A problem I’ve been wrestling with for a while is how to legibly plot spatial densities with wide dynamic range. Ink on paper is okay for about 10 dB of dynamic range; reversing the video and plotting white on black on a computer screen gives you about 20 dB; and by blooming high-density regions outward at the cost of spatial resolution, which is often an acceptable tradeoff, I think you can do 30 dB. But how can we efficiently calculate a bloom whose area accurately represents the total magnitude within?
In 3-D rendering, convolution surfaces are implicit surfaces of a function defined by convolving a “field function” with a “geometry function”, a generalization of metaballs. McCormack and Sherstyuk 1997 chose the field function 1/(1 + s²r²)² as an approximation of a Gaussian that is more tractable to solve in closed form.
Metaballs (implicit curves of Σᵢdᵢ²/rᵢ², where rᵢ is the distance to center point i) in 2-D have the advantage that if you put two of them on top of each other, the total area is twice the total area they would have if they were far apart. There's a transition region in the middle where they're nearly joined and a bridge joining them adds about 20% to their total area. Unfortunately, they are difficult to calculate efficiently.
Is there perhaps an efficient way to convolve metaballs, or an approximation thereof, with the points you want to plot, in order to get the size of the total area to fill with light?
(A different approach would be to use a vector CRT or laser-light display, which can natively reach 30dB of dynamic range without sacrificing spatial resolution — indeed, improving it.)