Local Morse theory for solutions to the heat equation and Gaussian blurring
We are interested in a problem that arises in computer vision. Here one wishes to be able to identify and manipulate the significant objects in a computer screen image. A particular method of doing this is to use ``Gaussian blurring'' applied to a ``pixel intensity'' function \(u_0(x)\) which describes the brightness of the various pixels on the screen. Its aim is to blur away nonimportant details of the image at various levels of scale. When Gaussian blurring is applied to an intensity function \(u_0(x)\), it yields a family \(u(x, t)\) of intensity functions parametrized by \(t\). The family \(u(x, t)\) is actually a solution to the heat equation and satisfying initial conditions \(u(x, 0)= u_0(x)\) for \(u_0: \mathbb{R}^n\to \mathbb{R}\) at least continuous. We would like to know what the generic properties of \(u\) are, i.e., for most initial functions \(u_0\), what local properties we can expect \(u\) to have. This would include understanding how regions of given intensity interact in the course of the blurring. For example, \(u(x, t)\) represents the amount of heat at the point \(x\) at time \(t\). Since heat diffuses, we would expect that the region where the heat content locally exceeded a given amount would expand so that if two different regions joined they must merge and further expand and, moreover, no new regions of relative higher intensity would be created. We shall see that both of these intuitive expectations are false; so we would like to understand locally what can happen. Our goal then is to describe the local structure of mappings \(u: U\subset \mathbb{R}^{n+ 1}_+= \mathbb{R}^n\times \mathbb{R}_+\to \mathbb{R}\) which are solutions to the heat equation. Since any other nonisotropic heat equation with constant coefficients can be reduced to the heat equation by a linear change of coordinates in \(\mathbb{R}^n\), the results will likewise be applicable for any constant coefficient heat equation.
scientific article; zbMATH DE number 716261