Contour tracing for geographical digital data

Abstract Our purpose is to trace a contour in the form of a polygon. In this research, we use a bicubic spline function for interpolation of the elevation data on a grid covering the area of concern. We construct the polygon as a data consisting of ordered contour points on sides of the grid. The contour enters a cell at an entry point and goes out at an exit point on its sides. The polygon is formed connecting these points. A problem occurs as to which two points should be connected when a cell of the grid has more than three contour points on its sides. As for existing methods of the differential geometry such as discretization using tangential increments, it is difficult to predetermine a suitable step size to arrive at a next contour point correctly if several contour components wind closely to each other within a cell. As a solution, we take an algebraic approach exploiting a simple fact that a bicubic function is viewed as a univariate cubic function with a parameter. From this perspective, we identify the exit point examining the behavior of the real roots of the cubic equation for the contour in terms of the numerical order. Our method enables us to faithfully trace the contour of bicubic spline functions which provide smoother and better fitting curves than bilinear spline functions used by the other authors. Computation time is exhibited in the numerical experiment for an island in Japan.