Computer Vision Foundations · free preview

Discretizing the Derivative

The continuous derivative becomes a small convolution kernel, the backward difference shifts by half a sample, and the centered difference trades resolution for alignment.

1. From limits to taps

From limits to taps

2. The half-sample shift

The half-sample shift

3. The Fourier verdict

The Fourier verdict

4. Two dimensions and history

Two dimensions and history

5. Where the backward difference files its answer

The kernel $d_0 = [1, -1]$ compares sample $n$ with sample $n-1$ . At which location does its output naturally belong, and what does that imply on a step edge?

6. What the central difference gives up

The central difference $d_1 = [1,0,-1]/2$ has response $j\sin(2\pi u/N)$ . What does this response do at the highest image frequency?

7. Running the backward difference

The signal reads 5 then 7 then 2 across three samples. Apply $d_0 = [1, -1]$ and write the first valid output value, the one that compares the second sample against the first.

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