$5\mathrm{.}2.$ Comparison of Pooling Methods $(15$ points$).$

Suppose an image computed just after the application of a two$-$dimensional convolution kernel in a convolutional neural network is as shown at the top of the next page.

$1$

$\backslash$table$[[$Input Image$],[3,4,-4,1,1,-4],[-2,1,2,-1,-4,3],[5,0,3,-4,-1,2],[3,4,2,-5,4,5],[2,3,-3,-1,2,-4],[1,0,2,-5,-4,4]]$

Determine the output images from the following operations, and draw them on the left below the words "Output Image". In addition, by placing $\text{'}$s$,$ indicate in the array below the words "Gradient Update Positions" which original pixel locations correspond to center positions that result in coefficients being updated via gradient calculations.

a$)$ Application of ReLU nonlinearity, followed by $(22)$ max$-$pooling with a stride of $2.$ Output Image

Gradient Update Positions

$\backslash$table$[[,,,,,],[,,,,,],[,,,,,],[,,,,,],[,,,,,],[,,,,,]]$

b$)$ Application of ReLU nonlinearity, followed by $(33)$ max$-$pooling with a stride of $3.$ Output Image

Gradient Update Positions

$\backslash$table$[[,,,,,],[,,,,,],[,,,,,],[,,,,,],[,,,,,],[,,,,,]]$

c$)$ Application of ReLU nonlinearity, followed by $(22)$ average$-$pooling with a stride of $2.$ Output Image

Gradient Update Positions

$\backslash$table$[[,,,,,],[,,,,,],[,,,,,],[,,,,,],[,,,,,],[,,,,,]]$

d$)$ Application of $(33)$ max$-$pooling with a stride of $2,$ followed by ReLU nonlinearity. Output Image

Gradient Update Positions

$\backslash$table$[[,,,,,],[,,,,,],[,,,,,],[,,,,,],[,,,,,],[,,,,,]]$

e$)$ Application of $(22)$ average$-$pooling with a stride of $3,$ followed by ReLU nonlinearity. Output Image

Gradient Update Positions

$\backslash$table$[[,,,,,],[,,,,,],[,,,,,],[,,,,,],[,,,,,],[,,,,,]]$