Spectral Properties of the Regularized Inversion of the Laplace Transform

Abstract Inversion of the Laplace integral equation, used in the laser scattering measurement of colloidal particle size distributions, presents sever numerical and experimental difficulties. In the presence of noise the variance of the inversion integral is infinite, indicating maximum uncertainty in the accuracy of the inversion. The regularized inversion of the Laplace intergral equation provides a convenient computational algorithm which requires no a priori knowledge of the unknown linewidth distribution. Using the eigenfunction decomposition of the Laplace kernel, the spectral properties of the regularized inversion may be seen. Regularized inversion represents a type of low pass filter which preserves the properties of the inversion spectrum at low frequencies, but provides a cutoff at a point controlled by the regularization parameter. This filtering reduces the variance of the inversion to a finite value. Regularized inversion is somewhere between optimal filtering and the abrupt truncation used in singular value decomposition and other similar methods. Two examples, a monodisperse and a bimodal linewidth distribution, are used to compare the performance of regularized inversion to that obtained through an optimal filter.