Computer Vision Foundations · free preview

The Discrete Fourier Transform

Complex exponentials form an orthogonal basis for finite images, and projecting onto them gives the DFT and its fast algorithm.

1. Waves become complex exponentials

Waves become complex exponentials

2. Orthogonality, the miracle that makes it work

Orthogonality, the miracle that makes it work

3. The transform pair

The transform pair

4. Keeping only the biggest coefficients

Keeping only the biggest coefficients

5. What orthogonality buys

Why does the orthogonality of the complex exponentials let the DFT read each coefficient independently of the others?

6. The DC term

An image is constant, every pixel equal to 10. What does its DFT look like?

7. The DC coefficient of a small image

A 4×44 \times 4 image has mean value 1010 in every pixel. Write the DC coefficient L[0,0]\mathscr{L}[0,0] as a number.

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