In the last two decades, many edge detection methods have been developed and widely used in image processing for edge detection and the hybrid compact-WENO finite difference (hybrid) schemes for solving the system of hyperbolic conservation laws with solutions containing both discontinuous and complex fine-scale structures. However, many edge detection methods include the problem-dependent parameters such as the high order multi-resolution (MR) analysis (Harten, JCP, 49 (1983)). Therefore, we combined the Tukey’s boxplot method with MR analysis (Gao et al., JSC, 73 (2017)) to overcome this problem in a sense. But the Tukey’s boxplot method needs to sort the data at the beginning of Runge-Kutta time integration method, which is relatively time-consuming and inefficient. In this study, we employ the Pauta criterion and remove the problem-dependent parameters in the MR analysis. Furthermore, two new edge detection approaches, which are based on second-order central difference scheme and Ren’s idea (Ren et al., JCP, 192 (2003)), are also proposed. The accuracy, efficiency and robustness of the hybrid scheme with the new edge detectors are verified by numerous classical one- and two-dimensional examples in the image processing and compressible Euler equations with discontinuous solutions.