Lagrange Detector in Image Processing

Edge detection is a basic operation in the field of image processing and computer vision. However, to the best of our knowledge, there is less mathematical work has been proposed beyond the first- and the second-order derivative operator for edge detection in the past decays. We propose a mathematical model called Lagrange detector for edge detection. Based on Lagrange polynomial interpolation theory, this detector calculates Lagrange remainder as the strength of edge and points at the features in various orders of discrete data or signal. Lagrange remainder combines the first-order derivation and the first- and second-order derivative of neighborhood by multiplication. We use the truncation error of polynomial interpolation to estimate Lagrange remainder. Lagrange detector performs well in detecting both outlines and tiny details. Furthermore, Lagrange detector can be used to detect high-frequency information like as corner, point, Moire pattern and etc. The research of Lagrange detector opens an new window for low level image processing, and will be used as the basis for further studies on image processing.