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2D slices from a BRDF

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I'm going to do some prefiltering of env. maps for realtime rendering like in this paper ( http://people.csail.mit.edu/green/papers/NonLinPreFilt_lowres.pdf ) Now I figured out how I can solve the minimalisation problem on page 4 but I'm strugling to get the right input. What I need are basically slices from 'an isotropic BRDF'. Taking a fixed outgoing direction I kinda need the BRDF slice for that fixed outgoing, parametrised by the incoming direction. Unfortunaltely I have little or no idea how to exactly get those slices. Anyone any idea?

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