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\noindent{\bf Math 155: Hw \# 7, due Friday March 3rd, or the following Monday}
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\noindent{\bf [1]} Consider the two-dimensional continuous case.
\noindent(a) Write the Inverse Fourier transform formula in 2D (express $f(x,y)$ function of $F(u,v)$ in the continuous case).
\noindent(b) Assume $f$ is twice differentiable. Using (a), find the Fourier transform of the mixed partial derivative $\frac{\partial^2 f}{\partial x\partial y}$, function of $F$.
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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 $f(x)=\sin(2\pi u_0x)$, 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 [3] } 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).
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\noindent{\bf [4]} 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 [5] } {\bf Periodic Noise Reduction Using a Notch Filter}
(a) Write a program that implements sinusoidal noise of the form given in the previous homework: $n(x,y)=A\sin(2\pi u_0 x+2\pi v_0y)$. 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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