CSE 788 Fractal Terrains

1. CSE 788Fractal Terrains

2. Terrain Map • Height Map z = f(x, y) x and y are sampled on a 2D integer grid • Real data : Satellite, Elevation maps • Synthetic : Texture map, Noise functions

3. Terrain Map • Connect samples into a mesh

4. Terrain Map • Natural looking • Model high detail • Size • Storage (Millions of polygons) • Rendering speed • Zoom-in, zoom-out (does not scale well) • Use textures to model detail? • Not real geometry (bump mapping at silhouettes) • Not scale independent • Repeated textures do not work well

5. Fake Terrain • Generate the heightfield • Random process • Reflects “realistic” terrain in some way

6. Random Terrain • Simple: • Terrain(x,y) = rand( MAX_HEIGHT ) • Results in random noise • Next step: • Smooth the terrain generated above • FIR filter:

7. Filtering • FIR (Finite Impulse Response) filter: • Pass a “window” over the data • Multiply data in window by weights • Result = d1*w1 + d2*w2+ … + dN*wN • Window + weights is called “filter kernel” • Larger window grabs more data

8. +dHeight Random Fault Lines • Repeatedly: • Create a line that crosses terrain • Add dHeight offset on one side of line • dHeighti = dHeight0 + (i/n)(dHeightn -dHeight0) • Select random endpoints to generate line • Compute “side of line” by plugging X,Y into line equation

9. More Realistic Faults • Apply FIR filter to faulted heightmap • Smooth out abrupt fault transitions: “Erosion”

10. 1000 iterations

11. Variations • Use a smoothstep function

12. Variations • Use a bump: • -d at distance R away • +d at zero. • Images from lighthouse3d • Used a cosine here.

13. Variations • Cosine Bump

14. Variations • Do not need to use a half-plane. Any function that splits space into two halves will work. • Radial functions • Etc.

15. Particle Deposition • Drop particles in areas where you want volcanic mountains • Particles flow downhill until they come to rest • Move the drop point to create interesting peaks

16. Particle Deposition • Flow computation • At given point, find lowest neighbor • If local geometry is not flat enough, move new particle on top of lowest neighbor • Roll down hill • “Flat enough” = 27 degrees or so • Check 3x3 pixel window or 5x5 to check neighbors.

17. Particle Deposition

18. Perlin Noise • Using a sampling of 2D perlin Noise provides smooth hills.

19. Terrain Coloring • Using a 1D texture map based on the altitude can provide many useful mapping.

20. Terrain Coloring • Striped 1D texture map.

21. Terrain Coloring • Using a 2D texture map provides richer detail, but is independent of the terrain.

22. Terrain Coloring • More advanced coloring is based on altitude and slope.

23. Rolling Hills • Scaling in one dimension gives smooth rolling hills.

24. Procedural Modeling With Fractals • Procedural Modeling • Compute geometry “on-the-fly” • Fractals • Model Natural Phenomena - Self Similarity • Mountains, fire, clouds, etc. • Scales to infinity • Add or “make up” natural looking details with mathematical tools

25. Fractals “Repetition of form over a variety of scales” • Mandelbrot set, Julia set

26. Two Fractal Properties • Self-similarity

27. Two Fractal Properties • Fractal Dimension • Euclidean dimensions : 1, 2, 3, 4, … • Fractal : 1.2342, 2.7656 • Measure of detail or roughness of a fractal D = (ln N)/(ln 1/s)

28. Midpoint Subdivision • Midpoint (recursive) subdivision

29. Midpoint Subdivision • Brownian Motion • Describes random movement of particles in a gas or fluid • Fractional Brownian Motion • Brownian Motion + Fractal Dimension • A useful model for natural phenomena

30. Fractional Brownian Motion • Fractional Brownian Motion • Equations are compute intensive • Approximate with “A family of 1D Gaussians ” • Zero mean • Standard Deviation : S = k2-iH H = fractal dimension (roughness)

31. Fractional Brownian Motion • Fractal dimension = roughness, I.e. H

32. Terrain Modeling Criteria • Input • Initial coarse mesh + stochastic parameters • Two criteria • Internal Consistency • Reproducibility : Model is independent of position and orientation • Associate “random numbers” to point index • External Consistency • Continuity between adjacent primitives

33. A B A B A B Fractal Mountains • Fournier & Fussell (1980) • Recursively subdivide geometry by random number d: –dHeight/2 < d < dHeight/2 • At each recursion: • dHeight *= 2-r • r=1 : self-similar • r>1 : large features early • r<1 : large features late

34. Triangle Subdivision • Subdivide a triangle into 4 triangles at edge midpoint

35. Quadrilateral Subdivision • Subdivide a quad into 4 quads at edge midpoints and a new center point.

36. Diamond-Square Subdivision • Alternate subdivision • Alleviates some creases on the grid.

37. Fractal Terrain

38. Mesh Subdivision • Square-square Subdivision • Addresses “creasing problem” (slope discontinuities) • Subdivide parametric patches

39. Mesh Subdivision • Displacement is scaled by the recursion level. • | b – a | -rn • When do you stop the recursion? • Pixel resolution • Displace upward, or towards the normal to the surface? • Implies that it is not a height-map. • Could use Geometry Images here though.

40. Mesh Subdivision • External Consistency • Avoid tears between neighboring polygons • How do we compute the normals? • Average polygon normals at each vertex. • Propagate normals down the recursion • Cheaper : use the original mesh normals only

41. Ridged Fractal Terrains • To create sharp peaks, add an absolute value and flip the surface upside down. • Or, reflect about a maximum value. • Provides a volcano-like effect.

42. Caldera

43. Multi-Fractal • This produces a ridged multi-fractal

44. Add a 1D Texture

45. More Controls • To create plateaus, add a min function to flatten out high areas.

46. More Controls • To create a plain, add a max function to flatten out low areas.

47. More Controls • Multiply by a Guassian to limit the mountain range.

48. More Controls • To create valleys, create a lower amplitude and rather smooth terrain to use as the max operator.

49. max (Fractal, Smooth Noise)

50. max (Fractal, Smooth Noise)