Volume rendering, transfer functions,pre-integrated classification...oh my!

I plan on refactoring the volume rendering framework of our software suite in the near future to allow us to extend functionality beyond basic classification w/ 1d transfer functions. One of the main issues we have is the “slicing” artifact(aliasing) that appears in vol rendering due to not enough slices (and possibly incorrect alpha values)
For DVR similar to the link found in this post:

For Iso-Surfaces, similar to the link found in this post:

Anyway, I ran across a post mentioning a paper by Klaus and his group:

that I’d like to try implementing. Everything seems to make sense except for one thing. As I mentioned earlier, to compute colors I use a 1d texture which represents my transfer function for classification. In the paper, the idea is to use a 2d texture that is basically a function of slab thickness. The Color and Alpha values in that texture are computed from “integral functions” (?) of the transfer function…I don’t quite understand what that means. How can we represent a transfer function (which previously,was points on a “ramp”, for example, stored in a texture) so that it can be integrated at runtime for a specific value of my density field? :confused:
Thanks for any help/clarification here!

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