The theory of the low-energy electron point source (LEEPS) microscope is reformulated in matrix form to readily account for multiple scattering. An algorithm is developed for the storage of the structure matrix and an iterative method is used to solve the matrix equation for the structure factor. Examples of small and large clusters of atoms are given to compare single and multiple scattering. A Kirchho 2Helmholtz transform is used for the reconstruction. We ®nd that in some cases the multiple scattering is too strong and reconstruction is not possible. We give examples which show that, even when multiple scattering is important, one can still obtain reconstructions that reveal the atomic structure both along and lateral to the optical axis. We also compare our results with those found in LEED. #