  Dwarf early-type galaxies (ETGs) display a rich diversity in their
photometric, structural, and dynamical properties. In this work, we address
their structural complexity by studying with deep imaging a sample of nine
dwarf ETGs from the Virgo galaxy cluster, characterized by having faint disk
features, such as bars and spiral arms, which lie mostly hidden within the
bright diffuse light of the galaxies. We present a new, robust method that aims
to identify and extract the disk substructure embedded in these dwarf ETGs. The
method consists in an iterative procedure that gradually separates a galaxy
image into two components; the bright, dominant diffuse component, and the much
fainter, underlying disk component. By applying it to the dwarf ETG sample, we
quantify their disk substructure and find that its relative contribution to the
total galaxy light ranges between 2.2 to 6.4% within two effective radii. We
test the reliability of the method, and prove that it is accurate in recovering
the disk substructure we introduce in mock galaxy images, even at low
disk-to-total light fractions of a few percent. As a potential application of
the method, we perform a Fourier analysis on the extracted disk substructures
and measure the orientation, length, and strength of the bars, and the pitch
angle and strength of the spiral arms. We also briefly discuss a scenario based
on the numerical simulations presented in our companion paper, Brought to Light
II: Smith et al. 2021, in which we investigate the origins of the substructure
in such dwarf systems.
