Objectives. Introduce Phong model Introduce modified Phong model Consider computation of required vectors Discuss polygonal shading.

Transcription

1 Shading II 1

2 Objectives Introduce Phong model Introduce modified Phong model Consider computation of required vectors Discuss polygonal shading Flat Smooth Gouraud 2

3 Phong Lighting Model A simple model that can be computed rapidly Has three components Ambient Diffuse Specular Uses four vectors To source (l) To viewer (v) Normal (n) Perfect reflector (r) 3

4 Ambient reflection Amount and color depend on both the color of the light(s) and the material properties of the object I a = k a L a 0<=k a <=1 reflection coef intensity of ambient light 4

5 Diffuse Reflection An ideal diffuse surface is, at the microscopic level, a very rough surface. Chalk is a good approximation to an ideal diffuse surface. Because of the microscopic variations in the surface, an incoming ray of light is equally likely to be reflected in any direction over the hemisphere. 5

6 Lambertian Surface Perfectly diffuse reflector Light scattered equally in all directions Amount of light reflected is proportional to the vertical component of incoming light reflected light ~ cos q cos q= l n if vectors normalized There are also three coefficients, k r, k b, k g that show how much of each color component is reflected 6

7 Lambert's Cosine Law Lambert's law determines how much of the incoming light energy is reflected. Tthe amount of energy that is reflected in any one direction is constant in this model. In other words, the reflected intensity is independent of the viewing direction. 7

8 Illumination effects Shaded using a diffuse-reflection model,from left to right k d =0.4, 0.55, 0.77, 0.85, 1.0. Shaded using a ambient and diffuse-reflection model, I a =I light = 1.0, k d = 0.4. From left to right k a =0.0, 0.15, 0.30, 0.45,

9 Ideal Specular Surfaces Normal is determined by local orientation Angle of incidence = angle of relection The three vectors must be coplanar 9

10 Specular Surfaces Most surfaces are neither ideal diffusers nor perfectly specular (ideal reflectors) Smooth surfaces show specular highlights due to incoming light being reflected in directions concentrated close to the direction of a perfect reflection specular highlight 10

11 Modeling Specular Relections Phong proposed using a term that dropped off as the angle between the viewer (v) and the ideal reflection increased (r) I r ~ k s I cos a f reflected shininess coef intensity incoming intensity absorption coef f 11

12 The Shininess Coefficient Values between 5 and 10 give surface that look like plastic Values of a between 100 and 500 correspond to metals cos a f -90 f 90 12

13 Spheres shaded using phong illumination model 13

14 Distance Terms The light from a point source that reaches a surface is inversely proportional to the square of the distance between them We can add a factor of the form 1/(c 1 +c 2 d L +c 3 d L2 ) to the diffuse and specular terms The constant and linear terms soften the effect of the point source 14

15 Light source attenuation I = I a k a +f att I light k d (N. L) f att =1/d L 2 f att = (1/min((c 1 +c 2 d L +c 3 d L2 ), 1)); Distance C

16 Adding up the Components For each light source and each color component, the Phong model can be written (without the distance terms) as I = k a I a + k d I d l n + k s I s (v r ) a For each color component we add contributions from all sources 16

17 add a distance term 17

18 Light Sources In the Phong Model, we add the results from each light source Each light source has separate diffuse, specular, and ambient terms to allow for maximum flexibility even though this form does not have a physical justification Separate red, green and blue components Hence, 9 coefficients for each point source I dr, I dg, I db I sr, I sg, I sb I ar, I ag, I ab 18

19 Material Properties Material properties match light source properties 9 absorbtion coefficients k dr, k dg, k db k sr, k sg, k sb k ar, k ag, k ab Shininess coefficient a 19

20 Modified Phong Model The specular term in the Phong model is problematic because it requires the calculation of a new reflection vector and view vector for each vertex Blinn suggested an approximation using the halfway vector that is more efficient 20

21 The Halfway Vector h is normalized vector halfway between l and v h = ( l + v )/ l + v 21

22 Let ψ be the angle between the normal and the halfway vector, φ be the angle between the viewer and the reflection angle, and θ be the angle between the normal and the light source. If all the vectors lie in the same plane, the angle between the light source and the viewer can be computer either as φ + 2θ or as 2(θ + ψ). Setting the two equal, we find φ = 2ψ. If the vectors are not coplanar then φ < 2ψ 22

23 Using the halfway vector Replace (v r ) a by (n h ) b b is chosen to match shineness Note that halway angle is half of angle between r and v if vectors are coplanar Resulting model is known as the modified Phong or Blinn lighting model Specified in OpenGL standard 23

24 Example Only differences in these teapots are the parameters in the modified Phong model 24

25 Computation of Vectors l and v are specified by the application Can computer r from l and n Problem is determining n how we determine n differs depending on underlying representation of surface OpenGL leaves determination of normal to application Exception for GLU quadrics and Bezier surfaces (Chapter 11) glnormal3f(nx, ny, nz) 25

26 Plane Normals Equation of plane: ax+by+cz+d = 0 we know that plane is determined by three points p 0, p 2, p 3 or normal n and p 0 Normal can be obtained by n = (p 2 -p 0 ) (p 1 -p 0 ) 26

27 Implicit function Implicit function f(x,y,z)=0 the gradient vector the approximation is the equation of a plane whose normal is the gradient. 27

28 Normal to Sphere Sphere f(p)=p p - 1 n = [ f/ x, f/ y, f/ z] T =p the normal at every point on the surface of the sphere points directly out of the sphere, that is, in a direction from the origin through the point. For the unit sphere, the normalized result is n=p 28

29 Parametric Form For sphere x=x(u,v)=cos u sin v y=y(u,v)=cos u cos v z= z(u,v)=sin u Tangent plane determined by vectors p/ u = [ x/ u, y/ u, z/ u] T p/ v = [ x/ v, y/ v, z/ v] T Normal given by cross product n = p/ u p/ v 29

30 General Case We can compute parametric normals for other simple cases Quadrics Parameteric polynomial surfaces Bezier surface patches (Chapter 11) In OpenGL, the current normal vector is part of the state glnormal3f(nx, ny, nz); glnormal3fv(pointer_to_normal); 30

31 the reflection vector R =Ncosq + S =Ncosq + Ncosq - L =2N (N.L) - L Calculating N. H instead of R. V, in which H = (L+V)/ L+V 31

32 Although the fixed-function Open GL pipeline uses the modified Phong model and thus avoids having to calculate the reflection vector programmable shaders can use the reflection vector. 32

33 Polygonal Shading Shading calculations are done for each vertex Vertex colors become vertex shades By default, vertex shades are interpolated across the polygon glshademodel(gl_smooth); If we use glshademodel(gl_flat); the color at the first vertex will determine the shade of the whole polygon 33

34 Flat shading Flat shading of polygonal mesh Step chart 34

35 Polygon Normals Polygons have a single normal Shades at the vertices as computed by the Phong model can be almost same Identical for a distant viewer (default) or if there is no specular component Consider model of sphere Want different normals at each vertex even though this concept is not quite correct mathematically 35

36 Smooth Shading We can set a new normal at each vertex Easy for sphere model If centered at origin n = p Now smooth shading works Note silhouette edge 36

37 Mesh Shading The previous example is not general because we knew the normal at each vertex analytically For polygonal models, Gouraud proposed we use the average of the normals around a mesh vertex n = (n 1 +n 2 +n 3 +n 4 )/ n 1 +n 2 +n 3 +n 4 37

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41 Gouraud V.S. Phong Gouraud shading Cheap but gives poor highlights Phong shading Slightly more expensive, but gives high quality highlights 41

42 Gouraud and Phong Shading Gouraud Shading Find average normal at each vertex (vertex normals) Apply modified Phong model at each vertex Interpolate vertex shades across each polygon Phong shading Find vertex normals Interpolate vertex normals across edges Interpolate edge normals across polygon Apply modified Phong model at each fragment 42

43 Comparison If the polygon mesh approximates surfaces with a high curvatures, Phong shading may look smooth while Gouraud shading may show edges Phong shading requires much more work than Gouraud shading Until recently not available in real time systems Now can be done using fragment shaders (see Chapter 9) Both need data structures to represent meshes so we can obtain vertex normals 43