Distinguishing whether a signal corresponds to a single source or a limited number of highly overlapping point spread functions (PSFs) is a ubiquitous problem across all imaging scales, whether detecting receptor-ligand interactions in cells or detecting binary stars. Super-resolution imaging based upon compressed sensing exploits the relative sparseness of the point sources to successfully resolve sources which may be separated by much less than the Rayleigh criterion. However, as a solution to an underdetermined system of linear equations, compressive sensing requires the imposition of constraints which may not always be valid. One typical constraint is that the PSF is known. However, the PSF of the actual optical system may reflect aberrations not present in the theoretical ideal optical system. Even when the optics are well characterized, the actual PSF may reflect factors such as non-uniform emission of the point source (e.g. fluorophore dipole emission). As such, the actual PSF may differ from the PSF used as a constraint. Similarly, multiple different regularization constraints have been suggested including the <i>l</i><sub>1</sub>-norm, <i>l</i><sub>0</sub>-norm, and generalized Gaussian Markov random fields (GGMRFs), each of which imposes a different constraint. Other important factors include the signal-to-noise ratio of the point sources and whether the point sources vary in intensity. In this work, we explore how these factors influence super-resolution image recovery robustness, determining the sensitivity and specificity. As a result, we determine an approach that is more robust to the types of PSF errors present in actual optical systems.
In this paper, we describe the use of various methods of one-dimensional spectral compression by variable selection as well as principal component analysis (PCA) for compressing multi-dimensional sets of spectral data. We have examined methods of variable selection such as wavelength spacing, spectral derivatives, and spectral integration error. After variable selection, reduced transmission spectra must be decompressed for use. Here we examine various methods of interpolation, e.g., linear, cubic spline and piecewise cubic Hermite interpolating polynomial (PCHIP) to recover the spectra prior to estimating at-sensor radiance. Finally, we compressed multi-dimensional sets of spectral transmittance data from moderate resolution atmospheric transmission (MODTRAN) data using PCA. PCA seeks to find a set of basis spectra (vectors) that model the variance of a data matrix in a linear additive sense. Although MODTRAN data are intricate and are used in nonlinear modeling, their base spectra can be reasonably modeled using PCA yielding excellent results in terms of spectral reconstruction and estimation of at-sensor radiance. The major finding of this work is that PCA can be implemented to compress MODTRAN data with great effect, reducing file size, access time and computational burden while producing high-quality transmission spectra for a given set of input conditions.