Discusses the mathematical foundations of statistical inference for building a three-dimensional model of the environment from image and sensor data that contain noise, a task required of autonomous robots guided by video cameras and sensors. Derives a theoretical accuracy bound for the optimization procedure to maximize the reliability of the estimation based on noise data, and also the practical computation schemes to attain that bound. Includes examples of both synthetic and real data to demonstrate that conventional methods are not optimal and how accuracy improves if truly optimal methods are used.