  We define a new framework that unifies the filtration and mapper approaches
from TDA, and present efficient algorithms to compute it. Termed the box
filtration of a PCD, we grow boxes (hyperrectangles) that are not necessarily
centered at each point (in place of balls centered at points). We grow the
boxes non-uniformly and asymmetrically in different dimensions based on the
distribution of points. We present two approaches to handle the boxes: a point
cover where each point is assigned its own box at start, and a pixel cover that
works with a pixelization of the space of the PCD. Any box cover in either
setting automatically gives a mapper of the PCD. We show that the persistence
diagrams generated by the box filtration using both point and pixel covers
satisfy the classical stability based on the Gromov-Hausdorff distance. Using
boxes also implies that the box filtration is identical for pairwise or higher
order intersections whereas the VR and Cech filtration are not the same.
  Growth in each dimension is computed by solving a linear program (LP) that
optimizes a cost functional balancing the cost of expansion and benefit of
including more points in the box. The box filtration algorithm runs in
$O(m|U(0)|\log(mn\pi)L(q))$ time, where $m$ is number of steps of increments
considered for box growth, $|U(0)|$ is the number of boxes in the initial cover
($\leq$ number of points), $\pi$ is the step length for increasing each box
dimension, each LP is solved in $O(L(q))$ time, $n$ is the PCD dimension, and
$q = n \times |X|$. We demonstrate through multiple examples that the box
filtration can produce more accurate results to summarize the topology of the
PCD than VR and distance-to-measure (DTM) filtrations. Software for our
implementation is available at https://github.com/pragup/Box-Filteration.
