We study a novel degenerate and singular elliptic operator $ \Delta_{\tau, \xi} $ , which is a weighted sum of one-Laplacian and infinite-Laplacian. We establish the well-posedness of the Neumann boundary value problem for the parabolic equation $ u_t=\Delta_{\tau, \xi } (u) $ in the framework of viscosity solutions. We also prove the consistency and the convergence of the numerical scheme for the finite difference method of the parabolic equation above. Numerical simulations show that this novel operator $\Delta_{ \tau, \xi } $ gives better results than both the Perona-Malik (Perona and Malik, 1990) and total variation (TV) methods (Chan and Shen, 2005) when applied to image enhancement.