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\noindent{\bf Math 155: Hw \# 7, due on Friday, March 1st}
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\noindent{\bf [1]} We have seen in continuous variables that ${\cal F}(\delta)\equiv 1$, thus $\delta$ and $1$ form a Fourier pair; we also have that ${\cal F}(1)=\delta$. Using the second property and the translation property, show that the Fourier transform of $f(x)=\sin(2\pi u_0x)$, where $u_0$ is a real number, is $F(u)=(i/2)\Big[\delta(u+u_0)-\delta(u-u_0)\Big]$.
\noindent(hint: you could express $f$ function of exponentials).
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\noindent{\bf [2] } We have seen in continuous variables that ${\cal F}(\delta)\equiv 1$, thus $\delta$ and $1$ form a Fourier pair; we also have that ${\cal F}(1)=\delta$. Using the second property and the translation property,
show that the Fourier transform of the continuous function $f(x,y)=A\sin(2\pi u_0 x+2\pi v_0y)$ is
$$F(u,v)=A\frac{i}{2}\Big[\delta(u+u_0,v+v_0)-\delta(u-u_0,v-v_0)\Big].$$
(hint: you could express $f$ function of exponentials as done in the 1D case).
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\noindent{\bf [3]} Assume that ${\cal F}(1)=\delta$ also holds in the discrete case (this can be shown). Using this property and the translation property, show that
the Fourier transform of the discrete function $f(x,y)=\sin(2\pi u_0 x+2\pi v_0y)$ is
$$F(u,v)=\frac{i}{2}\Big[\delta(u+Mu_0,v+Nv_0)-\delta(u-Mu_0,v-Nv_0)\Big].$$
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\noindent{\bf [4] } {\bf Periodic Noise Reduction Using a Notch Filter}
(a) Write a program that implements sinusoidal noise of the form:
$n(x,y)=A\sin(2\pi u_0 x+2\pi v_0y)$.
The input to the program must be the amplitude, A, and the two frequency components $u_0$ and $v_0$.
(b) Download image 5.26(a) of size $M\times N$ and add sinusoidal noise to it, with $v_0 = 0$. The value of A must be high enough for the noise to be quite visible in the image (for example, you can take $A=100$, $u_0=134.4$, $v_0=0$).
(c) Compute and display the degraded image and its spectrum (you may need to apply a log transform to visualize the spectrum).
(d) Notch-filter the image using a notch filter of the form shown in Fig. 5.19(c) to remove the periodic noise.
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\noindent{\bf [5] } (a) Recall the definition of the convolution $f*g(x,y)$ in continuous variables in 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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