Diffusion runs low on persistence fast

Interpreting an image as a function on a compact subset of the Euclidean plane, we get its scale-space by diffusion, spreading the image over the entire plane. This generates a 1-parameter family of functions alternatively deï¬ nedas convolutions with a progressively wider Gaussian kernel. We prove that the corresponding 1-parameter family ofpersistence diagrams have norms that go rapidly to zero astime goes to inï¬ nity. This result rationalizes experimentalobservations about scale-space. We hope this will lead totargeted improvements of related computer vision methods.