1. Wavelet Noise

2. Perlin Noise The loss-of-detail vs. Aliasing Problem
2D slice through a 3D band-limited function is not band-limited

3. Perlin Noise (cont’d)

4. Theory

5. Orthogonality
where

6. Orthogonality (cont’d) - what we want ? -
for all j>=0

7. Wavelet Noise
Downsample
Upsample -

8. Refinability and Upsampling
S1:={G(x)|G(x)=Σgiψ(2x-i)} i
ψ(x) refinable If

9. Refinability and Upsampling (cont’d)
The sequence (…,fi↑,…) is twice as long as the sequence (…,fk,…)

10. Wavelets and Downsampling
G(x) = G↓(x) + D(x)
↑ ↑ ↑
in S in S in S1 (also in W0)
G↓(x) is the least squares best approximation to G(x) in S0
D(x) is the least squares residual and contains the information is S1 that cannot be represented in S0
Given the coefficients gi for G(x), the coefficients gi↓ for G↓(x)
The sequence (…,gk,…) is twice as long as the sequence (…,gi↓,…)

11. Wavelets and Downsampling(cont’d)
It follows that
for all integers j>=0 ↑
This is what we want

12. Constructing noise bands
Creating R(x) using a random number generator
Compute R↓(x) by downsampling R(x)

13. Constructing noise bands (cont’d)
Computing R↓↑(x) by upsampling R↓(x)
Compute N(x) by subtracting R↓↑(x) from R(x)

14. Fractional scales & translates
Contribution

15. Noise in Multiple Dimensions

16. Fourier Slice Theorem Band Limited Function Fourier Transform
Projection Profile
1D IDFT

17. Projected Noise

18. Result

19. Rendered Image