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\noindent{\bf Math 155. Instructor: Luminita Vese. 

\ \hspace{15mm}Teaching Assistant: Jacob Moorman.}

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\noindent{\bf Homework $\#$ 6, due on Friday, February 23}

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\noindent{\bf [1]} (a) Show in discrete variables that 
$${\cal F}\Big(f(x,y)e^{2\pi i(u_0\frac{x}{M}+v_0\frac{y}{N})}\Big)=F(u-u_0,v-v_0),$$ where $F={\cal F}(f)$. 

(b) Using (a), deduce the formula used in shifting the center of the transform by multiplication with $(-1)^{x+y}$, when $u_0=M/2$ and $v_0=N/2$, with $M$ and $N$ even positive integers. 

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\noindent{\bf [2]} 

(a) Show, in discrete variables, the translation property 
$${\cal F}\Big(f(x-x_0,y-y_0)\Big)=F(u,v)e^{-2\pi i(x_0u/M+y_0v/N)},$$
where $F(u,v)={\cal F}(f(x,y))$. 

(b) Consider the linear difference operator $g(x,y)=f(x+1,y)-f(x,y)$. Obtain the filter transfer function, $H(u,v)$, for performing the equivalent process in the frequency domain. 

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\noindent{\bf [3]} Prove the validity of the discrete convolution theorem in one variable (you may need to use the translation properties). 

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\noindent{\bf [4]} Assume that $f(x)$ is given by the discrete IFT formula in one dimension. Show the periodicity property $f(x)=f(x+kM)$, where $k$ is an integer. 

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\noindent{\bf [5]} (a) Implement the Gaussian lowpass filter in Eq. (4.3-8), using a 
radius $D_0=25$, and apply the algorithm to Fig4.11(a).  

(b) Highpass the input image used in (a), using a highpass Gaussian filter
of radius $D_0=25$ (see eq. (4.4-4)). 

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