Solution of an Abel-type integral equation in the presence of noise by quadratic programming

Abstract

The well-known integral transform $$i(r) = - \frac{1}{\pi }\int\limits_{x = r}^1 {\frac{{dI(x)}}{{\sqrt {x^2 - r^2 } }},} 0 \leqq r \leqq 1,I(1) = 0$$ arising in spectroscopy, corresponds to half-order differentiation by substitutingr 2 = 1 ?s,x 2 = 1 ? t. Therefore noise is amplified by transforming the measured functionI intoi. Two undesirable effects may arise: (a) lack of smoothness ini (r), (b) intervals in whichi(r) < 0, although for physical reasons we should havei(r) ? 0. After developing a heuristic theory of noise amplification we present a fitting technique for approximate computation ofi(r), using the extra informationi(r) ? 0 as a restriction. This leads to a quadratic programming problem.