Fig. 5

Camera placement and imaging mechanism in the real world. The coordinate system \(XO{^{\prime}}Y\) represents the world coordinate system, and the coordinate system \({X}^{{^{\prime}}}{O}^{{^{\prime}}{^{\prime}}}{Y}^{{^{\prime}}}\) is the physical coordinate system. \(P(x,y)\) is the point to be measured on the ground, with \({P}_{x}\) as the projection of P on the X-axis, with a length of x, and \({P}_{y}\) as the projection of P on the Y-axis, with a length of y. The segment \(O{^{\prime}}P\), with length d, represents the straight-line distance from the point P to the camera. The optical axis \(O{O}^{{^{\prime}}{^{\prime}}}\) passes through the optical center \(O\) and is perpendicular to the image coordinate system \({X}^{{^{\prime}}}{O}^{{^{\prime}}{^{\prime}}}{Y}^{{^{\prime}}}\). α denotes the angle between the optical axis of the camera and the horizontal line. The distance \(OO{^{\prime}}{^{\prime}}\) from the optical center O to the origin \(O{^{\prime}}{^{\prime}}\) of the image coordinate system is the camera’s focal length f, and h represents the real-world height of the camera, specifically the distance \(OO{^{\prime}}\) from the optical center O to the world coordinate system on the ground. After imaging, the point P projects to \({P}^{{^{\prime}}}({x}^{{^{\prime}}},{y}^{{^{\prime}}})\) in the physical coordinate system, with \({P}_{x}^{{^{\prime}}}\) as the projection of \({P}^{{^{\prime}}}\) on the \({X}^{{^{\prime}}}\)-axis, having a length of \({x}^{{^{\prime}}}\), and \({P}_{y}^{{^{\prime}}}\) as the projection of \({P}^{{^{\prime}}}\) on the \({Y}^{{^{\prime}}}\)-axis, with a length of \({y}^{{^{\prime}}}\). Points \(M\) and N are the intersections of the extended line \({P}^{{^{\prime}}}{P}_{x}^{{^{\prime}}}\) and the extended line \({P}_{y}^{{^{\prime}}}{O}^{{^{\prime}}{^{\prime}}}\) with the horizontal plane where the optical center O is located. The ultimate goal of the positioning model is to convert \({P}^{{^{\prime}}}({x}^{{^{\prime}}},{y}^{{^{\prime}}})\) into \(P(x,y)\).