  While the direct detection of the dark-matter particle remains very
challenging, the nature of dark matter could be possibly constrained by
comparing the observed abundance and properties of small-scale sub-galactic
mass structures with predictions from the phenomenological dark-matter models,
such as cold, warm or hot dark matter. Galaxy-galaxy strong gravitational
lensing provides a unique opportunity to search for tiny surface-brightness
anomalies in the extended lensed images (i.e. Einstein rings or gravitational
arcs), induced by possible small-scale mass structures in the foreground lens
galaxy. In this paper, the first in a series, we introduce and test a
methodology to measure the power spectrum of such surface-brightness anomalies
from high-resolution Hubble Space Telescope (HST) imaging. In particular, we
focus on the observational aspects of this statistical approach, such as the
most suitable observational strategy and sample selection, the choice of
modelling techniques and the noise correction. We test the feasibility of the
power-spectrum measurement by applying it to a sample of galaxy-galaxy strong
gravitational lens systems from the Sloan Lens ACS Survey, with the most
extended, bright, high-signal-to-noise-ratio lensed images, observed in the
rest frame ultraviolet. In the companion paper, we present the methodology to
relate the measured power spectrum to the statistical properties of the
underlying small-scale mass structures in the lens galaxy and infer the first
observational constraints on the sub-galactic matter power spectrum in a
massive elliptical (lens) galaxy.
