Computer Vision Foundations · free preview

The Convolution Theorem

Convolution in space is multiplication in frequency, complex exponentials are the eigenfunctions of every linear shift-invariant system, and spectra make surgical edits possible.

1. The theorem itself

The theorem itself

2. Eigenfunctions of every filter

Eigenfunctions of every filter

3. The MIT columns

The MIT columns

4. The dual rule and windowing

The dual rule and windowing

5. Why filters cannot invent frequencies

A linear shift-invariant filter is applied to an image whose spectrum is zero beyond some maximum frequency. What can be said about the output spectrum?

6. Predicting the column peaks

A facade photo shows railings repeating every 8 pixels across a 256-pixel-wide image. Where do the bright spectral peaks land, and what does zeroing that band do?

7. Frequency of a repeating pattern

Vertical stripes repeat every $8$ pixels in an image $256$ pixels wide. Write the horizontal frequency $u$ of the spectral peak as a number.

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