Understanding the Behavior of Structure from Motion Problem with False Focal Length

After intensive research in the past two decades, the geometric and computational aspect of structure from motion (SFM) seems well studied. However, the state of the art is that a practical SFM algorithm that can handle general visual tasks in the real world is still unavailable. One of the contributing reasons is that the first step of SFM involves solving an ill-conditioned problem. In this paper, we present an expressions to describe the error behavior of egomotion estimation when the focal length is calibrated with error. The key results are independent of both the egomotion estimation as well as the calibration algorithms. We show the bas-relief valley will be rotated according to the error in the focal length, in a way that is dependent on the motion-scene configuration. One important suggestion is that, provided that one knows the rough range of the true focal length, setting a larger-than-true focal length helps to estimate the direction of translation better though possibly with larger biases.