Compute the gradient of the luminance channel (Y).

Calculate a new luminance channel by solving the partial differential equation of Poisson. This is done using a Gauss-Seidel solver at the moment, and this is why it doesn't converge at speeds that are sufficient for realtime use.

Convert back to RGB color space and write the result image into your favorite format.

Result

The original photograph is on the left, and the result is shown on the right side. This is a photo I took from Rovaniemi, Finland.

Summary

One operation that is possible when working on HDR tone mapping is the reconstruction, through mathematical means, of the photo's luminance through solving partial differential equations on its gradient.