$[$Question $6]$ Edge Detection $(50$ marks$)$

In this question, the goal is to implement a rudimentary edge detection process that uses a derivative of Gaussian, through a series of steps. For each step $($excluding step $1)$ you are supposed to test your implementation on the provided image, and also on one image of your own choice. Include the results in your report.

Step I $-$ Gaussian Blurring $(10$ marks$)$: Implement a function that returns a $2$D Gaus$-$ sian matrix for input size and scale $\backslash$sigma $.$ Please note that you should not use any of the existing libraries to create the filter, e$.$g$.$ cv$2.$getGaussianKernel$().$ Moreover, visualize this $2$D Gaussian matrix for two choices of $\backslash$sigma with appropriate filter sizes. For the visualization,

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you may consider a $2$D image with a colormap, or a $3$D graph. Make sure to include the color bar or axis values.

Step II $-$ Gradient Magnitude $(10$ marks$)$: In the lectures, we discussed how partial derivatives of an image are computed. We know that the edges in an image are from the sudden changes of intensity and one way to capture that sudden change is to calculate the gradient magnitude at each pixel. The edge strength or gradient magnitude is defined as:

g$($x$,$y$)=|$f$($x$,$y$)|=$qgx$2+$gy$2$

where gx and gy are the gradients of image f$($x$,$y$)$ along x and y$-$axis direction respectively.

Using the Sobel operator, gx and gy can be computed as:

$101121$ gx$=202$f$($x$,$y$)$andgy$=000$f$($x$,$y$)$

$101121$

Implement a function that receives an image f $($x$,$ y$)$ as input and returns its gradient g$($x$,$ y$)$ magnitude as output using the Sobel operator. You are supposed to implement the convolu$-$ tion required for this task from scratch, without using any existing libraries.

Step III $-$ Threshold Algorithm $(20$ marks$)$: After finding the image gradient, the next step is to automatically find a threshold value so that edges can be determined. One algorithm to automatically determine image$-$dependent threshold is as follows:

$1.$ Let the initial threshold $\backslash$tau $0$ be equal to the average intensity of gradient image g$($x$,$ y$),$

as defined below:

Phj$=1$ Pwi$=1$ g$($i$,$ j$)\backslash$tau $0=$ h$\backslash$times w

where h and w are the height and width of the image under consideration.

$2.$ Set iteration index i $=0,$ and categorize the pixels into two classes, where the lower class consists of the pixels whose gradient magnitudes are less than $\backslash$tau $0,$ and the upper class contains the rest of the pixels.

$3.$ Compute the average gradient magnitudes mL and mH of lower and upper classes, respectively.

$4.$ Set iteration i $=$ i $+1$ and update threshold value as: $\backslash$tau i $=$ mL $+$ mH

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$5.$ Repeat steps $2$ to $4$ until $|\backslash$tau i $\backslash$tau i$1|<=\backslash$epsi is satisfied, where $\backslash$epsi $->0$; take $\backslash$tau i as final threshold and denote it by $\backslash$tau $.$

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Once the final threshold is obtained, each pixel of gradient image g$($x$,$ y$)$ is compared with $\backslash$tau $.$ The pixels with a gradient higher than $\backslash$tau are considered as edge point and is represented as white pixel; otherwise, it is designated as black. The edge$-$mapped image E$($x$,$ y$),$ thus obtained is:

$(255,$ if g$($x$,$ y$)>=\backslash$tau $0,$ otherwise

Implement the aforementioned threshold algorithm. The input to this algorithm is the gra$-$ dient image g$($x$,$ y$)$ obtained from step II$,$ and the output is a black and white edge$-$mapped image E$($x$,$ y$).$

Step IV $-$ Test $(10$ marks$)$: Use the image provided along with this assignment, and also one image of your choice to test all the previous steps $($I to III$)$ and to visualize your results in the report. Convert the images to grayscale first. Please note that the input to each step is the output of the previous step. In a brief paragraph, discuss how the algorithm works for these two examples and highlight its strengths and$/$or its weaknesses.