===========
 Exercises
===========


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

   .. math::
      \nabla^2 f^s = f^s_{xx} + f^s_{yy}

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

   #. Give the mathematical expression for that kernel.
      
   #. 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**.

#. Show that the scale normalized version of the gradient norm is
   given by $s f^s_w$.

#. 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.