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Garland and Heckbert had a nice algorithm for this sort of thing in their 1995 paper, "Fast Polygonal Approximation of Terrains and Height Fields." The paper is mainly devoted to height fields, obviously, but at the end they demonstrate that their algorithm is also effective at triangulating color images for Gouraud-shading as well.

I'd be curious to know how this stacks up in terms of speed and quality.

EDIT: Oh yes, and there's also "Image Compression Using Data-Dependent Triangulations" and "Survey of Techniques for Data-dependent Triangulations Approximating Color Images", both by Lehner et al., 2007. I don't mean to discourage you here, just pointing out the bar to be beaten. It's a cool idea.

To the OP: There are also several other tools for scattered data approximation/interpolation developed in the last few decades, both mesh-based and mesh-free. Linear interpolation using barycentric coordinates on a triangulation is fast (and might be the most practical method for this particular use case), but nowhere near as good a result as you can get via other methods.

See e.g. http://scribblethink.org/Courses/ScatteredInterpolation/scat...

Not sure if that applies for my purposes, since I'm not actually using linear interpolation barycentric coordinates (I don't think that's possible). The barycentric coordinates supply the gradient within themselves.

I may have to read further, though. That's a lot of math.

What you’re calling a gradient is also known as linear interpolation.

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