  A new class of high-contrast image analysis algorithms that empirically fit
and subtract systematic noise has lead to recent discoveries of faint exoplanet
/substellar companions and scattered light images of circumstellar disks. These
methods are extremely efficient at enhancing the detectability of faint
astrophysical signal, but they do generally create systematic biases in their
observed properties. This paper provides a general solution for this
outstanding problem. We present the analytical derivation of a linear expansion
that captures the impact of astrophysical over-subtraction and/or
self-subtraction these image analysis techniques. We examine the general case
for which the reference images of the astrophysical scene move azimuthally
and/or radially across the field of view as a result of the observation
strategy. Our new method is based on perturbing the covariance matrix
underlying any least-squares speckles problem, and propagating this
perturbation through the data analysis algorithm. We then demonstrate practical
applications of this new algorithm. We first consider the case of the spectral
extraction of faint point sources in IFS data and illustrate, using public
Gemini Planet Imager commissioning data, that our novel perturbation-based
Karhunen-Lo\`eve Image Processing Forward Modeling (KLIP-FM) can indeed
alleviate algorithmic biases. We then apply KLIP-FM to the problem associated
with the detection of point sources. We show how it decreases the rate of false
negatives (e.g missed planets) while keeping the rate of false positives
unchanged when compared to classical least-squares fitting methods. This can
potentially have important consequences on the design of follow-up strategies
of ongoing direct imaging surveys.
