Fractals and Terrain Synthesis

WALL-E, 2008]

Proceduralism

• Philosophy of algorithmic content creation • Frees up artist time to concentrate on most important elements (hero characters, major locations) • Musgrave: "not one concession to the hated user"

Simulation and Optimization

• Simulation: – models through simulation of underlying process – control through initial settings – may be difficult to adjust rules of simulation • Optimization: – models through energy minimization – control through constraints, energy terms – may be difficult to design energy function

[Rusnell, Mould, and Eramian 2009]

Height Fields

• Each point on xy-plane has a unique height value • Convenient for graphics – simplifies representation (can store in 2D array) • Used for terrain, water waves • Drawback: not able to represent full range of possibilities

Height Fields and Texture

• Can use any texture synthesis process to generate height fields – simply interpret intensity as height, create mesh, render • The most successful processes have used fractals – self-similarity a feature of real terrains – self-similarity defining characteristic of fractals

Iterated Function Systems

• Show up frequently in graphics • L-systems replacement grammar a celebrated example • Capable of producing commonly cited fractal shapes – Sierpinski gasket – Menger sponge – Koch snowflake

Mandelbrot Set

• Said to “encode the Julia sets” • coloring of the complex plane for connectivity of quadratic Julia sets – say Jc is the set for z n+1 = z n 2 + c • Point c is in the Mandelbrot set if Jc is connected, not in the set otherwise • Partitions complex plane – “Mandelbrot separator” – fractal curve

Mandelbrot set calculation

• Turns out that it is quite straightforward to get the Mandelbrot set computationally: – for each pixel c: • let z0 = c • compute z = z 2 +c repeatedly, until – (a) |z| > 2 (diverges) – (b) iteration count exceeds constant (say 1000) • if diverged, color it according to the iteration number on which it diverged • if never diverged, color with some special color

Fractals

• Nonfractal complexity: arises from accretion of different kinds of detail – e.g., people: complex, but not self-similar • Fractal complexity: arises from repeating the same details – What detail to repeat?

– Perlin noise a suitable source of detail

Multiresolution Noise

• Different signals at different scales • Fractals: clouds, mountains, coastlines 1/2 1/4 1/8 1/16 sum

Multiresolution Noise

• FNoise(x,y,z) = sum((2^-i)*Noise(x*2^i…)) • Extremely common formulation – so common that many mistake it for the basic noise primitive

Fractional Brownian Motion

• aka fBm • requires parameter H (relative size of successive octaves – "roughness") val = 0; for (i = 0; i < octaves; i++) { val += Noise(point)*pow(2,-H*i); point *= 2; }

Fractional Brownian Motion

• aka fBm • requires parameter H (relative size of successive octaves – "roughness") val = 0; for (i = 0; i < octaves; i++) { val += Noise(point)*pow(2,-H*i); point *= 2; } why 2?

"Lacunarity" parameter

Lacunarity

• "Lacunarity" (from Latin "lacuna", gap) gives the spacing between octaves • Larger values mean fewer octaves needed to cover same range of scales – faster to compute – but individual octaves may be visible • Smaller values mean more densely packed octaves, richer appearance

Lacunarity

• Balance between speed and quality • Value of 2 the "natural" choice – but in genuinely self-similar fractals, may lead to visible artifacts as same features pile up • Transcendental numbers good – genuinely irrational, no piling at any scale • Values slightly over 2 offer good compromise of speed/appearance – e-1/2, π-1

Fractal ranges of scale

• Real fractals are band-limited: they have detail only at certain scales • Computed fractals also band-limited – practical limitations: don’t write code with infinite loops • Mandelbrot: fractal objects have 3+ scales

Midpoint Displacement

• Repeated subdivision: – begin with two endpoints; at each step, divide each edge and perturb the midpoint – In 2D: on alternate steps, divide orthogonal and diagonal edges • Among the first fractal terrain systems (Fournier/Fussell/Carpenter 1982) • Problems: seams from early points

Midpoint Displacement

Midpoint Displacement

Characteristics of fBm

• Homogeneous: the same everywhere • Isotropic: the same in all directions • Real terrains are neither – mountains differ from plains – direction can matter (e.g., rivers flow downhill) • Require multifractals

Multifractals

• Fractal dimension varies with location • Simple multifractal: multiplicative cascade val = 1; for (i = 0; i < oct; i++) { val *= (Noise(point)+offset)*pow(2,-H*i) point *= 2; }

Problems

• Multiplicative formation unstable (can diverge) • Extremely sensitive to value of offset • Control elusive

Hybrid multifractals

• In real terrains, higher areas are rougher (new mountains) and lower areas smoother (worn down, silted over) • Musgrave: weight of each octave multiplied by current value of function – near value=0 (“sea level”), higher frequencies damped – very smooth – higher values: more jagged – need to clamp value to prevent divergence

Ridges

• Simple trick to get ridges out of noise: – Noise values range from -1 to 1 – Take 1-|N(p)| – Absolute value reflects noise about y=0; negative moves reflections to top • Cellular texture (Voronoi regions) naturally has ridges, if distance interpreted as height

von Koch snowflake

L-Systems

• "Lindenmeyer systems", after Aristid Lindenmeyer (1960's) • Replacement grammar – set of tokens – rules for transformation of tokens – All rules applied simultaneously across string

L-Systems

• Very successful for modeling certain classes of structured organic objects – ferns – trees – seashells • Success has impelled others to apply the methods more widely – rust – entire ecosystems

L-System example

• Tokens: A, B • Rules – A → B – B → AB

L-System example

• Tokens: A, B • Rules – A → B – B → AB • Initial string: A • Sequence: A, B, AB, BAB, ABBAB… • Lengths are Fibonacci numbers (why?)

Geometric Interpretation

• Strings are interesting, but application to graphics requires geometric interpretation • Usual method: interpret individual tokens as geometric primitives

Turtle Graphics

• The language Logo (1967) – once widely used for education • Turtle has heading and position; when it moves, it draws a line behind it • Commands: – F, B: move forward/backward fixed distance – +,- : turn right/left fixed angle – [, ] : push or pop the current state – A : no-op

L-Systems and the Turtle

• Example replacement rules for the turtle: – F → F-F++F-F – everything else unchanged

von Koch snowflake

Branching

• 'Push' and 'pop' operators can produce branching: – A → F[+A][-A]FA – F → FF • A is an 'apex' – the tip of a branch • Each apex sprouts a new branch with buds midway along its length, while existing branches elongate

Turtle Graphics in 3D

• Turtle has orientation and position • Commands: – F, B: move forward/backward fixed distance – +,- : turn right/left fixed angle (yaw) – ^,& : turn up/down fixed angle (pitch) – \, / : roll right/left fixed angle – [, ] : push or pop the current state – A : no-op

Ternary Tree

• As usual, just one rule: • F → F[&F][/&F][\&F] • Each segment has three branches attached to its tip