light from point source light bucketaperture (diameter d) z0 point source (radiant intensity I) zi detector lens (or mirror)

assume: source and detector on optical axis object-to-sensor distance much greater than aperture167DR. ROBERT A. SCHOWENGERDT schow[email protected] 520 621-2706 (voice), 520 621-8076 (fax)

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diameter

z0 d

no transmission losses irradiance at lens aperture flux collected by aperture c = E A aperture d = E -------4 I d = ---- -------2 4 z02 2

E = I z0

2

(inverse square law)

flux at detector (no losses)

i = c

proportional to radiant intensity of source proportional to square of aperture diameter168DR. ROBERT A. SCHOWENGERDT schow[email protected] 520 621-2706 (voice), 520 621-8076 (fax)

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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 magnificationm = hi ho = zi zo

169DR. ROBERT A. SCHOWENGERDT schow[email protected] 520 621-2706 (voice), 520 621-8076 (fax)

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aperture (diameter d) z0 Lambertian source (radiance Lo, height ho, area Ao) zi image (height hi, area Ai) lens (or mirror)

flux collected by aperturec = Lo Ao A = L o A o ---2 zo L o A o d 2 = ------------ -------2 4 zo

NOTE: similarity to light bucket equation flux at detector (no losses)i = c

170DR. ROBERT A. SCHOWENGERDT schow[email protected] 520 621-2706 (voice), 520 621-8076 (fax)

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irradiance at detector

E i = c Ai c = ------------2 m Ao d = L o --------------2 2 m 4z o2

by definition of magnification (see Geometrical Optics)d E i = L o -------2 4z i2

define effective f-number N of sensor, then, E i = L o --------2 4N

N = zi d

Camera Equation

NOTE: proportional to radiance of source171DR. ROBERT A. SCHOWENGERDT schow[email protected] 520 621-2706 (voice), 520 621-8076 (fax)

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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 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22What is rationale for the above preset values of N?

172DR. ROBERT A. SCHOWENGERDT schow[email protected] 520 621-2706 (voice), 520 621-8076 (fax)

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Camera imaging reflecting object Irradiance on object E Reflectance of object

source

aperture (diameter d) irradiance E image (height hi, area Ai) lens (or mirror)

Lambertian reflector (reflectance o, height ho, area Ao)

From earlier derivation:

Eo E i = --------2 4N173

DR. ROBERT A. SCHOWENGERDT

[email protected]

520 621-2706 (voice), 520 621-8076 (fax)

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Camera imaging extended reflecting object

source

aperture (diameter d) irradiance E

image (height hi, area Ai) Lambertian reflector (reflectance o, height ho, area Ao) lens (or mirror)

Cos4 Law applies:Eo 4 E i ( ) = --------- ( cos ) 2 4N

With object irradiance and reflectance fixed, what is the only way to increase the174DR. ROBERT A. SCHOWENGERDT schow[email protected] 520 621-2706 (voice), 520 621-8076 (fax)

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amount of light collected by a camera?

175DR. ROBERT A. SCHOWENGERDT schow[email protected] 520 621-2706 (voice), 520 621-8076 (fax)