  In Topological Data Analysis, filtered chain complexes enter the persistence
pipeline between the initial filtering of data and the final persistence
invariants extraction. It is known that they admit a tame class of
indecomposables, called interval spheres. In this paper, we provide an
algorithm to decompose filtered chain complexes into such interval spheres.
This algorithm provides geometric insights into various aspects of the standard
persistence algorithm and two of its run-time optimizations. Moreover, since it
works for any filtered chain complexes, our algorithm can be applied in more
general cases. As an application, we show how to decompose filtered kernels
with it.
