The work of Stanley Osher

In this paper we briefly present some of Stanley Osher's contributions in the areas of high resolution shock capturing methods, level set methods, partial differential equation (PDE) based methods in computer vision and image processing, and optimization. His numerical analysis contributions, including the Engquist-Osher scheme, total variation diminishing (TVD) schemes, entropy conditions, essentially non-oscillatory (ENO) and weighted ENO (WENO) schemes and numerical schemes for Hamilton-Jacobi type equations have revolutionized the field. His level set contributions include new level set calculus, novel numerical techniques, fluids and materials modeling, variational approaches, high codimension motion analysis, geometric optics, and the computation of discontinuous solutions to Hamilton-Jacobi equations. As we will further detail in this paper, the level set method, together with his total variation contributions, have been extremely influential in computer vision, image processing, and computer graphics. On top of that, such new methods have motivated some of the most fundamental studies in the theory of PDEs in recent years, completing the picture of applied mathematics inspiring pure mathematics. On optimization, he introduced Bregman algorithms and applied them to problems in a variety of contexts such as image processing, compressive sensing, signal processing, and machine learning. Finally, we will comment on Osher's entrepreneurship and how he brought his mathematics to industry.