New Developments in Direct Shape Determination from Small-Angle Scattering. 2. Uniqueness

The problem of uniqueness of the low-resolution shape determination from small-angle scattering by isotropic monodisperse systems is considered. The particle shape is represented by the envelope function parameterized using spherical harmonics as described in a previous paper [Svergun & Stuhrmann (1991). Acta Cryst. A47, 736-744]. Computer simulations are made on the model bodies with sharp boundaries exactly represented by spherical harmonics. If the number of independent parameters describing the shape is 1 to 1.5 times the number of Shannon channels covered by the data set, the shape restoration is found to be unique and stable with respect to the random and systematic errors. The resolution limits of the straightforward shape determination are connected to the computational accuracy of the model intensities; with current algorithms, shapes described by 15 to 20 independent parameters can be uniquely determined. The results form a basis for an ab initio low-resolution shape determination in terms of spherical harmonics.