A robust iterative unfolding method for signal processing. (English) Zbl 1107.94004

László, András

J. Phys. A, Math. Gen. 39, No. 44, 13621-13640 (2006).

Summary: It is a common problem in signal processing to remove a non-ideal detector resolution from a measured probability density function of some physical quantity. This process is called unfolding (a special case is the deconvolution), and it would involve the inversion of the integral operator describing the folding (i.e. the smearing of the detector). Currently, there is no unbiased method known in the literature for this issue (here, by unbiased we mean those approaches which do not assume an ansatz for the unknown probability density function). There is a well-known series expansion (Neumann series) in functional analysis for perturbative inversion of specific operators on Banach spaces. However, operators that appear in signal processing (e.g. folding and convolution of probability density functions), in general, do not satisfy the usual convergence condition of that series expansion. This paper provides some theorems on the convergence criteria of a similar series expansion for this more general case, which is not yet covered by the literature.
The main result is that a series expansion provides a robust unbiased unfolding and deconvolution method. For the case of the deconvolution, such a series expansion can always be applied, and the method always recovers the maximum possible information about the initial probability density function, thus the method is optimal in this sense. A very significant advantage of the presented method is that one does not have to introduce ad hoc frequency regulations etc, as in the case of usual naive deconvolution methods. For the case of general unfolding problems, we present a computer-testable sufficient condition for the convergence of the series expansion in question. Some test examples and physics applications are also given. The most important physics example shall be (which originally motivated our survey on this topic) the case of \(\pi ^{0} \rightarrow \gamma + \gamma\) particle decay: we show that one can recover the initial \(\pi ^{0}\) momentum density function form the measured single \(\gamma\) momentum density function by our series expansion.

MSC:

94A12

Signal theory (characterization, reconstruction, filtering, etc.)