This is a joint talk with the Student Analysis Seminar. Note the different room.

• Jeff Calder
• October 13, Wednesday 4-5 p.m
• EH 4096

The Perona-Malik equation is arguably one of the most widely used partial differential equations (PDE) in image processing and computer vision. Although numerical schemes are (for the most part) stable, the continuum equation is an ill-posed forward/backward parabolic PDE. In an attempt to explain this paradoxical result and provide a rigorous theoretical foundation for the Perona-Malik equation, Catteé, Lions, Morel and Coll (1992) introduced a minor spatial regularization in the nonlinearity which allowed them to prove existence, uniqueness and $C^\infty$ regularity of solutions. In this talk, I will briefly explain why the Perona-Malik equation is ill-posed and then I will present the existence proof for the Catté regularization (uniqueness and regularity are standard proofs). The proof is a very nice application of Schauder’s fixed point theorem.