7.4. Exercises

1. The Laplacian of an image (at scale \(s\)) is given as:

   \[\nabla^2 f^s = f^s_{xx} + f^s_{yy}\]

   1. Prove that the Laplacian can be calculated with one convolution (from the zero scale image).

   2. Give the mathematical expression for that kernel.

   3. Make a 3D plot of the Laplacian convolution kernel. If you plot the negative version of the kernel you will understand why the Laplacian operator is sometimes called the Mexican hat operator.

2. Show that the scale normalized version of the gradient norm is given by \(s f^s_w\).

3. Show that for \(t>s\) it is true that \(s+\sqrt{t^2-s^2} < t\). See the first section of this chapter (at the end) for why this is important.