Lecture 4 Problems - CSCI 497P / 597P

1. For each of the following, decide whether it's possible to design a convolution filter that performs the given operation.

   • Max filter: the output pixel is the maximum value among the pixels in the input window

   • Threshold: the output pixel is

     • 255 if the input pixel is > 127
     • 0 otherwise
     • $y$ partial derivative: the output is a finite-differences approximation of the input image's vertical derivative $\frac{\partial}{\partial y} f(x, y)$¹ .

2. Compute the following convolution, assuming "repeat" padding and "same" output size.

   $$\begin{bmatrix} 0 & 0 & 1\\ 0 & 0 & 1\\ 0 & 1 & 1 \end{bmatrix} * \begin{bmatrix} 0 & 0 & 0\\ 1 & 0 & -1\\ 0 & 0 & 0 \end{bmatrix} = \begin{bmatrix} \ & \ & \ \\ \ & \ & \ \\ \hspace{1em} & \hspace{1em} & \hspace{1em} \end{bmatrix}$$

3. Will the answer to #2 change if you use the following non-square filter instead?

   $$\begin{bmatrix} 1 & 0 & -1\\ \end{bmatrix}$$

4. Verify that convolving the following approximate 3x3 Gaussian blur kernel with a centered horizontal derivative kernel results in the horizontal Sobel operator.

   $$\begin{bmatrix} 1 & 2 & 1\\ 2 & 4 & 2\\ 1 & 2 & 1 \end{bmatrix}$$