We consider the two-dimensional multi-target detection (MTD) problem of estimating a target image from a noisy measurement that contains multiple copies of the image, each randomly rotated and translated. The MTD model serves as a mathematical abstraction of the structure reconstruction problem in single-particle cryo-electron microscopy, the chief motivation of this study. We focus on high noise regimes, where accurate detection of image occurrences within a measurement is impossible. To estimate the image, we develop an expectation-maximization framework that aims to maximize an approximation of the likelihood function. We demonstrate image recovery in highly noisy environments, and show that our framework outperforms the previously studied autocorrelation analysis in a wide range of parameters.