L1-norm fitting of elliptic paraboloids with prior information for enhanced coniferous tree localization in ALS point clouds

Abstract

Airborne laser scanning (ALS) is an established tool for deriving various tree characteristics in forests. In some applications, an accurate pointwise estimate of the tree position is required. For dense data acquired by TLS or UAV-mounted scanners, this can be achieved by locating the stem, whose center coordinates are then used for deriving the planimetric tree position. However, in case of standard ALS data this is often not an option due to the low probability of obtaining stem hits in operational scenarios of forest mapping campaigns. This paper presents an alternative, indirect approach where the tree position is approximated as the center of a quadric surface which best represents the tree crown shape. The study targets coniferous trees due to their distinct crown shape which may be approximated by an elliptic paraboloid. It is assumed that individual tree point clusters are given and the task is to find the tree center for each cluster. We first consider the general problem of fitting an elliptic paraboloid with a known axis and an L1 residual norm error criterion, which is more robust to outliers compared to least-squares fitting. We formulate this problem as a quadratically constrained quadratic program (QCQP), and show how prior knowledge on the crown shape and center position can be incorporated. Next, a computationally simpler problem is considered where the paraboloid semiaxis lengths are constrained to be equal, and a corresponding linear program is constructed. Experiments on ALS datasets of forest plots from Bavaria, Germany and Oregon, USA reveal that a reduction in median tree position error of up to 20% can be attained compared to both least-squares fitting and other baseline techniques, resulting in an absolute error of ca. 22 cm on both datasets.