1. Introduction to fractional Brownian Motion for Terrain
CS 658

2. Brownian Motion time step = 2048 time step = 256 time step = 32

3. Brownian Motion this line is about 5 units long time step = 2048
then this line is about 3 units long
define this line to be 1 unit long

4. Brownian Motion this line is about 5 units long time step = 2048
then this line is about 3 units long
define this line to be 1 unit long
Note that the measured velocity of the particle changes! The velocity didn’t change, it’s one particle with one velocity, but the measured velocity did change.

5. Brownian Motion this line is about 5 units long time step = 2048
then this line is about 3 units long
define this line to be 1 unit long
Note that the measured velocity of the particle changes! The velocity didn’t change, it’s one particle with one velocity, but the measured velocity did change.
Measured velocity of the particle depends on the time scale at which you measure.

6. Linear Motion (no Acceleration)4 8 12 16
t = 0, L = 0
Assume it’s moving at 1 unit of space per unit of time.

7. Linear Motion (no Acceleration)4 8 12 16
t = 0, L = 0
t = 1, L = 1
t = 4, L = ???
Velocity:
Δt=1, velocity =
Δt=4, velocity =
Δt=16, velocity =

8. Linear Motion (no Acceleration)4 8 12 16
t = 0, L = 0
t = 1, L = 1
t = 4, L = 4
Velocity:
Δt=1, velocity = 1 m/s
Δt=4, velocity = 1 m/s
Δt=16, velocity = 1 m/s
Remember: no acceleration

9. Brownian Motion

10. Brownian Motion Location at a time a little after time t
Normally distributed
random variable.
Amount of time that has passed but tweaked so that velocity increases with small decreases in t (when t < 1)
Location at time t
Upper bound on velocity
Probability, P, that a particle has velocity below x,
when measured at time interval ∆t, is normally distributed.

11. Brownian Motion
Velocity of the particle adjusted so that the velocity increases with small changes in time.
This reflects the fact that in a short time period, a Brownian motion particle can go fast in a given direction.
H is an exponent which is in [0,1] (in nature) and is called the Hölder exponent.

12. Brownian Motion this line is about 5 units long time step = 2048
then this line is about 3 units long
define this line to be 1 unit long

13. Brownian Motion in the Frequency Domain
Look at 1D Brownian motion…

14. Brownian Motion in the Frequency Domain
Look at 1D Brownian motion…
… in the frequency domain
Slope = 1/f2
Amplitude (log)
Frequency (log)

15. Brownian Motion in the Frequency Domain
Slope = 1/f2
Amplitude (log)
Frequency (log)
This, by the way, is called “Brown noise”

16. Brownian Motion in the Frequency Domain
Low frequency, high amplitude
High frequency, low amplitude

17. Brownian Motion in the Frequency Domain
Slope = 1/f2
Amplitude (log)
Frequency (log)

18. Weierstrauss-Mandelbrot Equation
Slope = 1/f2
Amplitude (log)
Frequency (log)
Sample along this line, randomize things a bit, sum the resulting sine waves and get fBm.

19. Influence of the Holder Exponent
From O. Deussen et al. “Digital Design of Nature” Springer 2005

20. Influence of the Holder Exponent
And you can convince yourself that
H=2.0 looks like rolling hills.
H=1.0 looks like the Alps.
From O. Deussen et al. “Digital Design of Nature” Springer 2005

21. Weierstrass-Mandelbrot functions

22. Weierstrass-Mandelbrot functions
Amplify the signal
Change the period
Normally distributed
random variable
Random phase shift
Frequency
Sum of sine functions