A data-independant transformation to discretize a range of values non-uniformly I am sure this is trivial, but I am looking for a transformation that nonuniformly discretizes all values of a range into several bins. The bins should be variant and I'd like them to be smaller around the middle of my distribution and grows larger around both ends. Say, instead of one fixed \delta as the step size for a uniform binning as below:

I'd like to have the following binning:
 
Where around the middle of the distribution, I have $\delta_{min}$ and they grow gradually (linearly) to $\delta_{max}$ on both ends. 
Do we have a transformation/regularization which does the following without the need to know the value of each element inside the bin and can generate all variable deltas at runtime? For instance, for the uniform binning I can write $d=\frac{maxValue- minValue}{\#ofBins}$ and I can update the values of all members of each bin (W) to the middle of each bin.
 A: Use quantiles.
The lowest 10% are the first bin, the next 10% are the second bin ...
If you hypothesize a data distribution, you can also use quantiles of the distribution for such a binning.
For example to split a standard normal distribution into 11 bins, you would use:
$-\infty$ -1.28155 -0.84162 -0.52440 -0.25335 0.00000
0.25335 0.52440 0.84162 1.28155 $\infty$
A: The most general solution uses a nondecreasing function $F:(0,1]\to[0,1]$ and the desired interval $(a,b]$ of values to bin.
To create $n$ bins, divide the unit interval $(0,1]$ into $n$ non-overlapping sections
$$B_i = \left(\frac{i-1}{n}, \frac{i}{n}\right],\ i=1, 2, \ldots, n$$
and assign any number $x$ with $a\lt x \le b$ into bin $i$ where $i$ is the unique value with
$$F_{a,b}(x) = F\left(\frac{x-a}{b-a}\right) \in B_i.$$
When $x$ is uniformly distributed between $a$ and $b,$ the expected proportion of values in bin $i$ therefore is $F_{a,b}(i/n)- F_{a,b}((i-1)/n).$
In the plots of the question, the blue graphs represent the derivative of $F$ and these expected proportions, by the Fundamental Theorem of Calculus, are the relative areas between the vertical lines.  You may construct such an $F,$ then, by stipulating what you want the expected "concentration" of points at each value of $x$ to be--a non-negative number $f(x)$--and computing
$$F_{a,b}(x) = C\int_a^x f(t)\mathrm{d}t.$$
The value of $C$ is chosen to make $F_{a,b}(b) = 1.$
For examples, the usual uniform binning corresponds to the identity function $F$ on $(0,1]$ for which $f(x)=1.$ Logarithmic binning corresponds to $F_{a,b}(x) = \log(x/a)/log(b/a),$ for which $f_{a,b}(x) = 1/(ax\log(b/a)).$
