International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. B. ch. 2.5, p. 324

Another method of phase determination, which is best suited to refining or extending a partial phase set, is the Hoppe–Gassmann density modification procedure (Hoppe & Gassmann, 1968; Gassmann & Zechmeister, 1972; Gassmann, 1976). The procedure is very simple but also very computerintensive. Starting with a small set of (phased) , an initial potential map is calculated by Fourier transformation. This map is then modified by some realspace function, which restricts peak sizes to a maximum value and removes all negative density regions. The modified map is then Fouriertransformed to produce a set of phased structure factors. Phase values are accepted via another modification function in reciprocal space, e.g. , where p is a threshold quantity. The new set is then transformed to obtain a new and the phase refinement continues iteratively until the phase solution converges (judged by lower crystallographic R values).
The application of density modification procedures to electroncrystallographic problems was assessed by Ishizuka et al. (1982), who used simulated data from copper perchlorophthalocyanine within the resolution of the electronmicroscope image. The method was useful for finding phase values in reciprocalspace regions where the transfer function . As a technique for phase extension, density modification was acceptable for test cases where the resolution was extended from 1.67 to 1.0 Å, or 2.01 to 1.21 Å, but it was not very satisfactory for a resolution enhancement from 2.5 to 1.67 Å. There appear to have been no tests of this method yet with experimental data. However, the philosophy of this technique will be met again below in the description of the the maximum entropy and likelihood procedure.
References
Gassmann, J. (1976). Improvement and extension of approximate phase sets in structure determination. In Crystallographic computing techniques, edited by F. R. Ahmed, pp. 144–154. Copenhagen: Munksgaard.Google ScholarGassmann, J. & Zechmeister, K. (1972). Limits of phase expansion in direct methods. Acta Cryst. A28, 270–280.Google Scholar
Hoppe, W. & Gassmann, J. (1968). Phase correction, a new method to solve partially known structures. Acta Cryst. B24, 97–107.Google Scholar
Ishizuka, K., Miyazaki, M. & Uyeda, N. (1982). Improvement of electron microscope images by the direct phasing method. Acta Cryst. A38, 408–413.Google Scholar