Modeling. Michael Kazhdan ( /657) HB FvDFH Modeling Seashells, Fowler et al. 1992

Transcription

1 Modeling Michael Kazhdan ( /657) HB FvDFH Modeling Seashells, Fowler et al. 1992

2 Modeling How do we... Represent 3D objects in a computer? Construct such representations quickly and/or automatically with a computer? Manipulate 3D objects with a computer? H&B Figure Fowler H&B Figure 10.83b

3 Model Construction Interactive modeling tools CAD programs Subdivision/Spline surface editors Active scanners CAT/MRI, light scanners, touch probe, etc. Passive scanners Stereo, motion, etc. Procedural generation Sweeps, fractals, grammars

4 Interactive Modeling Tools User constructs objects with drawing program Menu commands, direct manipulation, etc. CSG, parametric surfaces, quadrics, etc. Cosmoworlds, SGI

5 Model Construction Interactive modeling tools CAD programs Subdivision/Spline surface editors Active scanners CAT/MRI, light scanners, touch probe, etc. Passive scanners Stereo, motion, etc. Procedural generation Sweeps, fractals, grammars

6 Active Scanners Acquire geometry of objects with active sensors CAT/MRI Light scanners Robotic arm etc.

7 Depth Color Active Scanners Acquire geometry of objects with active sensors CAT/MRI Light scanners Robotic arm etc. x p (x c, y c )

8 Active Scanners Acquire geometry of objects with active sensors CAT/MRI Light scanners Robotic arm etc.

9 Active Scanners Triangulation (in 2D): To figure out the position of a point we need: 1. The position of the camera 2. The position of the light source Project the light onto the surface Find where the lit point projects onto the camera Cast a ray from the lit pixel, through the camera

10 Active Scanners Triangulation (in 2D): To figure out the position of a point we need: 1. The position of the camera 2. The position of the light source Project the light onto the surface For Find the where lit point the to project The lit point position projects of the lit point onto the appropriate pixel, onto the camera it has to lie somewhere has to on be at the intersection the Cast ray. a ray from the of lit the pixel, two through rays the camera The lit point is also constrained to lie on the ray from the light source.

11 Laser Range Scanning Example: 70 scans Stanford Graphics Laboratory

12 Active Scanners Acquire geometry of objects with active sensors CAT/MRI Light scanners Touch probe etc.

13 Model Construction Interactive modeling tools CAD programs Subdivision/Spline surface editors Active scanners CAT/MRI, light scanners, touch probe, etc. Passive scanners Stereo, motion, etc. Procedural generation Sweeps, fractals, grammars

14 Passive Scanners Infer 3D geometry from (passive) images Stereo Motion etc.

15 Passive Scanners Infer 3D geometry from (passive) images Stereo Motion etc. This is similar to the approach for scanners, but instead of triangulating using a light and a camera, we triangulate using two cameras. The challenge is to determine pixel pair correspondences across the two images.

16 Passive Scanners Infer 3D geometry from (passive) images Stereo Motion etc. Image courtesy of

17 Passive Scanners Infer 3D geometry from (passive) images Stereo Motion etc. In this case we need to solve simultaneously for the surface points and the camera position. Image courtesy of

18 Model Construction Interactive modeling tools CAD programs Subdivision/Spline surface editors Active scanners CAT/MRI, light scanners, touch probe, etc. Passive scanners Stereo, motion, etc. Procedural generation Sweeps, fractals, grammars

19 Procedural Modeling Goal: Describe 3D models algorithmically Best for models resulting from... Repeating processes Self-similar processes Random processes Advantages: Automatic generation Concise representation Parameterized classes of models

20 Procedural Modeling Sweeps Fractals Grammars

21 Sweeps Given a 3D sweep curve H(Θ) and a 2D generating curve C(Ψ), we define the sweep surface S(Θ, Ψ) as the sweep of C along H: C(Ψ) S(Θ, Ψ) H(Θ) In this example, the sweep curve is simply used to translate the generating curve: S Θ, Ψ = H Θ + C(Ψ) We can define more complex sweep surfaces.

22 Example: Seashells Create 3D polygonal surface models of seashells Modeling Seashells, Deborah Fowler, Hans Meinhardt, and Przemyslaw Prusinkiewicz, Computer Graphics (SIGGRAPH 92), Chicago, Illinois, July, 1992, p Fowler et al. Figure 7

23 Example: Seashells Sweep generating curve around helico-spiral axis Generating Curve

24 Example: Seashells Sweep generating curve around helico-spiral axis As the generating curve is swept, it is: Translated Scaled Generating Curve Rotated

25 Example: Seashells Sweep generating curve around helico-spiral axis Helico-Spiral definition: H Θ = cos Θ r Θ, z Θ, sin Θ r Θ Angle: Θ Radius: r Θ = e λθ Height: z Θ = e μθ If C(Ψ) is the generating curve, we Fowler et al. Figure 1 can try to represent the surface as: S θ, Ψ = H Θ + C u Ψ, C v Ψ, 0 r Θ

26 Example: Seashells Sweep generating curve around helico-spiral axis Helico-Spiral definition: H Θ = cos Θ r Θ, z Θ, sin Θ r Θ Angle: Θ Radius: r Θ = e λθ Height: z Θ = e μθ This doesn t rotate the generating curve around the axis of the helix! If C(Ψ) is the generating curve, we can try to represent the surface as: S θ, Ψ = H Θ + C u Ψ, C v Ψ, 0 r Θ Fowler et al. Figure 1

27 Example: Seashells Sweep generating curve around helico-spiral axis Helico-Spiral definition: H Θ = cos Θ r Θ, z Θ, sin Θ r Θ Angle: Θ Radius: r Θ = e λθ Height: z Θ = e μθ Instead, we compute a local frame u Θ, v Θ, w Θ at each point on the sweep curve and describe the surface w.r.t. this frame: S Θ, Ψ = H Θ + u Θ C u Ψ + v Θ C v Ψ r(θ) Fowler et al. Figure 1

28 Example: Seashells Sweep generating curve around helico-spiral axis u Θ and v Θ define the plane that is Helico-Spiral definition: perpendicular to the curve H at Θ: H Θ = cos Θ r Θ, z Θ, sin Θ r Θ u(θ) is the curve normal Angle: v Θ is the Radius: r Θ = e λθ curve bi-tangent (perp. to u Θ and the curve tangent) Height: z Θ = e μθ Instead, we compute a local frame u Θ, v Θ, w Θ at each point on the sweep curve and describe the surface w.r.t. this frame: S Θ, Ψ = H Θ + u Θ C u Ψ + v Θ C v Ψ r(θ) Fowler et al. Figure 1

29 Example: Seashells Generate different shells by varying parameters Different helico-spirals Fowler et al. Figure 2

30 Example: Seashells Generate different shells by varying parameters Different generating curves Fowler et al. Figure 3

31 Example: Seashells Generate many interesting shells with a simple procedural model! Fowler et al. Figures 4,5,7

32 Procedural Modeling Sweeps Fractals Grammars

33 Fractals Defining property: Self-similar with infinite resolution Mandelbrot Set H&B Figure

34 Fractals Useful for describing natural 3D phenomenon Terrain Plants Clouds Water Feathers Fur etc. H&B Figure 10.80

35 Fractal Generation Deterministically self-similar fractals Parts are scaled copies of original Statistically self-similar fractals Parts have same statistical properties as original

36 Deterministic Fractal Generation General procedure: Initiator: start with a shape Generator: replace subparts with scaled copy of original H&B Figure 10.68

37 Deterministic Fractal Generation Apply generator repeatedly Koch Curve H&B Figure 10.69

38 Deterministic Fractal Generation Useful for creating interesting shapes! Mandelbrot Figure X

39 Deterministic Fractal Generation Useful for creating interesting shapes! Mandelbrot Figure 46

40 Deterministic Fractal Generation Useful for creating interesting shapes! H&B Figures 75 & 109

41 Fractal Generation Deterministically self-similar fractals Parts are scaled copies of original Statistically self-similar fractals Parts have same statistical properties as original

42 Statistical Fractal Generation General procedure: Initiator: start with a shape Generator: replace subparts with a self-similar

43 Statistical Fractal Generation General procedure: Initiator: start with a shape Generator: replace subparts with a self-similar random pattern Random Midpoint Displacement

44 Statistical Fractal Generation Example: terrain H&B Figure 10.83b

45 Statistical Fractal Generation Useful for creating mountains H&B Figure 10.83a

46 Statistical Fractal Generation Useful for creating 3D plants H&B Figure 10.82

47 Statistical Fractal Generation Useful for creating 3D plants H&B Figure 10.79

48 Procedural Modeling Sweeps Fractals Grammars

49 Grammars Generate description of geometric model by applying production rules S A B AB Ba a Ab b AB BaB BaAb AbaAb

50 Grammars Useful for creating plants T Start Tree Tree Branch Tree leaf Branch [Tree] cylinder = Leaf = Cylinder = Tree = Branch = [Tree] T

51 Grammars Useful for creating plants Start Tree Tree Branch Tree leaf Branch [Tree] cylinder T BT = Leaf = Cylinder = Tree = Branch = [Tree] BT

52 Grammars Useful for creating plants Start Tree Tree Branch Tree leaf Branch [Tree] cylinder c T BT BT = Leaf = Cylinder = Tree = Branch = [Tree] cbt

53 Grammars Useful for creating plants Start Tree Tree Branch Tree leaf Branch [Tree] cylinder T BT c BT [T] BT = Leaf = Cylinder = Tree = Branch = [Tree] c[t]bt

54 Grammars Useful for creating plants Start Tree Tree Branch Tree leaf Branch [Tree] cylinder = Leaf T BT c BT [T] BT [BT]c BT = Cylinder = Tree = Branch = [Tree] c[bt]cbt

55 Grammars Useful for creating plants Start Tree Tree Branch Tree leaf Branch [Tree] cylinder = Leaf = Cylinder T BT c BT [T] BT [BT]c BT [c l] [T] BT = Tree = Branch = [Tree] c[cl]c[t]bt

56 Grammars Useful for creating plants Start Tree Tree Branch Tree leaf Branch [Tree] cylinder = Leaf = Cylinder T BT c BT [T] BT [BT]c BT [c l] [T] BT [BT]c BT = Tree = Branch = [Tree] c[cl]c[bt]cbt

57 Grammars Useful for creating plants Start Tree Tree Branch Tree leaf Branch [Tree] cylinder = Leaf = Cylinder = Tree T BT c BT [T] BT [BT]c BT [c l] [T] BT [BT]c BT [c BT] [T] BT = Branch = [Tree] c[cl]c[cbt]c[t]bt

58 Grammars Useful for creating plants Start Tree Tree Branch Tree leaf Branch [Tree] cylinder = Leaf = Cylinder = Tree = Branch = [Tree] T BT c BT [T] BT [BT]c BT [c l] [T] BT [BT]c BT [c BT] [T] BT [T] BT][BT]c l c[cl]c[c[t]bt]c[bt]cl

59 Grammars Useful for creating plants Start Tree Tree Branch Tree leaf Branch [Tree] cylinder = Leaf = Cylinder = Tree = Branch = [Tree] T BT c BT [T] BT [BT]c BT [c l] [T] BT [BT]c BT [c BT] [T] BT [T] BT][BT]c [BT] c l] [c l] l c[cl]c[c[bt]cl]c[cl]cl

60 Grammars Useful for creating plants Start Tree Tree Branch Tree leaf Branch [Tree] cylinder = Leaf = Cylinder = Tree = Branch = [Tree] c[cl]c[c[cl]cl]c[cl]cl T BT c BT [T] BT [BT]c BT [c l] [T] BT [BT]c BT [c BT] [T] BT [T] BT][BT]c [BT] c l] [c l] [c l] l

61 Grammars Useful for creating plants H&B Figure 10.77

62 Summary Interactive modeling tools CAD programs Subdivision/Spline surface editors Active scanners CAT/MRI, light scanners, touch probe, etc. Passive scanners Stereo, motion, etc. Procedural generation Sweeps, fractals, grammars Constructing 3D models is hard!