Set-based modeling and observer design for planar structure from motion

Sets that are defined either from closed contours or from a set of points are generic descriptors for several kinds of objects in binary images. In this paper, we derive a novel model in the space of sets and design an observer for the proposed model to estimate the depth and orientation of planar objects from a camera. This problem is well known as "structure from motion." When the objects are only partially projected on the image plane of the camera, our model makes object depth and orientation estimation possible without feature tracking and matching between distinct image frames (i.e., the so-called correspondence problem), which is an advantage over the image moments-based model. However, the proposed model fails in some situations (the failures can be seen as brief instabilities), and it is not always continuous. To compensate for these drawbacks, we designed a fast observer based on L₁ control theory with a binary signal for the proposed model. Stability analysis with respect to a certain asymptotic instability ratio is also presented in this paper. The effectiveness of the proposed model and observer is demonstrated through simulations and experimental results.