1. CS248 Final Review

2. CS248 Final Thurs, December 12, 7-10 pm, Gates B01, B03 Mainly from material in the second half of the quarter – will not include material from last part of last lecture (volume rendering, image-based rendering) Review session slides available from class website Office hours as regularly scheduled

3. CS248 Final Review Contents Image warping, texture mapping Perspective Visibility Lighting / Shading

4. Texture Mapping Coordinate systems – [u,v,q] => [x o, y o z o, w o ] => [x w, y w z w, w w ] => [x, y, w] – Assuming all transforms are linear, then – [A][u, v, q]’ = [x, y, w] Common mappings – forward mapping (scatter), texture->screen – backward mapping (gather)

5. Texture Warps Rotation, translation perspective Minification (decimation) – unweighted average: average projected texel elements that fall within a pixel’s filter support – area-weighted average: average based on area of texel support

6. Texture Warps Magnification – Unweighted – Area-weighted – bilinear interpolation = texel = pixel

7. Textures 1.Mipmapping 1. multi-resolution texture 2. bilinear interpolation at 2 closest resolutions to get 2 color values 3. linear interpolate 2 color values based on actual resolution 2.Summed area tables 1. fast calculation of prefilter integral in texture space

8. Viewing: Planar Projections Perspective Projection – rays pass through center of projection – parallel lines intersect at vanishing points Parallel Projection – center of projection is at infinity – oblique – orthographic How many vanishing points are there in an image produced by parallel projection ?

9. Specifying Perspective Views Observer position (eye, center of projection) Viewing direction (normal to picture plane) Clipping planes (near, far, top, bottom, left, right)

10. Viewing: OpenGL Pipeline Object Space Eye Coordinates Projection Matrix Clipped to Frustum Homogenize to normalized device coordinates Window coordinates

11. Visibility 1.6 visible-surface determination algorithms: 1. Z-buffer 2. Watkins 3. Warnock 4. Weiler-Atherton 5. BSP Tree 6. Ray Tracing

12. Things to know how does it work what are the necessary preconditions? asymptotic time complexity how can anti-aliasing be done? how can shading be incorporated? well-suited for hardware? parallelizable? ease of implementation best-case/worst-case scenarios

13. Z-buffer Project all polygons to the image plane, at each pixel, pick the color corresponding to closet polygon

14. Watkins Scanline + depth – progressing across scanline, if pixel is inside two or more polygons, use depth to pick – process interpenetrating polygons, add those events

15. Warnock Subdivision Start with area as original image – subdivide areas until either: all surfaces are our outside the area only one inside, overlapping or surrounding a surrounding surface obscures all other surfaces *

16. Weiler-Atherton Subdivision Cookie-cutter algorithm:clips polygons against polygons – front to back sort of list – clip with front polygon

17. BSP Trees Provides a data structure for back-to- front or front-to-back traversal – split polygons according to specified planes – create a tree where edges are front/back, leaves are polygons

18. Ray Tracing “Ray Casting” – for each pixel, cast a ray into the scene, and use the color of the point on the closest polygon – Parametric form of a line: u(t) = a+(b-a)t a b (0,0) x y t

19. Ray Tracing Sphere: |P-P c | 2 – r 2 = 0 Plane: N P = -D Can you compute the intersection of a ray and a plane? A ray and a sphere?

20. Ray Tracing Point in polygon tests – Odd, even rule draw a line from point to infinity in one direction count intersections: odd = inside, even = outside – Non-zero winding rule counts number of times polygon edges wind around a point in the clockwise direction winding number non zero = inside, else outside

21. Lighting Terminology – Radiant flux: energy/time (joules/sec = watts) – Irradiance: amount of incident radiant flux / area (how much light energy hitting a unit area, per unit time) – Radiant intensity (of point source): radiant flux over solid angle – Radiance: radiant intensity over a unit area

22. Lighting Point to area transport – Computing the irradiance to a surface – Cos falloff: N L – E = F att x I x (N L)

23. Lighting Lambertian (diffuse) surfaces – Radiant intensity has cosine fall off with respect to angle – Radiance is constant with respect toangle – Reason: the projected unit area ALSO gets smaller as a cosine fall off! – F att x I x K d x (N L) N V I  length = cos(t) Radiance intensity: intensity/solid angle N V

24. Lighting BRDF = Bidirectional Reflectance Distribution Function – description of how the surface interacts with incident light and emits reflected light – Isotropic Independent of absolute incident and reflected angles – Anisotropic Absolute angles matter – Don’t forget the generalizations to the BRDF! Spatially/spectrally varying, florescence, phosphorescence, etc.

25. Lighting Phong specular model – Isn’t true to the physics, but works pretty well – reflected light is greatest near the reflection angle of the incident light, and falls off with a cosine power – L spec = K s x cos n (a), a= angle between viewer and reflected ray – how do you compute the reflected ray vector? NL R V

26. Lighting Local vs. infinite lights – Understand them! Know how to draw the goniometric diagrams for various light/viewer combinations N H model – H is the halfway vector between the viewer and the light – What is the difference in specular highlight? N V R HL

27. Shading Gouraud shading – Compute lighting information (ie: colors) at polygon vertices, interpolate those colors – Problems? Misses highlights need high resolution mesh to catch highlights mach bands!

28. Shading Angle interpolation – interpolate normal angles according to the implicit surface – compute shading at each point of the implicit surface – CORRECT! But very expensive

29. Shading Phong shading – Compute lighting normals at all points on the polygon via interpolation, and do the lighting computation on the interpolated normals (of the polygon) – Problems? Difference with angle interpolation? Implicit surfacePolygon approximation N1 N2

30. Lighting and Shading Know the OpenGL 1.1, 1.2 light equations

31. Exotic uses of textures Environment/reflection mapping Alphas for selecting between textures/shading parameters Bump mapping Displacement mapping Object placement 3d textures

32. Good Luck! Good Luck on the Final! More review questions at: http://graphics.stanford.edu/courses/cs248-99/final_review