Mathematical Methods for Data Inversion

Abstract

The study of ill-posed linear inverse problems, characterised by Fredholm equations of the first kind, has important applications in many areas of science and technology. Problems of this type introduce some loss of information between what we may call a generalised “object” and its “image” under the linear mapping. This loss of information often makes the attempted inversion of image to object very diffcult. In practice the image is usually a discrete set of observations but the mapping may be retained essentially as a finite-rank integral operator using a fine discretisation throughout in the object space. This gives a very useful “automatic” interpolation of the recovered object at a sampling rate much higher than the conditioning of the problem would otherwise allow. Sub-micron particle sizing by Photon Correlation Spectroscopy is a problem where this type of inversion difficulty arises. Work is presented here which attempts to add in some of the lost information by making use of the a-priori constraints of positivity and known moments. This is achieved by the dual method of quadratic programming using some spectacular new mathematical results in this field.