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We investigate methods of estimating a background image frame for subtraction from a data frame for use when a more suitable measured background frame is not available. We define background as any signal component that is not attributable to the phenomenon currently under investigation. We describe a technique that is based on pixel-by-pixel least-squares regression of images for computing a background frame from available data. We argue that the same technique can be a useful quality-assurance tool for evaluating instrument performance. For example, it can help to separate image structure resulting from the reading process from structure resulting from the characteristics of the detector itself. We demonstrate that background estimation can be nontrivial by comparing the results of different background estimation procedures by using data obtained from a CCD array detector. We investigate the temperature-dependent contributions of the detector and readout electronics to the total signal as a demonstration of the diagnostic capabilities of least-squares image regression.