2001, Denis Zorin. Subdivision Surfaces

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1 200, Denis Zorin Subdivision Surfaces

2 Example: Loop Scheme What makes a good scheme? recursive application leads to a smooth surface 200, Denis Zorin

3 Example: Loop Scheme Refinement rule 200, Denis Zorin

4 Example: Loop Scheme Two geometric rules: even (update old points) odd (insert new) n , Denis Zorin

5 Control Points Vertices of initial mesh define the surface each influences finite part of surface 200, Denis Zorin

6 Triangles and Quads 200, Denis Zorin

7 Subdivision and Splines Uniform splines can be computed using subdivision quartic box spline rules: , Denis Zorin

8 Subdivision and Splines For splines, the control mesh is regular 200, Denis Zorin

9 Extraordinary Vertices Triangle meshes Quad meshes êéöìä~ê î~äéååé=s î~äéååé=q Éñíê~çêÇáå~êó î~äéååé= S î~äéååé= Q 200, Denis Zorin

10 Constructing the Rules Start with spline rules define rules for: bñíê~çêçáå~êó îéêíáåéë _çìåç~êáéë `êé~ëéë=éíåk n , Denis Zorin

11 Constructing the Rules invariance under rotations and translations small support smoothness and Fairness 200, Denis Zorin

12 Affine Invariance ëìäçáîáçé íê~åëñçêã q íê~åëñçêã q ëìäçáîáçé 200, Denis Zorin

13 Affine Invariance Coefficients of masks must sum to n p = X a i p i 3 3 Çáëéä~ÅÉãÉåí X ai (p i + t) = ³ X ai t + p 200, Denis Zorin

14 Boundaries Three types of points: ëãççíü ÅçåîÉñ ÅçêåÉêë ÅçåÅ~îÉ convex/concave need different rules 200, Denis Zorin

15 Concave Corner Problem Standard rule produces folds special concave corner rule 200, Denis Zorin

16 Vertex Types Four basic types interior boundary smooth convex concave 200, Denis Zorin

17 Boundaries and Creases Boundaries and creases are alike boundary independent of the interior ÄçìåÇ~êáÉë 2 2 ÅêÉ~ëÉë 200, Denis Zorin

18 Crease Examples 200, Denis Zorin

19 Subdivision Schemes Primal Dual (no interpolation) ^ééêçñk `~íãìääj `ä~êâ fåíéêék hçääéäí aççjp~äáåi jáçéçöé iççé _ìííéêñäó aóåjiéîáåjiáì EåçåJäáåÉ~êF 200, Denis Zorin

20 Loop Scheme: Boundaries Extraordinary vertices not C rulesofhoppeetal. problem: depend on valence; boundary is not a cubic spline. fix: modify rules for interior neighbors eçééé=éí=~äk çìê=êìäéë 200, Denis Zorin

21 Loop Scheme: Boundaries eçééé=éí=~äk ëãççíü çìê=êìäéë ³ +cos ¼ K 2 4 cos ¼ K , Denis Zorin

22 Loop Scheme: Corners Endpoint interpolation interpolation rule for corner vertex modified rules for odd vertices next to the boundary concave corners require more 200, Denis Zorin

23 Loop Scheme: Corners Corner rules, st step for convex, use µ> ¼ for concave, Kuse µ< ¼ K 4 ( + cosµ) 2 4 cosµ 200, Denis Zorin

24 Loop Scheme: Corners Corner rules, 2 nd step really need it only for concave pull the surface toward the tangent plane: add a combination of tangents í~åöéåíë 200, Denis Zorin

25 Subdivision Schemes Primal Dual (no interpolation) ^ééêçñk `~íãìääj `ä~êâ fåíéêék hçääéäí aççjp~äáåi jáçéçöé iççé _ìííéêñäó aóåjiéîáåjiáì EåçåJäáåÉ~êF 200, Denis Zorin

26 Catmull-Clark Scheme Primal, quadrilateral, approximating tensor-product bicubic splines , Denis Zorin

27 Catmull-Clark Scheme Reduction to a quadrilateral mesh do one step of subdivision with special rules: only quads remain 200, Denis Zorin

28 Catmull-Clark Scheme Extraordinary vertices = 4K = 3 2K K K K K K K 200, Denis Zorin

29 Catmull-Clark Scheme Boundaries, creases, corners cubic spline (same as Loop!) need to fix rules for C -continuity 3 32 ( + cosµ) cosµ , Denis Zorin

30 200, Denis Zorin Lighting model

31 Local lighting model Describes interaction of the light with the surface. Almost never truly based on physics: perception plays a greater role. Visible light: electromagnetic waves, with wavelengths 400nm (violet) - 700nm (red); intensity can vary over many orders of magnitude. Computer model: only three frequencies : RGB, intensity varies over a small range, typically only 255 discrete values/ color. 200, Denis Zorin

32 Physics vs. graphics Computer graphics terms are somewhat confusing and disagree with physics: Graphics: color of an object; physics: reflection spectrum (i.e. fraction of light of each frequency that gets reflected). Graphics: intensity or color of a light ray; physics: radiance distribution (measured in watts/steradian/meter 2 /meter) Graphics: intensity or color of a point light source; physics: intensity spectrum of a point light source (almost in agreement!) measured in watts/steradian/meter. 200, Denis Zorin

33 Illumination model Two main components: light source characteristics position intensity for each freq. (color) often, different intensity can be specified for different colors directional distribution surface properties reflectance for each freq. (color) different reflectance can be specified for diffuse and specular light 200, Denis Zorin

34 Reflection geometry V: direction to the eye surface N: normal R: reflected diection L: direction to the light source R=2(N,V)N-V 200, Denis Zorin

35 Illumination components In the model commonly used in graphics applications, there are several components diffuse relection: intensity does not depend on the direction to the viewer specular: simulates relective surfaces and specular highlights depending on the direction to the viewer ambient: a crude approximation to the illumination created by the light diffusely reflected from surfaces 200, Denis Zorin

36 Diffuse component Diffuse surfaces are surfaces following the Lambert s law: the energy of the light reflected from a surface in a direction D is proportional to the cosine of the angle between the normal and D. As intensity (radiance) is proportional to the energy times cross section of the ray, it does not depend on the view direction, but is proportional to the cosine of the angle between the normal and the direction to the light. L diff =k diff (L,N) 200, Denis Zorin Acosθ θ small area A

37 Specular component Specular component approximates behavior of shiny surfaces. If a surface is an ideal mirror, the light from a source reaches the eye bouncing of a fixed point of the surface, only if the direction to the light coincides with the reflected direction to the eye: N V L=R ideal mirror sphere 200, Denis Zorin

38 Specular reflection For non-ideal reflectors, the reflection of the light is still the brightest when L=R but decays, rather than disappears, as the angle between L and R increases. One way to achieve this effect is to use cosine of the angle to scale the reflected intensity: L spec =k spec (R,L) p N R Phong exponent V α 200, Denis Zorin

39 Specular reflection p=0 p= p=2 p = 0 p = 25 p = , Denis Zorin

40 Metal vs. plasitic Natural look of metallic surfaces is difficult to simulate, but the first approximation is obtained using proper highlight color. For plastic objects, highlights are close in color to the color of the light. For metals, to the color of the surface. Assuming white lights, for plastic set k spec =[c,c,c], where c is a constant, for more metalliclooksetk spec =k diff 200, Denis Zorin plastic look metallic look

41 Ambient component Not all light illuminating a surface comes from light sources, or reflections of light sources in ideal mirrors; however, the light diffusely reflected from other surfaces is difficult to take into account, especially for real-time rendering. It is approximated by the ambient component: a constant is added to all objects. To have more control over ambient contribution, surfaces can be assigned ambient reflectivity. L amb =k amb I amb 200, Denis Zorin

42 Complete equation Itotal = k ambiamb + Ii diff i + all lights ( p k (L,N) k (L,R) ) spec i intensity of i-th light direction to i-th light If we are ray tracing for rendering, in summation only visible lights are present, and there are two additional terms: contribution from the reflected ray and transmitted ray. If we are using Z-buffering, then all active light sources are regarded as visible. 200, Denis Zorin

43 Attenuation In real life, radiance reaching us from a light source decreases with distance as /r 2 (the stars are much less bright than the sun). However, due to the nature of approximations used in graphics, the inverse-square law typically results in pictures that are too dark; the fix is to allow the programmer to control how fast the decay is. I i in the formulas is replaced not by I i /r 2,as physics suggest, but by I i a + br + br 2 and the most common choice of constants is a =, b = c = 0, that is, no attenuation! 200, Denis Zorin

44 OpenGL model Phong exponent called shininess. Several additions: emmision; ambient, diffuse, specular light intensities Setting material parameters (K diff,k spec,k amb,p) GLfoat mat_diffuse[3], mat_spec[3], mat_amb[3]; GLfloat shininess;... glmaterialfv(gl_front,gl_diffuse,mat_diffuse); glmaterialfv(gl_front,gl_specular,mat_spec); glmaterialfv(gl_front, GL_AMBIENT,mat_amb); glmaterialf(gl_front, GL_SHININESS,shininess); 200, Denis Zorin

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