We address the problem of recognition and growth of ice nuclei in simulation of supercooled bulk water. Bond orientation order parameters based on the spherical harmonics analysis are shown to be ineffective when applied to ice nucleation. Here we present an alternative method which robustly differentiates between hexagonal and cubic ice forms. The method is based on accumulation of the maximum projection of bond orientations onto a set of predetermined vectors, where different terms can contribute with opposite signs with the result that the irrelevant or incompatible molecular arrangements are damped out. We also introduce an effective cluster size by assigning a quality weight to each molecule in an ice-like cluster. We employ our cluster analysis in Monte Carlo simulation of homogeneous ice formation. Replica-exchange umbrella sampling is used for biasing the growth of the largest cluster and calculating the associated free energy barrier. Our results suggest that the ice formation can be seen as a two-stage process. Initially, short tetrahedrally arranged threads and rings are present; these become correlated and form a diffuse ice-genic network. Later, hydrogen bond arrangements within the amorphous ice-like structure gradually settle down and simultaneously 'tune-up' nearby water molecules. As a result, a well-shaped ice core emerges and spreads throughout the system. The process is very slow and diverse owing to the rough energetic landscape and sluggish molecular motion in supercooled water, while large configurational fluctuations are needed for crystallization to occur. In the small systems studied so far the highly cooperative molecular rearrangements eventually lead to a relatively fast percolation of the forming ice structure through the periodic boundaries, which inevitably affects the simulation results.