Repeat the above Problem $1$ using a harmonic mean filter.

Repeat Problem $5\mathrm{.}1$ using a contraharmonic mean filter with $Q=-1.$

In answering the following, refer to the contraharmonic

filter in Eq$.$

$($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 when the wrong polarity is chosen for $Q.$

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

$5.$ With reference to Eq$.(5-33){?}^{H_{NR}(u,v)}=pro{d}_k=1^0{H}_k(u,v)H_{-k}(u,v),$ obtain equations for:

$($a$)$ An ideal notch filter transfer function.

$($b$)$ A Gaussian notch filter transfer function.

$($c$)$ A Butterworth notch filter transfer function. $6.$ A linear, space invariant system has the impulse response

$h(x,y)=(x-a,y-b)$

where a and $b$ are constants, and $x$ and $y$ are discrete quantities. Answer the following, assuming negligible noise in each case.

$($a$)$ What is the system transfer function in the frequency domain?

$($b$)$ What would the spatial domain system response be to a constant input, $f(x,y)=K.$

$($c$)$ What would the spatial domain system response be to an impulse input,

$f(x,y)=(x,y)?$