o Represent 3D objects in a computer? o Manipulate 3D objects with a computer? o CAD programs o Subdivision surface editors :)

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1 Modeling How do we... o Represent 3D objects in a computer? Procedural Modeling Adam Finkelstein Princeton University COS 426, Spring 2003 o Construct such representations quickly and/or automatically with a computer? o Manipulate 3D objects with a computer? H&B Figure Fowler H&B Figure 10.83b Modeling How do we... o Represent 3D objects in a computer? o Construct such representations quickly and/or automatically with a computer? o Manipulate 3D objects with a computer? Model Construction o CAT, MRI, laser, magnetic, robotic arm, etc. H&B Figure Fowler H&B Figure 10.83b Interactive Modeling Tools User constructs objects with drawing program o Menu commands, direct manipulation, etc. o CSG, parametric surfaces, quadrics, etc. Interactive Modeling Tools Example: Mechanical CAD Cosmoworlds, SGI H&B Figure 9.9

2 Model Construction o Laser, magnetic, robotic arm, etc. Scanning tools Acquire geometry of objects with active sensors o CAT/MRI o Laser range scanner o Magnetic sensor o Robotic arm Lorensen Stanford Graphics Laboratory Scanning tools Acquire geometry of objects with active sensors o CAT/MRI o Laser range scanner o Magnetic sensor o Robotic arm Depth Color Laser Range Scanning Example: 70 scans o Volumetric reconstruction Xp (Xc,Yc) Stanford Graphics Laboratory Scanning tools Acquire geometry of objects with active sensors o CAT/MRI o Laser range scanner o Magnetic sensor o Robotic arm Scanning tools Acquire geometry of objects with active sensors o CAT/MRI o Laser range scanner o Magnetic sensor o Robotic arm

3 Model Construction o Laser, magnetic, robotic arm, etc. Computer Vision Infer 3D geometry from images o Stereo o Motion o Constraints Computer Vision Infer 3D geometry from images o Stereo o Motion o Constraints Computer Vision Infer 3D geometry from images o Stereo o Motion o Constraints Debevec96 Model Construction o Laser, magnetic, robotic arm, etc. Procedural Modeling Goal: o Describe 3D models algorithmically Best for models resulting from... o Repeating processes o Self-similar processes o Random processes Advantages: o Automatic generation o Concise representation o Parameterized classes of models

4 Procedural Modeling Sweeps Fractals Grammars 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 Sweep generating curve around helico-spiral axis Connect adjacent points to form polygonal mesh Helico-spiral definition: Θi+ 1 = Θi + Θ ri + 1 = ri λr z = z λ i+ 1 i z Fowler et al. Figure 1 Fowler et al. Figure 6 Model is parameterized: o Helico-spiral: z 0, λ z, r 0, λ r,ν θ, θ o Generating curve: shape, N c, λ c Generate different shells by varying parameters Fowler et al. Figure 1 Different helico-spirals Fowler et al. Figure 2

5 Generate different shells by varying parameters Generate many interesting shells with a simple procedural model! Different generating curves Fowler et al. Figure 3 Fowler et al. Figures 4,5,7 Procedural Modeling Sweeps Fractals Grammars Fractals Defining property: o Self-similar with infinite resolution Mandelbrot Set H&B Figure Fractals Useful for describing natural 3D phenomenon o Terrain o Plants o Clouds o Water o Feathers o Fur Fractal Generation Deterministically self-similar fractals o Parts are scaled copies of original Statistically self-similar fractals o Parts have same statistical properties as original H&B Figure 10.80

6 Deterministic Fractal Generation General procedure: o Initiator: start with a shape o Generator: replace subparts with scaled copy of original Deterministic Fractal Generation Apply generator repeatedly H&B Figure Koch Curve H&B Figure Deterministic Fractal Generation Useful for creating interesting shapes! Deterministic Fractal Generation Useful for creating interesting shapes! Mandelbrot Figure X Mandelbrot Figure 46 Deterministic Fractal Generation Fractal Generation Useful for creating interesting shapes! Deterministically self-similar fractals o Parts are scaled copies of original Statistically self-similar fractals o Parts have same statistical properties as original H&B Figures 75 & 109

7 Statistical Fractal Generation General procedure: o Initiator: start with a shape o Generator: replace subparts with a self-similar random pattern Statistical Fractal Generation Example: terrain Random Midpoint Displacement H&B Figure 10.83b Statistical Fractal Generation Useful for creating mountains Statistical Fractal Generation Useful for creating 3D plants H&B Figure 10.83a H&B Figure Statistical Fractal Generation Useful for creating 3D plants Procedural Modeling Sweeps Fractals Grammars H&B Figure 10.79

8 Grammars Generate description of geometric model by applying production rules S A B AB Ba a Ab b ab bab baab abaab... Grammars Useful for creating plants Tree Branch Tree Leaf Branch Cylinder [ Tree ] C[CL]C[C[CL][CL]]C[[CL][CL]] C[*]C[*][*] Grammars Useful for creating plants Summary o CAT, MRI, Laser, magnetic, robotic arm, etc. Constructing 3D models is hard! H&B Figure Jurasic Park