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SKIING WITH LAMBERT  --  SNOW, ILLUMINATION AND MACHINE VISION  (continued)




4. Lost Structure -- How Snow Turns Invisible
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Snow looks white, and sometimes even blue; o.k. .
But why do we sometimes see its surface structure (para. 1, fig. 1_a), and sometimes we do not (para. 1, fig. 1_b)?

Well, this is where we obviously need LAMBERT. (Johann Heinrich LAMBERT, german physicist and philosopher, 1728 - 1777.)



4.1 Lambert and Simple Mathematics
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Key words:
Lambert | luminous intensity I | angle | luminance L



First, we need to know what "LUMINOUS INTENSITY I " is. You can find the definition in "Measuring Light" para. 5.2.2.

LAMBERT found that a perfectly diffuse radiator (or reflector) emits a LUMINOUS INTENSITY  I  that depends on the ANGLE  (epsilon)  between the emitted ray and the surface normal:

I[(epsilon)] = I(0) * cos (epsilon) (eqn. 1)

where I(0) = luminous intensity in normal direction i.e., luminous intensity at ANGLE (epsilon) = 0

The function of (eqn. 1) is shown in fig. 4_a.

fig. 4_a: I = I(0) * cos (epsilon); 12 kByte

fig. 4_a: LAMBERT's Cosine Law
The black circle represents the function.
If you allow all possible directions of  I[(epsilon)]  in space, then the black circle becomes a full sphere.


A radiator (or reflector) of this kind is called a "LAMBERTian radiator".

So far, we looked at the area where light comes from. Now, let us consider the receiving end, that is our eye.
The brightness that our eye assigns to a surface element, is very closely related to the quantity "LUMINANCE  L  " in cd/m^2 .
LUMINANCE  L  is defined in "Measuring Light" para. 5.2.3 as
L = I / A1 (eqn. 2)

Where  A1  is the area of the emitting surface.

Now consider fig. 4_b concerning the effective size of a surface element  A1  upon which our eye looks at various ANGLEs  (epsilon) :

fig. 4_b: area A1, seen at angle (epsilon); 4 kByte
fig. 4_b: area A1, seen at angle (epsilon)

This effective size for the eye will shrink according to ANGLE (epsilon) :
A1[(epsilon)] = A1 * cos (epsilon) (eqn. 3)


So, if we look at a LAMBERT radiator under an ANGLE (epsilon) , we find with (eqn. 2) and (eqn. 3):
L[(epsilon)] = I / A1[(epsilon)] = I / [A1 * cos (epsilon)] (eqn. 4)


And, together with LAMBERT's cosine law from (eqn. 1):
L[(epsilon)] = I / [A1 * cos (epsilon)]
             = I(0) * cos (epsilon) / [A1 * cos (epsilon)]
L[(epsilon)] = I(0) / A1


(eqn. 5)


This function is absolutely independent of ANGLE (epsilon) that is, not varying over all ANGLEs!
The function is shown in fig. 4_c.
fig. 4_c: area A1, seen at angle (epsilon); 4 kByte

fig. 4_c: LAMBERT's Cosine Law, transformed to Luminances  L
The black half-circle represents the function.
If you allow all possible directions of  L[(epsilon)]  in space, then the black half-circle becomes a half-sphere.


Expressed in words:
A LAMBERTian radiator is always to be seen with the same luminance  L , no matter from which direction of (half-)space you look at it.




Continued: 4.2 Snow and its Different Illuminants


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Last modified Nov. 30st, 2009