We propose an automated calibration method with a closed-form solution for three-dimensional (3-D) integral imaging with randomly distributed sensors of unknown positions in 3-D space. In the method presented, a robust feature-detection algorithm is applied to the elemental image set, and the matching features are used to derive the relations between the corresponding sensors. Once all the sensor positions are determined, the 3-D objects are reconstructed with a ray backprojection reconstruction scheme. The 3-D objects can also be reconstructed from multiple perspectives in this method. To the best of our knowledge, this is the first report on 3-D integral imaging using automated sensor position estimation with a closed-form solution. This approach can provide 3-D object reconstruction without any known sensor positions in a 3-D space.