Illustrating this problem, the crosses in Fig. 5d represent all the grid points that made positive contributions in any of the preliminary sequence of low-resolution fits (which covers all grid points of the high-resolution grid, as described above). All the dots (red and blue) are the positive solution components of the exact high-resolution solution. They are colored blue if they coincide with a cross, i.e., have been correctly identified in the first stage as being part of the solution, and they are colored red if they were never part of any low-resolution fit and were therefore excluded from the second-stage grid. If the analysis proceeds with the second-stage grid (i.e., preconstraining the analysis to the values indicated by crosses in Fig. 5d) , we arrive at the solution shown in Fig. 5c . This is very different from the true high-resolution solution shown in Fig. 5a . Thus, the second stage cannot correct for the errors that occur from a naïve subdivision of grids in (I). Fig. 4 Magnitude of the elements of the normal matrix P calculated for the 10 9 100 model shown in Fig. 1 . P is symmetrical and has 1,000 9 1,000 values, plotted here by row and column numbers as indicated in the abscissa and ordinate of the picture, and the values P kl j j are plotted using the color scale. In principle, the nomenclature indexing the 10 9 100 grid points for the f r 9 s grid to form the vector of 1,000 parameters is arbitrary. a Here, all grid points are sorted by increasing s-value, i.e., (s 1 , f r,1 ), (s 1 , f r,2 ),…(s 1 , f r,10 ), (s 2 , f r,1 ),…(s 2 , f r,10 ),…(s 100 , f r,10 ). As can be discerned from the smooth appearance, the matrix elements are not strongly dependent on the f r -value. b The same matrix can be reordered to reflect subdivision along ten regular subgrids c, each of the form (s 10(c-1)?1 , f r,1 ), (s 10(c-1)?1 , f r,2 ),…(s 10(c-1)?1 , f r,10 ), (s 10(c-1)?2 , f r,1 ), (s 10(c-1)?2 , f r,2 ),…(s 10(c-1)?2 , f r,10 ),…(s 10(c-1)?10 , f r,1 ), (s 10(c-1)?10 , f r,2 ),… (s 10(c-1)?10 , f r,10 ) with c = 1…10. Each of the subgrids represents an evenly spaced 10 9 10 grid with origin shifted by D 2 s = 0.1505 S. c The idea that one could determine a high-resolution size-and-shape distribution from merging the results obtained separately in fits with subgrids corresponds to the assumption that there be no correlation between points from the different grids, i.e., that P can be subdivided into the ten submatrices P (c) . For the present data, this corresponds to the solution of the problem with a normal matrix as shown in c. Clearly, this is very different from the true matrix shown in b b Eur Biophys J (2010) 39: 1261-1275 1269 Brookes mentions multiple stages of the sequence (I) and (II) with different mesh sizes D 2 s and D 2 f r , and an ''iterative refinement'' of the procedure that utilizes in stage (I) the coarse starting grids that have been extended with populated points from the results of stage (II) of the previous iteration (Brookes et al. 2009 ). The same fundamental concerns apply to this iteration. To the extent that the results from (II) may not contain the grid points of the exact solution, it is unclear how the inclusion of these additional grid points would aid in the recognition of the correct solution. Even if the added grid points in (I) do represent part of the correct solution, it is not certain that they would be correctly maintained as part of the solution after (II). Empirically, the Demeler laboratory reports convergence of this iteration series in the absence, but not in the presence, of systematic noise corrections to the data (Brookes et al. 2006) . Even if the iteration does converge, it is unclear whether it is convergent to the correct solution.