Here's a little spreadsheet to very simply illustrate how I'm thinking about
Hold down F9 and watch the iterations of random sampling cycle.
Lower the tightness value to increase the clustering at the higher values.
sqrt(x) is a nice little function ranging between 0 & 1 over 0 to 1.
The probability ranges from 0 to 2/3.
I calculate the function, and its probability and that's the top graph.
Next, the probability gets normalized then raised to a fractional power
representing the tightness term.
That importance-modified probability corresponds to an x value, which itself
corresponds to an F(x) value.
That gets plotted on the lower graph.
sin (x*(pi/2)) would give a (radially symmetric) spherical coordinate.
That's my implementation of employing the probability. Choosing a suitable
function to evaluate that has the same shape as the BRDF is challenge 1, and
choosing it so that it can be easily and quickly integrated is the second
Perhaps some sort of sigmoid shape would give you the peak and the F'(x)=0 at
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Download 'probability.xlsx.zip' (33 KB)