Blur as Low-Pass Filtering

Why averaging means blur

The box filter

Separability and the integral image

DC gain and brightness preservation

A $3 \times 3$ kernel has every tap equal to $1$, unnormalized. What does a constant image of value $6$ look like after convolution with it, away from the borders?

A $7 \times 7$ box blur can run either as one two-dimensional convolution or as the separable cascade $\text{box}_{3,0} \circ \text{box}_{0,3}$. How many multiplications per output pixel does each variant take, and which is cheaper?

An unnormalized $3 \times 3$ box filter has all nine taps equal to $1$. Compute its DC gain, the sum of all its taps, and write the result as an integer.