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I am trying to use the techniques discussed in this paper to implement a BTF into PBRT. I am stuck on how to actually interpolate the weights of the four closest sampling directions.

At the BTF, I have converted $w_o$ and $w_i$ from cartesian to spherical coordinates in order to do a lookup in the table of BTFs. The samples are taken with the following properties:

$\theta$ light (and view angles) range from 0 to 75 in 15 degree increments. $\Delta\phi$ at each $\theta$ goes from 0 to (360 - $\Delta\phi$) at -, 60, 30, 20, 18 and 5 degree increments respectively. In a table:

$\theta$ $\Delta\phi$ (0 - (360 - $\Delta\phi$))
0 -
15 60
30 30
45 20
60 18
75 5

How can I determine the four closest sampling directions given $w_o$ and $w_i$? Retrieving the values/weights seems trivial, but I don't really have a clue on how to determine the four nearest directions and just don't know where to start.

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A simple table could be a good choice. There are a few approach's the suggestion I'll put here is just my take on the description, but it can be tweaked to your needs. In psuedocode:

const float table[6] = {0,60,30,20,18,5}; int table_index = asInt(floor(theta/15.0)); float result = table[table_index] * (theta > range);

The above is implemented as the step function in increments of 15. The last bit of the result is a range check that can be used to zap out of range values. (theta>range) evaluates to zero or one.

If this is a total miss on what you need let me know and I'll tweak it to help get closer.

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