CS 445 / 645: Introductory Computer Graphics

1. CS 445 / 645: Introductory Computer Graphics Light

2. Administrivia • Assignment 3 Due • Sunday, October 21st at Midnight • Midterm Exam • Tuesday, October 23rd • See web page for reading material • Review in class today

3. Lighting Models • Lighting Models • Ambient • Normals don’t matter • Lambert/Diffuse • Angle between surface normal and light • Phong/Specular • Surface normal, light, and viewpoint • Rendering Equation (Kajiya)

4. Shading Models (Direct lighting) • Flat Shading • Compute Phong lighting once for entire polygon • Gouraud Shading • Compute Phong lighting at the vertices and interpolate lighting values across polygon • Phong Shading • Compute averaged vertex normals • Interpolate normals across polygon and perform Phong lighting across polygon

5. Shading Models (Indirect lighting) • Ray Tracing Radiosity

6. Radiosity • Ray tracing models specular reflection and refractive transparency, but still uses an ambient term to account for other lighting effects • Radiosity is the rate at which energy is emitted or reflected by a surface • By conserving light energy in a volume, these radiosity effects can be traced

7. Radiosity • Radiosity of patch ‘i’ is • ei is the rate at which light is emitted • ri is patch i’s reflectivity • Fj,i is the form factor • fraction of patch j that reaches patch i as determined by orientation of both patches and obstructions • Use Aj/Ai (Area of patch j / Area of patch I) to scale units to total light emitted by j as received per unit area by i

8. Form Factors • Compute n-by-n matrix of form factors to store radiosity relationships between each light patch and every other light patch Hij = 1 or 0 depending on occlusion

9. Form Factor – Another example • Spherical projections to model form factor • project polygon Aj on unit hemisphere centered at (and tangent to) Ai • Contributes cosqj / r2 • Project this projection to base of hemisphere • Contributes cosqi • Divide this area by area of circle base • Contributes p

10. Form Factor – Another Model • Hemicube allows faster computations • Analytic solution of hemisphere is expensive • Use rectangular approximation, hemicube • cosine terms for top and sides are simplified • Dimension of 50 – 200 squares is good

11. Solving for all Patches • One patch defined by: • Symmetry: AiFi,j = AjFj,I • Therefore: • And: • Use matrix algebra to solve for Bi’s

12. Radiosity • Radiosity is expensive to compute • Get your PhD by improving it • Emitted light and viewpoint can change • Light angles and object positions cannot • Computing form factors is expensive • Specular reflection information is not modeled

13. View-dependent vs View-independent • Ray-tracing models specular reflection well, but diffuse reflection is approximated • Radiosity models diffuse reflection accurately, but specular reflection is ignored • Advanced algorithms combine the two

14. Texture Mapping • Limited ability to generate complex surfaces with geometry • Images can convey the illusion of geometry • Images painted onto polygons is called texture mapping

15. Texture Mapping • Texture map is an image, two-dimensional array of color values (texels) • Texels are specified by texture’s (u,v) space • At each screen pixel, texel can be used to substitute a polygon’s surface property • We must map (u,v) space to polygon’s (s, t) space T V U S

16. Texture Mapping • (u,v) to (s,t) mapping can be explicitly set at vertices by storing texture coordinates with each vertex • How do we compute (u,v) to (s,t) mapping for points in between • Watch for aliasing • Watch for many to one mappings • Watch for perspective foreshortening effects and linear interpolation

17. Bump Mapping • Use textures to modify surface geometry • Use texel values to modify surface normals of polygon • Texel values correspond to height field • Height field models a rough surface • Partial derivative of bump map specifies change to surface normal

18. Bump Mapping

19. Displacement Mapping • Bump mapped normals are inconsistent with actual geometry. Problems arise (shadows). • Displacement mapping actually affects the surface geometry

20. Assignment 3 – Plane Rendering • Plane defined by Ax + By + Cz + D = 0 • (A, B, C) defines normal to the plane • -D specifies translation from origin • Ex: 1x + 0y + 0z + 4 = 0 • All values of y and z work. X must be –4 • y-z plane at x = -4 • y-z plane has normal of (1, 0, 0)

21. Plane Rendering • Imagine first rendering plane of desired size centered on x-z plane • We must then rotate and translate this plane so its normal aligns with the (A, B, C) vector and the plane is –D from the origin U = Normal of x-z plane (0, 1, 0) V = Normal of clip plane (A, B, C)

22. Plane Rendering • Angle between U and V is computed with the dot product: u.dot.v = ||V|| ||U|| cos (n) Rotate -n degrees Rotate about VxU • Vector to rotate about is VxU:[vyuz-vzuy, -vxuz+vzux, vxuy-vyux]

23. Clipping Plane Rendering • So we have a glRotate to execute • glRotatef (-n, VxUx, VxUy, VxUz) • And we have a glTranslate to execute • glTranslatef (0, -D, 0) • And we have a command to draw a polygon • glBegin (GL_POLYGON) • glVertex3fv(front-left); glVertex3fv(front-right); • glVertex3fv(back-right); glVertex3fv(back-left); • So just put them in the right order • An exercise for the reader

24. Review • Hearn and Baker • Chapter 2 • Chapter 3 (84-102, 117-126) • Chapter 5 • Chapter 6 (225-248) • Chapter 11 (No Quaternions) [Recently added] • Chapter 12 (432-456) • Chapter 14 • Chapter 15 • Appendix 1 – 4 • OpenGL Red Book • Chapter 1 - 3

25. Review • If you’re going to print out all the slides from my lectures, please try to keep our costs down. • Render multiple slides to a page. • Convert black backgrounds to white to reduce toner usage • Make sure you convert fonts to black • Doesn’t PowerPoint have a mode that does this automatically?

26. Review • Vector arithmetic • Add, subtract • Dot (to determine angle between and length of projection (when normalized vectors) • Cross product (Right-hand rule) • Coordinate space • Right-handed coordinate system • openGL camera initialized to look down –z axis

27. Review • OpenGL • No significant coding required • I might ask you to tell me what a function does • One you’ve used for the projects • I might ask you to tell me what order to put the transformation commands • Direction of rotations • Ordering of polygon vertices • Back-face rendering • Rendering Pipeline (model, light, view, clip, project) • Line Drawing, Circle Drawing • Difference Equations, symmetry • Triangle Rasterization

28. Review • Polygon rasterization • Example Problems • Active Edge Table • Clipping • Parametric equations of lines • Slope-intercept equation of line • Outcode • Line-plane intersection

29. Review • Homogeneous Coordinate System • What does w do? • Modeling Transformations • Rotate, translate, scale, rotate about arbitrary axis • Viewing Transformations • Orthographic (parallel) and Perspective • Understand how matrix formed by compositing many modeling and viewing transformations effect the vertices of a model and its rendering

30. Review • Color • Metamers • Relative changes versus absolute values • RGB sensitivity • Color spaces • Intuitive spaces, distance between similar colors • Lighting Equations • Dependent on what factors • Calculate the reflection vector • Know how parameters effect images

31. Review • Shading • Flat, Gouraud, Phong • Improvements versus cost • Indirect lighting • Ray tracing, radiosity • Viewer dependent/independent • Costs/benefits