Consider the following two candidate horizontal derivative filters. \[ \begin{bmatrix} 1 & -1 & 0\\ \end{bmatrix} \] \[ \begin{bmatrix} 1 & 0 & -1\\ \end{bmatrix} \]
- Why is the negative number to the right of the positive one?
- If we wanted to accurately calculate the slope with correct scale, how would we need to modify the above kernels?
- What are the relative merits of each of these filters?
Compute the following convolution, which results in a new filter kernel, and describe the effect of this new kernel in words. \[ \begin{bmatrix} 1 & 2 & 1\\ 2 & 4 & 2\\ 1 & 2 & 1 \end{bmatrix} * \begin{bmatrix} 0 & 0 & 0\\ 1 & 0 & -1\\ 0 & 0 & 0 \end{bmatrix} = \begin{bmatrix} \ & \ & \ \\ \ & \ & \ \\ \hspace{1em} & \hspace{1em} & \hspace{1em} \end{bmatrix} \]
Using the language of “low-” and “high-frequency” image content, explain why sharpening is not the inverse of blurring, and what it accomplishes instead.
Consider the original image of beans on the left, and the
processed version on the rihgt. Describe what has changed in terms of
frequency content. 
What’s the maximum frequency (expressed in full periods per pixel) representable in a 1D image (i.e., a row of pixels)? What does such an image look like?
What’s the minimum frequency representable in a 1D image? What does such an image look like?
If you walked far away from the image described in #5 until you couldn’t distinguish individual pixels, what would it look like?
The sobel filter detects edges in a fairly narrow (heh) sense. If your image is sufficiently high-resolution, a fairly prominent edge might still run across many pixels. For example, at original resolution, the bright left edge of Beans’ ear in the original photo above is probably 10+ pixels wide, so a 3x3 sobel edge detector (or a Canny edge detector) wouldn’t fire very strongly on that edge. Can you devise an approach to edge detection that would find edges at a range of spatial frequencies, rather than just the finest details at the original image’s resolution?