A Variable Density Sampling Scheme for Compressive Fourier Transform Interferometry.

Fourier Transform Interferometry (FTI) is an appealing Hyperspectral (HS) imaging modality in that applications demand high spectral resolution such as fluorescence microscopy. However, the effective resolution of FTI is indeed limited due to the durability of biological elements when exposed to illuminating light, since over-exposed elements become unable to fluoresce, i.e., photo-bleaching phenomenon. 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. In this paper we propose two variants of the FTI imager, i.e., Coded Illumination-FTI (CI-FTI) and Coded Aperture FTI (CA-FTI), based on the theory of compressed sensing. These schemes efficiently modulate light exposure temporally (in CI-FTI) or spatio-temporally (in CA-FTI). Leveraging a variable density sampling strategy, we provide optimal illumination and aperture coding strategies, so that the light exposure imposed on a biological specimen is minimized while the spectral resolution is preserved. Our theoretical analysis consists of two parts: (i) the trade-off between exposure intensity and the quality of the acquired HS volume for a fixed spectral resolution; (ii) the best HS volume quality for a fixed spectral resolution and constrained exposure intensity budget. Our contributions can be adapted to an FTI imager without hardware modifications. The reconstruction of HS volumes from compressive sensing FTI measurements relies on an $\ell_1$-norm minimization problem promoting a 3D sparsity prior. Numerically, we support the proposed methods on synthetic and experimental FTI data under various scenarios. In particular, the biological HS volume can be reconstructed with a three-to ten-fold reduction in the exposure intensity.