SIMG-782 HOMEWORK 4

1. A function f(x,y) has a Fourier transform F(u,v) which has the property that
   F(u,v)=F(-u,v)=F(u,-v)=F(-u,-v). What properties must f(x,y) have to make this possible?. Sketch an example of such a function.
2. Find the Fourier transform of the function f(x,y) which equals one on the unit square centered on (0,0) and equals zero elsewhere.
3. Let f(x,y) have the Fourier transform F(u,v). What is the Fourier transform of g(x,y)=f(ax,by)? Use this result along with the result of problem 2 to find the Fourier transform of a function g(x,y) that is a rectangle of size a along the horizontal edge and b along the vertical edge and centered on (0,0).
4. Find the Fourier transform of the cross function shown in Figure 1. The figure is centered on the origin.

5. Let f(x,y) have Fourier transform F(u,v). What is the Fourier transform of g(x,y)=f(x-p,y-q)? Clearly, g(x,y) is just a translated version of f(x,y).

6. Use the results of problems 3 and 5 to find the Fourier transform of a pair of parallel boxes located symmetrically about the origin as shown in Figure 2. The boxes have width a and height b and are spaced by c. Simplify your result so that it contains no complex terms. Note that, from problem 1, we can predict that such a simplification is possible.

7. Show that if a function f(x,y) can be written as a product f(x,y)=g(x)h(y) then F(u,v) can be found from the Fourier transforms G(u) and H(v).

8. Find the Fourier transform of the Gaussian function in two dimensions

where a>0, b>0. It is known that the Fourier transform of is

9. Write an IDL program that will reconstruct a function f(x,y) from a set of samples taken on a grid with spacing (dx,dy) by using interpolation with the 2D sinc function. Test your program by constructing an 11x11 grid of samples of the 2D Gaussian function with a=1, b=2 and then using these points in your program. Note that you will have to determine a sampling grid that will give you good reconstruction.