The paper introduces shock filters for image enhancement, which use nonlinear time-dependent partial differential equations (PDEs) and their discretizations. The PDEs evolve the initial image \( u_0(x, y) \) into a steady-state solution \( u_m(x, y) \) through \( u(x, y, t) \) for \( t > 0 \). The solutions satisfy a maximum principle and preserve the total variation (TV) of the initial data. The processed image is piecewise smooth, with jumps occurring across zeros of an elliptic operator (edge detector). The algorithm is fast and easy to program. The authors discuss the challenges of deconvolution in the presence of noise and the limitations of linear and Fourier-based methods for processing piecewise smooth images. They propose a feature-oriented approach, inspired by the numerical solution of nonlinear hyperbolic equations, to enhance images by preserving edges and discontinuities. In the one-dimensional case, they present a PDE \( u_t = -|u_t| F(u_{xx}) \) and its discrete approximation, which preserves TV and extremal points. The scheme is shown to be monotone and accurate, with numerical experiments demonstrating its effectiveness. For two-dimensional images, they consider the PDE \( u_t = -\sqrt{u_x^2 + u_y^2} F(\mathcal{L}(u)) \), where \( \mathcal{L}(u) \) is a second-order elliptic operator used as an edge detector. The authors derive a local maximum principle and show that the scheme preserves extremal points and TV. Numerical results on standard images from the USC IPI Image Data Base demonstrate the effectiveness of the shock filter in enhancing images, including resolution beyond the original optical limit and preserving fine details. The paper concludes with a discussion of the limitations of the method when significant information is lost due to blurring and the need for more sophisticated shock filters in such cases.The paper introduces shock filters for image enhancement, which use nonlinear time-dependent partial differential equations (PDEs) and their discretizations. The PDEs evolve the initial image \( u_0(x, y) \) into a steady-state solution \( u_m(x, y) \) through \( u(x, y, t) \) for \( t > 0 \). The solutions satisfy a maximum principle and preserve the total variation (TV) of the initial data. The processed image is piecewise smooth, with jumps occurring across zeros of an elliptic operator (edge detector). The algorithm is fast and easy to program. The authors discuss the challenges of deconvolution in the presence of noise and the limitations of linear and Fourier-based methods for processing piecewise smooth images. They propose a feature-oriented approach, inspired by the numerical solution of nonlinear hyperbolic equations, to enhance images by preserving edges and discontinuities. In the one-dimensional case, they present a PDE \( u_t = -|u_t| F(u_{xx}) \) and its discrete approximation, which preserves TV and extremal points. The scheme is shown to be monotone and accurate, with numerical experiments demonstrating its effectiveness. For two-dimensional images, they consider the PDE \( u_t = -\sqrt{u_x^2 + u_y^2} F(\mathcal{L}(u)) \), where \( \mathcal{L}(u) \) is a second-order elliptic operator used as an edge detector. The authors derive a local maximum principle and show that the scheme preserves extremal points and TV. Numerical results on standard images from the USC IPI Image Data Base demonstrate the effectiveness of the shock filter in enhancing images, including resolution beyond the original optical limit and preserving fine details. The paper concludes with a discussion of the limitations of the method when significant information is lost due to blurring and the need for more sophisticated shock filters in such cases.