Computer Vision Foundations · free preview

What the Box Filter Gets Wrong

The box filter's gain is not monotone in frequency, it lets the fastest alternating wave pass while nulling a slower one, and repeated boxes build triangles rather than boxes.

1. The gain that ripples

The gain that ripples

2. A wave the box cannot see

A wave the box cannot see

3. Boxes do not compose

Boxes do not compose

4. Consequences for repeated blurs

Consequences for repeated blurs

5. Reading the ripple

For the three-tap box [1,1,1][1,1,1] on a signal of length NN, which statement about the gain H[u]=sin⁡(3πu/N)/sin⁡(πu/N)H[u] = \sin(3\pi u/N)/\sin(\pi u/N) is correct?

6. What two boxes make

An image is blurred with the kernel [1,1,1][1,1,1] and then blurred again with the same kernel, with no normalization in between. What is the combined kernel of the two passes?

7. The wave that survives

The alternating wave [1,−1,1,−1][1, -1, 1, -1] passes through the kernel [1,1,1][1,1,1]. Compute the output value at the position aligned with the third input sample, that is the sum 1×1+1×(−1)+1×11 \times 1 + 1 \times (-1) + 1 \times 1, and write the ratio of this output amplitude to the input amplitude 11.

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