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Discrete linear and nonlinear inverse problems arise from many different imaging systems, exhibiting inherent ill-posedness wherein solution sensitivity to data perturbations prevails. This sensitivity is exacerbated by errors arising from imaging system components (e.g., cameras, sensors, etc.), necessitating the development of robust regularization methods to attain meaningful solutions. This talk starts with a review of distinct imaging systems and their mathematical formalism and subsequently introduces regularization techniques tailored for linear inverse problems. Then, we will look into the variable projection method, a powerful tool to address separable nonlinear least squares problems.