  Fourier Transform Interferometry (FTI) is an appealing Hyperspectral (HS)
imaging modality for many applications demanding high spectral resolution,
e.g., in fluorescence microscopy. However, the effective resolution of FTI is
limited by the durability of biological elements when exposed to illuminating
light. Overexposed elements are subject to photo-bleaching and become unable to
fluoresce. In this context, the acquisition of biological HS volumes based on
sampling the Optical Path Difference (OPD) axis at Nyquist rate leads to
unpleasant trade-offs between spectral resolution, quality of the HS volume,
and light exposure intensity. We propose two variants of the FTI imager, i.e.,
Coded Illumination-FTI (CI-FTI) and Structured Illumination FTI (SI-FTI), based
on the theory of compressive sensing (CS). These schemes efficiently modulate
light exposure temporally (in CI-FTI) or spatiotemporally (in SI-FTI).
Leveraging a variable density sampling strategy recently introduced in CS, we
provide near-optimal illumination strategies, so that the light exposure
imposed on a biological specimen is minimized while the spectral resolution is
preserved. Our analysis focuses on two criteria: (i) a trade-off between
exposure intensity and the quality of the reconstructed HS volume for a given
spectral resolution; (ii) maximizing HS volume quality for a fixed spectral
resolution and constrained exposure budget. Our contributions can be adapted to
an FTI imager without hardware modifications. The reconstruction of HS volumes
from CS-FTI measurements relies on an $l_1$-norm minimization problem promoting
a spatiospectral sparsity prior. Numerically, we support the proposed methods
on synthetic data and simulated CS measurements (from actual FTI measurements)
under various scenarios. In particular, the biological HS volumes can be
reconstructed with a three-to-ten-fold reduction in the light exposure.
