On space-spectrum uncertainty analysis for spectrally programmable cameras

We introduce and analyze the concept of space-spectrum uncertainty for certain commonly used designs of spectrally programmable cameras. Our key finding states that, it is not possible to simultaneously acquire high-resolution spatial images while programming the spectrum at high resolution. This phenomenon arises due to a Fourier relationship between the aperture used for resolving spectrum and its corresponding diffraction blur in the spatial image. We show that the product of spatial and spectral standard deviations is lower bounded by λ 4 π ν 0 femto square-meters, where ν0 is the density of groves in the diffraction grating and λ is the wavelength of light. Experiments with a lab prototype validate our findings and its implication for spectral programming.