Before revision:  For such data sets, one must first classify their images into homogeneous groups , where each group corresponds to the same underlying structure , followed by reconstruction of a three-dimensional model from each of the homogeneous groups. This task has been proven to be extremely difficult .
After revision:  For such data sets, one must first classify their images into homogeneous groups , where each group corresponds to the same underlying structure . The fundamental importance of the heterogeneity problem in cryo-electron microscopy has drawn many research efforts, and resulted in significant progress in classification algorithms for heterogeneous data sets. While these algorithms are extremely useful and effective in practice, they lack rigorous mathematical analysis and performance guarantees .