The premise underlying (I) is that the results from independent application of different grids can be meaningfully combined. Following the idea of the Demeler laboratory that low-resolution subgrids can be ''refined into a grid of any desired resolution'' through their combination scheme, let us consider that putative final regular high-resolution grid, which would have mesh size Ds = D 2 s and Df r = D 2 f r . As shown above, one can actually solve the size-and-shape distribution problem directly using the standard algorithm with this full-sized high-resolution grid with even mesh size, via the normal equation (5) with the K 9 K matrix P and K 9 1 vector d, where K is the total number of species of the two-dimensional grid. In our example of Fig. 1 , K = 1,000 for the 10 9 100 grid that is of sufficient resolution to describe all aspects of the experiment. Now going backwards, one may consider our high-resolution grid to be represented by a total of C different equal-sized subgrids, each referenced with index c (e.g., ten grids of 10 9 10 resolution), such that merging all grid points of the subgrids produces the high-resolution grid. For each subgrid, one can solve the distribution with the normal matrix method, and it is easy to show that the relevant matrix equations are P ðcÞ c ðcÞ ¼ d ðcÞ ; where P ðcÞ are square submatrices of P of size (K/C) 9 (K/C) and d ðcÞ are subvectors of d of size 1 9 (K/C). One can use a nomenclature for the elements of the high-resolution grid such that the points are ordered in sequence of the low-resolution subgrids.