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In many Fourier imaging applications, the presence of unaccounted for amplitude or phase errors in the Fourier domain data can lead to a degraded system impulse response and high sidelobes in the image domain. Historically, many methods for data-driven correction of these effects have been proposed, and numerical optimization of nonquadratic, p-norm image quality metrics has recently emerged as a robust solution. This paper presents a tutorial examination of the sources of image sidelobes in Fourier imaging applications, and studies the effectiveness of p-norm regularization algorithms under various experimental conditions. Several observations are made, including comments on robustness to noise and methods for tapered window design and energy-constrained sparse aperture imaging. Image examples are presented as experimental validation.