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\noindent{\bf Math 155: Instructor Luminita Vese. Teaching Assistant: Jacob Moorman.}

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$\bullet$ Midterm on Wednesday, February 14, 2018, 1.00-1.50pm (lecture room).

$\bullet$ Sections covered for the midterm (refering to the 3rd edition of the textbook): Chapter 2 Sections 2.3.4, 2.4.1-2.4.3. Chapter 3 Sections 3.1, 3.2 (except bit-plane slicing), 3.3, 3.4, 3.5, 3.6. Chapter 4: Sections 4.2, 4.4, 4.5, 4.6. 

$\bullet$ Solutions to exercises marked with an asterisk are posted on the authors web-page, at 

http://www.imageprocessingplace.com/root$_{-}$files$_{-}$V3/problem$_{-}$solutions.htm

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\noindent {\bf Homework $\#$ 5, due on Friday, February 16}

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\noindent{\bf [1]} Recall that the 1D DFT is 
$$F(u)=\sum_{x=0}^{M-1}f(x)e^{-2\pi iux/M}.$$

(a) Assuming the formula above, show the identity 
$$f(x)=\frac{1}{M}\sum_{u=0}^{M-1}F(u)e^{2\pi i u x/M},$$
using the following orthogonality of exponentials 
$$\sum_{u=0}^{M-1}e^{-2\pi iuy/M}e^{2\pi iux/M}=
\left\{
\begin{array}{ll}
M&\mbox{ if }x=y\\
0 &\mbox{ otherwise}.
\end{array}
\right.
$$

(b) Show now the converse of (a): assume given $f(x)$ function of 
$F(u)$ in the discrete case, and show the identity for $F(u)$ (use the same orthogonality of exponentials). 

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\noindent{\bf [2] } Show that the continuous 2D Fourier transform is a linear process. 

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\noindent{\bf [3] } Compute in continuous variables the Fourier transform of the 
function
$$f(x)=\left\{
\begin{array}{l}
A, \mbox{ if }0\leq x\leq K,\\
0, \mbox{ otherwise, }
\end{array}
\right. 
$$
where $A$ and $K$ are positive constants. Evaluate $F(0)$. 

\medbreak 

\noindent{\bf [4] } Consider again the 2D continous Fourier transform and its inverse (denote by $H(u,v)$ the 2D Fourier transform of the spatial filter $h(x,y)$). 
Show that if the transform $H(u,v)$ is real and symmetric, 
i.e. if 
$$H(u,v)=\overline{H(u,v)}=\overline{H(-u,-v)}=H(-u,-v),$$ 
then the corresponding spatial domain filter $h(x,y)$ is also real and symmetric. 

\medbreak

\noindent{\bf [5]} (Computational Project) {\bf Fourier Spectrum and Average Value} 

(a) Use in Matlab ``help fft'' and ``help fft2'' to learn the commands for computing discrete Fourier transforms. Sample codes using the Fourier transform in 1D and 2D are posted on the class webpage. 

(b) Download Fig5.26a and compute its (centered) Fourier spectrum.

(c) Display the spectrum.

(d) Using your algorithm, obtain the average value of the input image. 





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