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. 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?