CS248 Final Review

CS248 Final Monday, December 6, 3:30 to 6:30 pm, Gates B01 Closed book, closed notes 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

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

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]

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

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

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

Questions 1. What are some of the problems associated with Mipmaps? 2. What are some of the problems associated with SAT?

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?

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

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

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

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

Z-buffer Project all polygons to the image plane, at each pixel, pick the color corresponding to closest polygon What has to be done to render transparent polygons?

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

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

Weiler-Atherton Subdivision Cookie-cutter algorithm: clips polygons against polygons – front to back sort of list – clip with front polygon Why is this so difficult?

BSP Trees/List Priority 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

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

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

Sample question (2000) Q. As every scout knows, you can start a fire on a sunny day by holding a magnifying glass between the sun and a piece of paper placed on the ground. – Is the radiance of the sun as seen from the focal point of the lens more, less, or the same as the radiance as seen from the same point in the absence of the magnifying glass? – Is the irradiance due to the sun at the focal point more, less, or the same as the irradiance at the same point in the absence of the magnifying glass?

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

Lighting Lambertian (diffuse) surfaces – Radiant intensity has cosine fall off with respect to angle – Radiance is constant with respect to angle – 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

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.

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 NL R V

Lighting 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

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!

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

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

Lighting and Shading Know the OpenGL 1.1, 1.2 light equations (what terms mean what)

Good Luck! Good Luck on the Final!

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

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