$14.$ Assume an image I$($x$,$ y$)$ represented in a $2$D spatial domain, where x

and y are spatial coordinates.

$($a$)$ Define low pass and high pass filters in the context of spatial

domain filtering. Provide mathematical expressions for a generic

low pass filter L$($x$,$ y$)$ and a high pass filter H$($x$,$ y$),$ discussing

their roles in modifying the frequency components of I$($x$,$ y$).$

$($b$)$ Elaborate on the principle of convolution in spatial filtering. Explain

how the convolution of I$($x$,$ y$)$ with L$($x$,$ y$)$ and H$($x$,$ y$)$ alters

the spatial frequency content of the image. Use the convolution

theorem to relate spatial filtering to frequency domain operations.

$($c$)$ Discuss the effects of applying low pass filtering on the image$$s

details, edges, and noise. Include an analysis of how such filtering

influences image sharpness and the visibility of fine details.

$($d$)$ Analyze the consequences of high pass filtering on the image$$s appearance.

Focus on the enhancement of edges and textures, and

the potential introduction of artifacts such as ringing or amplification

of noise.

$($e$)$ Consider a scenario where both low pass and high pass filters are

applied sequentially to an image. Theorize about the resultant

image characteristics and potential applications of this combined

filtering approach in image processing tasks