Question

Let I be a grayscale image where all the even rows have a value of 0 and all the odd rows have a value of 100 . Note that the first row in the image is assumed to be row 0 . Let G be a 3×3 smoothing filter designed to average entries using equal weights in a manner that will not increase the overall image intensity. Convolve the filter G with the image I and find the value at location (1,1) of the convolved image. Divide this value by 10 and round. Use the resulting value as your answer.

Answer 1

To prove that I + I' = Io, use Malus's law for intensities through polarizing filters, and apply trigonometric identities to show the sum equals the original intensity, Io.

Step-by-step explanation:

The question asks us to prove that the sum of the intensities of light transmitted through two polarizing filters at angles θ and 90.0°-θ equals the original light intensity I0. According to Malus's law, the intensity I after passing through a filter oriented at an angle θ with respect to the direction of polarization of the light is given by I = I0cos²θ. For the second configuration, the intensity I' after passing through the filter oriented at an angle 90.0°-θ would use the identity cos(90.0°-θ) = sinθ, resulting in I' = I0sin²θ. Adding both intensities, we get I + I' = I0cos²θ + I0sin²θ = I0(cos²θ + sin²θ). Using the trigonometric identity cos²θ + sin²θ = 1, the sum becomes I + I' = I0, thus proving the equation.