A gaze tracking system, leaving a low power mode in response to an activation signal, captures an initial burst of eye pictures in short time by restricting the image area of a sensor, with the purpose of enabling an increased frame rate. Subsequent eye pictures are captured at normal frame rate. The first gaze point value is computed memorylessly based on the initial burst of eye pictures and no additional imagery, while subsequent values may be computed recursively by taking into account previous gaze point values or information from previous eye pictures. The restriction of the image area may be guided by a preliminary overview picture captured using the same or a different sensor. From the gaze point values, the system may derive a control signal to be supplied to a computer device with a visual display.