TITLE: partially reconstruct information of function convoluted with boxcar kernel
QUESTION [5 upvotes]: the function (f) I want to reconstruct partially could look like this:

The following properties are known:

It consists only of alternating plateau (high/low).
So the first derivation is zero respectively undefined at the edges.

The function was convoluted with a kernel fulfilling the following conditions:

It is a boxcar function
Its center is at x=0
Its integral is 1.

I want to reconstruct only the positions of the edges of the original function (f) from the convolution result (c). So just these positions are of interest to me:

If the convolution kernel width (k) is less than the minimum plateau width (b, 40 in the example above) of f, c looks as follows:

(The width of the box car convolution kernel here is k=31.)
In that case it is easy to reconstruct the edge positions:
I look for (possibly broad) extrema, and in between to neighbours [e1_x, e1_y] and [e2_x, e2_y] (one of them is a minimum and one a maximum of course), I search the x0 fulfilling: c(x0) = (e1_y + e2_y) / 2.
The reconstructed edge positions look like that:

But if k > b my approach fails:

(k=57)
Is there a possibility to calculate the original edge positions in f, if g (and so k) and c are known, also for the k>b cases?

REPLY [2 votes]: Just adding what I found as a solution for me:
I already stated that it is easy to reconstruct the edges for cases with k < b:

For cases with b < k < 2*b one can use the commutativity of the convolution operator: fg = gf
Then the wrongly measured distance between the two edges is K, but the interesting value (B) can just be found in the x length of the slope (diagonal part of the result). :-)
I also found out that there are already fine implementations for deconvolution with completely known kernels out there. This one i.e. works like a charm in my case: http://docs.scipy.org/doc/scipy/reference/generated/scipy.signal.deconvolve.html