This paper describes the development and experimental validation of algorithms for a novel laser vision system (LVS), suitable for measuring the relative posture from both solid and mesh-like targets in underwater environments. The system was developed in the framework of the AQUABOT project, a research project dedicated to the development of an underwater robotic system for inspection of offshore aquaculture installations. In particular, an analytical model for three-medium refraction that takes into account the nonlinear hemispherical optics for image rectification has been developed. The analytical nature of the model allows the online estimation of the refractive index of the external medium. The proposed LVS consists of three line-lasers within the field of view of the underwater robot camera. The algorithms that have been developed in this work provide appropriately filtered point-cloud datasets from each laser, as well as high-level information such as distance and relative orientation of the target with respect to the ROV. In addition, an automatic calibration procedure, along with the accompanying hardware for the underwater laser vision system has been developed to reduce the calibration overhead required by regular maintenance operations for underwater robots operating in seawater. Furthermore, a spatial image filter was developed for discriminating between mesh and non-mesh-like targets in the LVS measurements. Finally, a set of experiments was carried out in a controlled laboratory environment, as well as in real conditions at offshore aquaculture installations demonstrating the performance of the system.