A Variable Density Sampling Scheme for Compressive Fourier Transform Interferometry

Fourier Transform Interferometry (FTI) is an appealing Hyperspectral (HS) imaging modality in the applications demanding 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 biological elements suffer from photo-bleaching, i.e., a photochemical alteration. In this context, the acquisition of biological HS volumes based on Nyquist sampling rate of Optical Path Difference (OPD) axis leads to unpleasant trade-offs between spectral resolution, quality of 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, that efficiently modulate light exposure temporally (in CI-FTI) or spatiallytemporally (in CA-FTI). Leveraging a variable density sampling strategy we provide an optimum illumination and aperture coding, so that the light exposure imposed on a biological specimen is minimized while the spectral resolution is preserved. Our theoretical analysis consists in two parts, (i) the trade-off between exposure intensity and the quality of 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 modification. The reconstruction of HS volumes from compressive sensing FTI measurements relies on an `1-norm minimization problem promoting 3D HS sparsity prior. Numerically, we support the proposed methods on synthetic data and simulated compressed sensing measurements (from actual FTI measurements) under various scenarios. In particular, the HS volume of biological samples that are linearly mixed with fluorochrome spectral signatures can be reconstructed with a 3-10 fold reduction of exposure.