Symmetrization of illumination nonuniformity by thermal radiation in spherical hohlraum targets has been studied systematically for indirectly driven inertial confinement fusion. Numerical calculations have shown that the effect of X-ray re-emission on the illumination uniformity is quasi-linear. On the basis of a linear theory it is found that the nonuniformity of each mode of the X-ray source is considerably smoothed out by three different effects. The first is the geometrical effect, which accounts for the configuration and the number of X-ray converters. The second is the effect of single emission, which depends only on the hohlraum structure (area ratio). The third is the effect of multiple re-emission, which is equal to the reciprocal of the average circulation number of radiation in a hohlraum target. The author gives practical solutions regarding the required number of converters, Nc, in particular for heavy ion fusion systems. It is shown that Nc = 6 (hexahedron) is a necessary and sufficient condition to ensure tolerable symmetrical illumination (⩾ 1-2% rms)