Tutorial 1

1. Consider an image of dimension 100 x 100.  Compare the memory requirements to represent the data using
   
   
   1. a structured point dataset
   2. an unstructured point dataset
   
   
   
2. When designing an interpolation kernel, the various issues that one must consider are:
   
   
   • the width of the kernel (i.e., the number of sampled points to be covered by the kernel);
   • any additional free parameters to be associated with the kernel function;
   • the kernel function itself.
   
   In the last issue above, it is important that the kernel function is so designed that the sum of all the function values evaluated at the sampled points is 1, as these function values would be used as weights in the interpolation process. Ideally, the kernel function should be continuous and smooth also.
   • The cubic interpolation kernel that is defined below:

     f(x)  =	(a + 2) |x|3 - (a + 3) |x|2 + 1	if  0 <= |x| <= 1
     	a |x|3 - 5a |x|2 + 8a |x| - 4a	if  1 <= |x| <= 2
     	0	if  2 <= |x|

     has a free parameter, a.
     
     1. Verify that the kernel gives w₀ + w₁ + w₂ + w₃ = 1, where w₀, w₁, w₂, and w₃ are the weights at the sampled points x₀=-1.5, x₁=-0.5, x₂=0.5, and x₃=1.5.
        
     2. Is the total weight equal to 1 regardless of the free paramerer a?
   • The B-spline interpolation kernel is given below:

     f(x)  =	1/2 |x|3 - |x|2 + 2/3	if  0 <= |x| <= 1
     	-1/6 |x|3 + |x|2 - 2|x| + 4/3	if  1 <= |x| <= 2
     	0	if  2 <= |x|

     1. Show that the kernel function is C¹.
     2. Show that the sum of the weights at x₀=-1.5, x₁=-0.5, x₂=0.5, and x₃=1.5 is 1.
   
   
   
3. Show that the linearly interpolated attribute value, f(Q), of any point Q inside the triangle formed by 3 sampled points P₀, P₁, and P₂ is
   f(Q) = w₀f(P₀) + w₁f(P₁) + w₂f(P₂)
   where
   w₀ = (1-r-s)
   w₁ = r
   w₂ = s
   
   The two scalars r and s are defined as follows: r = ||P₀Q₁|| / ||P₀P₁ || ,  s = ||P₀Q₂|| / ||P₀P₂||,   where QQ₁ is parallel to P₀P₂   and QQ₂ is parallel to P₀P₁,  as shown in the diagram below: