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\noindent{\bf Math 155: Homework \# 8, due on Friday, March 9}

\medbreak 

%\noindent{\bf [1] } The two subimages shown were extracted from the top, right corners 
%of Figs. 5.7(c) and (d), respectively. Thus, the subimage on the left is the
%result of using an arithmetic mean filter of size 3x3; the other subimage is
%the result of using a geometric mean filter of the same size. 

%(a) Explain why the subimage obtained with geometric mean filtering is less
%blurred. {\it Hint:} Start your analysis by examining a 1-D step edge profile.

%(b) Explain why the black components in the right image are thicker. 

%\begin{figure}
%\begin{center}
%\includegraphics[scale=1]{Prob5.10left.ps}
%\includegraphics[scale=0.62]{Prob5.10right.ps}
%\end{center} 
%\end{figure}

\noindent{\bf [1] } Download from the class web page the image Fig5.07(b).jpg 
(X-Ray image corrupted by Gaussian noise). 

(a) Write a computer program to implement the arithmetic mean filter of size 3x3. Apply the program to the image Fig5.07(b).jpg  

(b) Write a computer program to implement the geometric mean filter of size 3x3. Apply the program to the image Fig5.07(b).jpg  

(c) Explain your results. Evaluate the SNR (signal-to-noise-ratio) for both results in (a) and (b) (before denoising and after denoising). Note, higher SNR, better denoised image. Let $\hat f$ be the denoised image, and $f$ the clean true image. Then $SNR=10\log_{10}\frac{\sum_{x,y}(\hat f)^2}{\sum_{x,y}(f-\hat f)^2}$. To evaluate the SNR before denoising, substitute $\hat f$ by $g$ in the above formula. 

\noindent{\bf [2] } Refer to the contraharmonic filter given in Eq. (5.3-6).

(a) Explain why the filter is effective in eliminating pepper noise when $Q$
is positive. 

(b) Explain why the filter is effective in eliminating salt noise when $Q$
is negative. 

(c) Explain why the filter gives poor results (such as the results shown in 
Fig. 5.9) when the wrong polarity is chosen for $Q$. 

(d) Discuss the behavior of the filter when $Q=-1$. 

(e) Discuss (for positive and negative $Q$) the behavior of the filter in areas
of constant gray levels. 

\noindent{\bf [3] } (a) Download from the class web page the image Fig5.08(a).jpg 
(X-Ray image corrupted by pepper noise). Write a computer program that will 
filter this image with a 3x3 contraharmonic filter of order 1.5. 

(b) Download from the class web page the image Fig5.08(b).jpg 
(X-Ray image corrupted by salt noise). Write a computer program that will 
filter this image with a 3x3 contraharmonic filter of order -1.5. 

\noindent{\bf [4] } (a) Recall the definition of the convolution $f*g(x,y)$ in continuous variables and two dimensions. 

(b) Show that $\nabla ^2(f*g)=f*(\nabla ^2g)$, where $\nabla ^2$ denotes the Laplace operator in $(x,y)$. 

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