Re: Exposure function

Sorry for spamming. Just a little update ;)

I wrote:
If you're intrested in what I have come up so far, read on:

I scrapped the old function I had, because the slope of the linear part
wasn't constant among curves corresponding to different brightnesses. I
came up with a new one. This is also based on a sigmoid. The rationale was
again that in semilog-x scale the curve looks quite a bit like sigmoid. So
because I want the curve to be sigmoid

sigm(t) = 1/(1+exp(-w))

under semilog scale I substituted the parameter w = log(t) and obtained

f(t) = 1/(1+1/t).

This can be parametrized easily by two parameters

f(t,a,b) = 1/(1+1/(b*t^a)) = t^a / (t^a + 1/b).

this is pretty close to the curve given in the japanese website earlier
(http://www.asahi-net.or.jp/~rt6k-okn/ddp/digital.htm). Parameter 'a' is
camera dependant (I hope) and determines the slope of the linear part.
Parameter 'b' corresponds to the brightness (shifts the curve).

However, I probably need a third parameter for the curvature of the
shoulder (i.e., how smoothly the function approaches the saturation
point). Right now the curves seem to have too high curvature (i.e., they
approach the asymptote too fast):

http://www.kolumbus.fi/jahu/images/curves.png

It was actually quite easy to modify the slope and curvature independently.
I used an additional exponent g1 (note 'a' has changed to 'g0'):

s = slope, c = "curvature", b = brightness
g0 = sqrt(s*c)
g1 = sqrt(s/c)
( 1 )^g1
f(t,g0,g1,b) = (-----------------------)
( 1 + 1/( b * t^g0 ) )

However, this is not the function the camera is "using", because if I fit
the slope "accurately" i'll get inaccurate "curvature" (curvature in
quotation marks as it probably isn't the curvature by definition) and vice
versa. Thus the shape of the function must be slightly wrong, although many
times it fits amazingly well.

Perhaps it isn't really important if the curvature has a large error. Maybe
it is more important to fit the linear part correctly in order to solve the
shift which corresponds to the brightness. However, the slope might be
quite sensitive to noise, especially when large portion of the linear part
is close to 0.

--
Jani Huhtanen
Tampere University of Technology, Pori
.

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