Visual motion ambiguities of a plane in 2-D FS sonar motion sequences

Sonar is the most common imaging modality in underwater, and high-resolution high data rate 2-D video systems have been emerging in recent years. As for visually guided terrestrial robot navigation and target-based positioning, the estimation of 3-D motion by tracking features in recorded 2-D sonar images is also a highly desirable capability for submersible platforms. Additionally, theoretical results dealing with robustness and multiplicity of solution constitute important fundamental findings due to nature of sonar data, namely, high noise level, narrow field of view coverage, scarcity of robust features, and incorrect matches. This paper explores the inherent ambiguities of 3-D motion and scene structure interpretation from 2-D forward-scan sonar image sequences. Analyzing the sonar image motion transformation model, which depends on the affine components of the projective transformation (or homography) of two plane views, we show that two interpretations are commonly inferred. The true and spurious planes form mirror images relative to the zero-elevation plane of the sonar reference frame. Even under each of pure rotation or translation, a spurious motion exists comprising both translational and rotational components. In some cases, the two solutions share certain motion components, where the imaged surface becomes parallel to a plane defined by two of the sonar coordinate axes. A unique solution exists under the very special condition where the sonar motion aligns the imaged plane with the zero-elevation planes. We also derive the relationship between the two interpretations, thus allowing closed-form computation of both solutions.