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This paper provides a comprehensive analysis of exactly what visual information about the world is embedded within a single image of an eye. It turns out that the cornea of an eye and a camera viewing the eye form a catadioptric imaging system. We refer to this as a corneal imaging system. Unlike a typical catadioptric system, a corneal one is flexible in that the reflector (cornea) is not rigidly attached to the camera. Using a geometric model of the cornea based on anatomical studies, its 3D location and orientation can be estimated from a single image of the eye. Once this is done, a wide-angle view of the environment of the person can be obtained from the image. In addition, we can compute the projection of the environment onto the retina with its center aligned with the gaze direction. This foveated retinal image reveals what the person is looking at. We present a detailed analysis of the characteristics of the corneal imaging system including field of view, resolution and locus of viewpoints. When both eyes of a person are captured in an image, we have a stereo corneal imaging system. We analyze the epipolar geometry of this stereo system and show how it can be used to compute 3D structure. The framework we present in this paper for interpreting eye images is passive and non-invasive. It has direct implications for several fields including visual recognition, human-machine interfaces, computer graphics and human affect studies.