An image acquired by a camera consists of measured intensity values which are related to scene radiance by a function called the camera response function. Knowledge of this response is necessary for computer vision algorithms which depend on scene radiance. One way the response has been determined is by establishing a mapping of intensity values between images taken with different exposures. We call this mapping the intensity mapping function. In this paper, we address two basic questions. What information from a pair of images taken at different exposures is needed to determine the intensity mapping function? Given this function, can the response of the camera and the exposures of the images be determined? We completely determine the ambiguities associated with the recovery of the response and the ratios of the exposures. We show all methods that have been used to recover the response break these ambiguities by making assumptions on the exposures or on the form of the response. We also show when the ratio of exposures can be recovered directly from the intensity mapping, without recovering the response. We show that the intensity mapping between images is determined solely by the intensity histograms of the images. We describe how this allows determination of the intensity mapping between images without registration. This makes it possible to determine the intensity mapping in sequences with some motion of both the camera and objects in the scene.