Shock capturing, level sets, and PDE based methods in computer vision and image processing: a review of Osher’s contributions

In this paper we review the algorithm development and applications in high resolution shock capturing methods, level set methods, and PDE based methods in computer vision and image processing. The emphasis is on Stanley Osher’s contribution in these areas and the impact of his work. We will start with shock capturing methods and will review the Engquist–Osher scheme, TVD schemes, entropy conditions, ENO and WENO schemes, and numerical schemes for Hamilton–Jacobi type equations. Among level set methods we will review level set calculus, numerical techniques, fluids and materials, variational approach, high codimension motion, geometric optics, and the computation of discontinuous solutions to Hamilton–Jacobi equations. Among computer vision and image processing we will review the total variation model for image denoising, images on implicit surfaces, and the level set method in image processing and computer vision.