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Many vision applications require precise measurement of scene radiance. The function relating scene radiance to image brightness is called the camera response. We analyze the properties that all camera responses share. This allows us to find the constraints that any response function must satisfy. These constraints determine the theoretical space of all possible camera responses. We have collected a diverse database of real-world camera response functions (DoRF). Using this database we show that real-world responses occupy a small part of the theoretical space of all possible responses. We combine the constraints from our theoretical space with the data from DoRF to create a low-parameter Empirical Model of Response (EMoR). This response model allows us to accurately interpolate the complete response function of a camera from a small number of measurements obtained using a standard chart. We also show that the model can be used to accurately estimate the camera response from images of an arbitrary scene taken using different exposures. The DoRF database and the EMoR model can be downloaded at http://www.cs.columbia.edu/CAVE.