Computer Vision Foundations · free preview

Decimation

Decimation as a linear operator, its Fourier-domain meaning as a superposition of shifted spectral copies, and the two-coefficient averaging identity.

1. Keeping every k-th sample

Keeping every k-th sample

2. What the spectrum becomes

What the spectrum becomes

3. The same fact as a matrix identity

The same fact as a matrix identity

4. Reading the identity as a warning

Reading the identity as a warning

5. Meaning of the superposition formula

According to L↓k[u,v]=1k2∑s,rL[u−sN/k,v−rM/k]\mathscr{L}_{\downarrow k}[u,v] = \frac{1}{k^2}\sum_{s,r}\mathscr{L}[u - sN/k, v - rM/k], what does one coefficient of the decimated spectrum contain?

6. Verifying the matrix identity

For N=16N = 16, the identity gives L↓2[0]=0.5 L[0]+0.5 L[8]\mathscr{L}_{\downarrow 2}[0] = 0.5\,\mathscr{L}[0] + 0.5\,\mathscr{L}[8]. Where does the index 8 come from?

7. The partner index

A signal of length N=16N = 16 is decimated by k=2k = 2. The decimated DC coefficient obeys L↓2[0]=0.5 L[0]+0.5 L[j]\mathscr{L}_{\downarrow 2}[0] = 0.5\,\mathscr{L}[0] + 0.5\,\mathscr{L}[j]. Write the partner index jj.

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