A New Fixed-Cost Approximation for Ellipse–Ellipse Intersection: A Case Study in Tree-Crown Delineation Post-Processing

The intersection area between two arbitrarily rotated ellipses is a recurring geometric primitive in imaging, computer vision, and robotics. In the general case, its evaluation is often associated with intersection-point recovery, topology-dependent case handling, adaptive refinement, or dense boundary approximation. This study presents a fixed-cost computational framework for ellipse–ellipse intersection based on a rotated-frame slice formulation. A coordinate-frame rotation expresses one ellipse in axis-aligned form while representing the second as a general conic. This yields a hybrid formulation that reduces the intersection area to a one-dimensional overlap integral of vertical slice height over the admissible horizontal interval. The integral is evaluated using fixed-order quadrature, with optional sine mapping to improve conditioning near grazing configurations. Numerical evaluation on 100,000 synthetic ellipse pairs shows that the proposed formulation reaches a low-error regime earlier than polygonal approximation while remaining substantially faster across the tested range. The formulation is further examined through a tree-crown delineation case study on the BAMFORESTS dataset, a benchmark forest dataset of very-high-resolution UAV imagery. In this case study, ellipse proxies derived from axis-aligned and oriented bounding box detections are used for overlap computation during non-maximum suppression (NMS). Using ellipse-proxy overlap during NMS preserves nearly the same peak F1 score of 0.785 while modestly shifting NMS behavior toward lower thresholds and producing slightly broader near-peak intervals.