math - partially reconstruct information of function convoluted with boxcar kernel -

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the function (f) want reconstruct partially this: original function f

the following properties known:

  • it consists of alternating plateau (high/low).

  • so first derivation 0 respectively undefined @ edges.

the function convoluted kernel fulfilling following conditions:

  • it boxcar function

  • its center @ x=0

  • its integral 1.

i want reconstruct positions of edges of original function (f) convolution result (c). these positions of interest me: interesting edge positions in f

if convolution kernel width (k) less minimum plateau width (b, 40 in example above) of f, c looks follows: boxcar convolution result c k=31 (the width of box car convolution kernel here k=31.)

in case easy reconstruct edge positions: (possibly broad) extrema, , in between neighbours [e1_x, e1_y] , [e2_x, e2_y] (one of them minimum , 1 maximum of course), search x0 fulfilling: c(x0) = (e1_y + e2_y) / 2.

the reconstructed edge positions that: successfully reconstructed edge positions

but if k > b approach fails: failing in reconstructing edge positions (k=57)

is there possibility calculate original edge positions in f, if g (and k) , c known, k>b cases?


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