We consider an acquisition system constituted by an array of sensors scanning an image. Each sensor produces a sequence of readouts, called a time series. In this framework, we discuss the image estimation problem when the time series are affected by noise and by a time shift. In particular, we introduce an appropriate data model and consider the least squares (LS) estimate, showing that it has no closed form. However, the LS problem has a structure that can be exploited to simplify the solution. In particular, based on two known techniques, namely, separable nonlinear LS and alternating LS, we propose and analyze several practical estimation methods. As an additional contribution, we discuss the application of these methods to the data of the photodetector array camera and spectrometer, which is an infrared photometer onboard the Herschel satellite. In this context, we investigate the accuracy and the computational complexity of the methods, using both true and simulated data.