Geometric properties estimation from discrete curves using discrete derivatives

Accurate geometric properties estimation from discrete curves is an important problem in many application domains, such as computer vision, pattern recognition, image processing, and geometric modeling. In this paper, we propose a novel method for estimating the geometric properties from discrete curves based on derivative estimation. We develop derivative estimation by defining the derivative of a discrete function at a point, which will be called the discrete derivative. Similarly, the second and higher order discrete derivatives at that point are also defined, and their convergence is demonstrated by theory analysis. These definitions of the different order discrete derivatives provide a simple and reliable way to estimate the derivatives from discrete curves. Based on the discrete derivatives, classical differential geometry can be discretized, and the geometric properties are estimated from discrete curves by using differential geometry theory. The proposed method is independent of any analytic curve and estimates the geometric properties directly from discrete data points, which makes it robust to the geometric shapes of discrete curves. Another advantage of the proposed method is the robustness to noise because of the calculation characteristics of the discrete derivatives. The proposed method is evaluated and compared with other existing methods in the experiments with both synthetic and real discrete curves. The test results show that the proposed method has good performance, and is robust to noise and suitable for different curve shapes.