Image gradient is a key technique used to detect edges in an image. Edges represent areas where

there are significant changes in intensity or color, and the gradient provides a quantitative measure

of these changes. By analyzing the magnitude and direction of the image gradient, edge detection

algorithms can identify and highlight boundaries.

a$)$ Given a grayscale image I$($x$,$y$)$ the gradient magnitude $($x$,$y$)$ is often used in edge

detection. Express the mathematical formula for the gradient magnitude in terms of the

partial derivatives.

b$)$ In the context of image processing, the Sobel operator is commonly used to approximate

the partial derivatives for edge detection. If Gx and Gy represent the convolutions of the

image with the Sobel kernels in the horizontal and vertical directions, respectively, express

the formula for the gradient magnitude $$I$$ using Gx and Gy

c$)$ The gradient magnitude $$I $$ is used to identify regions of significant intensity change in

an image. How does the choice of threshold influence the sensitivity and specificity of edge

detection? Provide a brief explanation.

d$)$ Given an image I$($x$,$y$),$ suppose the gradient magnitude $$I $$ is calculated, and a pixel has

a high gradient magnitude. What does this imply about the intensity variation at that pixel

in the original image? Explain the relationship between high gradient magnitude and image

features.

e$)$ The gradient magnitude can be calculated using different gradient operators, such as the

Sobel, Prewitt. Compare and contrast the Sobel and Prewitt operators in terms of their

convolution kernels and their impact on edge detection. How do their gradients differ?