Introduction to fractional Brownian Motion for Terrain CS 658

Brownian Motion time step = 2048 time step = 256 time step = 32 http://en.wikipedia.org/wiki/Image:Brownian_hierarchical.png

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

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.

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.

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.

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 =

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

Brownian Motion

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.

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.

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

Brownian Motion in the Frequency Domain Look at 1D Brownian motion… http://www.stat.umn.edu/~charlie/Stoch/brown.html

Brownian Motion in the Frequency Domain Look at 1D Brownian motion… … in the frequency domain Slope = 1/f2 Amplitude (log) Frequency (log) http://www.stat.umn.edu/~charlie/Stoch/brown.html

Brownian Motion in the Frequency Domain Slope = 1/f2 Amplitude (log) Frequency (log) This, by the way, is called “Brown noise” http://www.stat.umn.edu/~charlie/Stoch/brown.html

Brownian Motion in the Frequency Domain Low frequency, high amplitude High frequency, low amplitude http://www.stat.umn.edu/~charlie/Stoch/brown.html

Brownian Motion in the Frequency Domain Slope = 1/f2 Amplitude (log) Frequency (log) http://www.stat.umn.edu/~charlie/Stoch/brown.html

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. http://www.stat.umn.edu/~charlie/Stoch/brown.html

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

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

Weierstrass-Mandelbrot functions

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