For the implementation of a dynamic Radontransform I'm looking for a set of functions $f_\alpha:\mathbb{R}\to\mathbb{R}$ with very special condtions, where $\alpha$ is in an arbitrary interval of $\mathbb{R}$. The conditions are, that for every $\alpha$ in that interval holds
- $f_\alpha$ is a diffeomorphism
- $f_\alpha$ has an explicit inverse
- $f_\alpha([-2,2])\subset [-2,2]$
- $f_\alpha$ is not linear for "most" alphas ($f_\alpha(x)=\alpha x+1$ is not valid, but $f_\alpha$ can be linear for a few values of alpha)
I already found $f_\alpha(x)=\alpha(\beta(\alpha)x^3+x)$ for certain intervals of $\alpha$ and $\beta$. The inverse is a but ugly but can be given with cardanos method.
I would extremly appreciate any suggestion of other functions.
NOTE: Ideally you can chose one $\alpha$ in that intervall, such that $f_\alpha(x)=x$
Does $f_\alpha$ have to be a diffeomorphism on the whole of $\Bbb R$, or just on $(-2,2)\to(-2,2)$? If it's the latter, you could use $$ f_\alpha(x) = 4\left(\frac {x+2}{4}\right)^\alpha - 2, $$ for $\alpha\in(0,\infty)$. This is easy to invert (in fact $\left(f_\alpha\right)^{-1} = f_{1/\alpha}$), and $f_1(x) = x$. It's of course just a shifted and scaled version of $x\mapsto x^\alpha:[0,1]\to[0,1]$.
The following family might be more elegant/useful, in that $f_\alpha$ is symmetric around the line $y=-x$: $$ f_\alpha(x) = 4\left(1-\left(\frac{2-x}{4}\right)^\alpha\right)^{\frac1\alpha} - 2. $$ for $\alpha\in(0,\infty)$. They're modelled after the unit circle in $(p=\alpha)$-norm. We have $$ f_\alpha^{-1}(x) = -4\left(1-\left(\frac{2+x}{4}\right)^\alpha\right)^{\frac1\alpha} + 2, $$ and $f_1(x) = x$.