The quantum theory of atoms in molecules (QTAIM), developed by Bader and co-workers, is one of the most popular ways of extracting chemical insight from the results of quantum mechanical calculations. One of the basic tasks in QTAIM is to locate the critical points of the electron density and calculate various quantities (density, Laplacian, etc.) on them since these have been found to correlate with molecular properties of interest. If the electron density is given analytically, this process is relatively straightforward. However, locating the critical points is more challenging if the density is known only on a three-dimensional uniform grid. A density grid is common in periodic solids because it is the natural expression for the electron density in plane-wave calculations. In this article, we explore the reconstruction of the electron density from a grid and its use in critical point localization. The proposed reconstruction method employs polyharmonic spline interpolation combined with a smoothing function based on the promolecular density. The critical point search based on this reconstruction is accurate, trivially parallelizable, works for periodic and non-periodic systems, does not present directional lattice bias when the grid is non-orthogonal, and locates all critical points of the underlying electron density in all tests studied. The proposed method also provides an accurate reconstruction of the electron density over the space spanned by the grid, which may be useful in other contexts besides critical point localization.