Local Morse Theory for Gaussian Blurred Functions

When Gaussian blurring is applied to an intensity function f 0 (x), it yields a family f(x,t) of intensity functions parametrised by t, which is a solution to the heat equation $$ \frac{{\partial f}}{{\partial t}} = \Delta \left( f \right) = \sum\limits_{i = 1}^n {\frac{{{\partial ^2}f}}{{\partial x_i^2}}} \quad on\quad {R^n} \times {R_ + }$$ (11.1) and satisfying initial conditions f(x,0) = f 0(x) for f 0 : R n → R.