Example for an Unfolding Problem
What means unfolding of measured data ?
Fotos being out of focus and blurred or tape-recordings with background noise are typical examples for imperfect representated information.
In fotography two types of influences have to be considered. There is one type which can be avoided by correct application of technique (e.g. movement during shooting or wrong choice of focus).The other type deals with the limited resolution.
The question arises whether such influences can be corrected later by a suitable mathematical program.
Typical examples may be found in astrophysics or in tomography, where it is tried to improve the resolution afterwards, taking into account some model-assumptions. The procedure performed in practice is shown in the following.
Let us assume the camera has moved during taking a photo, or let us assume that there was a movement of the paper at the copying machine during copying by 30 pixels in y-diretion and by 10 pixels in x-direction.
The result of this movement is shown in the following picture:
Since we know the coordinates and the velocity of the movement, we may try to reconstruct the picture.
In the following a simulation of this process is shown.
The grey value G(x,y) of a certain pixel is the sum of the grey values of the pixels in the neighbourhood in a well defined way, defined by the transmission function T(x-x',y-y') (sometimes called response function). There is a integral representation:
and the function F(x',y') has to be determined from the integral equation by a suitable mathematical procedure (unfolding procedure).
The unfolding solution was obtained using one of the codes of the HEPRO package. Here, the GRAVELW code was used. The results after 3 iterations, after 10 iterations show how the original can be reconstructed.
Result after 3 iterations:
Result after 10 iterations:
Original :
