I was doing a research about retrive element densities from emission lines intensity observed by spacecraft. Follow the symbols of https://en.wikipedia.org/wiki/Abel_transform, suppose that intensity function $F(x,y)$ is
$$ F(x,y)=\int_{|y|}^{\infty} \frac{r f(r)}{(r^2-y^2)^{5/2}} dr+\int_{|y|}^{\sqrt{x^2+y^2}} \frac{r f(r)}{(r^2-y^2)^{5/2}} dr $$
where ${x,y}$ is the location of spacecraft and $f(r)$ is the density function. The power 5/2=1/2+2 is combined by 1/2 in normal Abel Tranform and 2 from the "The Inverse Square Law" since the light is dim when source is far away.
Suppose I know several values of $F(x,y)$ at $N$ points: $\{\{x_i,y_i,F(x_i,y_i)\}|i=1,2,3...,N\}$, then what should be my best estimation for $f(r)$?
For a special case, on the limit $x\rightarrow\infty$, $$ F(\infty,y)=2\int_{|y|}^{\infty} \frac{r f(r)}{(r^2-y^2)^{5/2}} dr $$ how could I retrive $f(r)$ from $\{\{y_i,F(\infty,y_i)\}|i=1,2,3...,N\}$?