The Heat Equation with Us Forever
What good comes from applying the heat equation to a digital image? The obvious answer is nothing: the heat equation just mixes colors and blurs images. Likewise could a solution to the Laplace equation produce decent images? Again, the a priori answer is no since a solution to the Poisson equation is smooth, and hence results in blurriness.
Contradicting this doxa, I'll show that long-pursued aims in image processing and in image analysis are solved with the simplest linear PDEs: the heat equation and the Poisson equation. Three examples are treated.
All three are basic questions about images and their perception. Thus, one could do a crash course in image processing with just the Laplacian. In the talk I'll go over such a course.
The first problem is shape recognition. A recent algorithm invented by David Lowe gives an wonderfully complete solution. At its core lies the heat equation.
The second example is Land's retinex theory, which delivers optimally contrasted images. Lo and behold the restored image is the solution of a Poisson equation!
My third example is image denoising. Untold methods have been suggested for this puzzling problem, including all sorts of nonlinear heat equations. Yet the problem is elegantly solved by the heat equation.
Jean-Michel Morel, ENS Cachan, France