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Subject: BRDF stochastic sampling


I'm a student, currently working on rendering materials using an
isotropic Ward BRDF.
For that, I use photon mapping and I need to sample this model

I use the formula from Ward's paper :
delta = atan( alpha*sqrt( -log(u) ) )

but I need the probability distribution function from which it comes from.

So I inverted the formula to obtain (assuming that "u" has been replaced
by "1-u" as u is uniform) the cumulative pdf :
u = 1 - exp( -tan(delta)2 / alpha2 )

I differentiated it to obtain the real pdf and obtained :
pdf = 2*tan(delta)/(alpha2 * cos(delta)2 ) * exp(- tan(delta)2 / alpha2)

At this point I have several problems :
- surfing the web, I've found a different formula for Ward pdf (for
example in http://www.iit.bme.hu/~szirmay/scook.pdf ) :
pdf = 1/(alpha2 * pi * cos(delta)3 ) * exp(- tan(delta)2 /
for which I don't know why it is different...
- to obtain a reference image, I was thinking of the reject method...
but exp(- tan(delta)2/m2) / (4*pi*alpha2) from Ward brdf is far from
being a pdf, even with alpha less than 0.2 (the integral is between 0.6
to 1.3 for the few values of alpha that I have tested)

So, could you explain from which pdf the stochastic sampling formula of
Ward BRDF comes ?

Thanks in advance !!!

Nicolas Bonneel


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