Lecture 3 Problems - CSCI 497P / 597P

1. Recall that $f \otimes w$ means cross-correlating image $f$ with filter $w$. Compute the following cross-correlation using same output size and zero padding.

   $$\begin{bmatrix} 0 & 1 & 0\\ 0 & 1 & 0\\ 0 & 1 & 0 \end{bmatrix} \otimes \begin{bmatrix} 1 & 2 & 1\\ 2 & 4 & 2\\ 1 & 2 & 1 \end{bmatrix}$$

    

2. Perform the same convolution as above, but use repeat padding.

3. Perform the same convolution as above, but use valid output size.

4. Describe in words the result of applying the following filter using cross-correlation. If you aren't sure, try applying it to the image above to gain intuition.

5. For each of the following, determine 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

   • Partial derivative: the output is a finite-differences approximation of the input image's vertical derivative $\frac{\partial}{\partial y} f(x, y)$. Note: don't worry if you haven't seen partial derivatives! We'll review them next class.