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Billboard Clouds

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I've been trying to implement Billboard Clouds, as described in: but I'm having some problems following some parts of it, most notably the appendix in bc03.pdf, where they calculate the range [rho_min,rho_max]. rho_plus_minus_ij = +- epsilon . Mi is the vertex and dj is the directional vector of a neighbour, but what is V? Then I'm a little unsure about the calculation of rho_min and rho_max: "We union these ranges on j and intersect them on i, that is rho_min = maximinj(rho_minus_ij) and rho_max = minimaxj(rho_plus_ij). My current rho_min and rho_max calculation looks like:
float rho_minus[3][4];
float rho_plus[3][4];

// Calculate the elements somehow...

float min_values[3] = { rho_minus[0][0], rho_minus[1][0],rho_minus[2][0] },
			max_values[3] = { rho_plus[0][0], rho_plus[1][0], rho_plus[2][0] };
for( uint i=0;i<3;++i ) 
    for( uint j=1;j<4;++j ) 
	if( rho_minus[j] < min_values) min_values = rho_minus[j];
	if( rho_plus[j] > max_values) max_values = rho_plus[j];
rho_min = MAX(MAX(min_values[0],min_values[1]),min_values[2]);
rho_max  = MIN(MIN(max_values[0],max_values[1]),max_values[2]);

Any help is appreciated.

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Sorry, I'm not actually answering your question.

I had a look at the forest rendering thesis first and then later found MemoryEfficientBillboardCloudsForBTFTexturedObjects.pdf and am currently implementing the hierarchical face clustering approach in the latter (but avoiding the BTFs) - it seems like a better method to me - though it does require calculating OBBs (oriented bounding boxes) which involves PCA (principle component analysis) and possibly calculating 3D convex hulls.

The Hough space approach (in forest rendering thesis) seems to require a bunch of extra measures to avoid using too much texture memory for the billboards.

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