CS248 Final ReviewCS248 Final Review

CS248 FinalCS248 Final

• Wednesday, December 10, 7-10 pm, Gates B01

• 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

CS248 Final Review CS248 Final Review ContentsContents

• Image warping, texture mapping• Perspective• Visibility• Lighting / Shading

Texture MappingTexture Mapping

• Coordinate systems– [u,v,q] => [xo, yo zo, wo] => [xw, yw zw,

ww] => [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)

Texture WarpsTexture 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 WarpsTexture Warps

• Magnification– Unweighted– Area-weighted– bilinear interpolation

= texel

= pixel

TexturesTextures

1.Mipmapping1.multi-resolution texture2.bilinear interpolation at 2 closest

resolutions to get 2 color values3.linear interpolate 2 color values

based on actual resolution

2.Summed area tables1.fast calculation of prefilter integral in

texture space

QuestionsQuestions

• 1. What are some of the problems associated with Mipmaps?

• 2. What are some of the problems associated with SAT?

Viewing: Planar Viewing: Planar ProjectionsProjections

• 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 Specifying Perspective ViewsViews

• Observer position (eye, center of projection)

• Viewing direction (normal to picture plane)

• Clipping planes (near, far, top, bottom, left, right)

Viewing: OpenGL PipelineViewing: OpenGL Pipeline

• Object Space• Eye Coordinates• Projection Matrix• Clipped to Frustum• Homogenize to normalized device

coordinates• Window coordinates

Why do we not just uniformly scale Z coordinates?

VisibilityVisibility

1.6 visible-surface determination algorithms:1.Z-buffer2.Watkins3.Warnock4.Weiler-Atherton5.BSP Tree6.Ray Tracing

Things to knowThings to know

how does it workwhat are the necessary preconditions?asymptotic time complexityhow can anti-aliasing be done?how can shading be incorporated?well-suited for hardware?parallelizable?ease of implementationbest-case/worst-case scenarios

Z-bufferZ-buffer

• Project all polygons to the image plane, at each pixel, pick the color corresponding to closet polygon

What has to be done to render transparent polygons?

WatkinsWatkins

• Scanline + depth– progressing across scanline, if pixel is

inside two or more polygons, use depth to pick

– process interpenetrating polygons, add those events

Warnock SubdivisionWarnock 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 Weiler-Atherton SubdivisionSubdivision

• Cookie-cutter algorithm:clips polygons against polygons– front to back sort of list– clip with front polygon

Why is this so difficult?

BSP TreesBSP 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

Ray TracingRay 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

Ray TracingRay Tracing

• Sphere: |P-Pc|2 – r2 = 0

• Plane: N • P = -D• Can you compute the intersection

of a ray and a plane? A ray and a sphere?

Ray TracingRay 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

LightingLighting

• 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)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?

LightingLighting

• Point to area transport– Computing the irradiance to a surface– Cos falloff: N • L

– E = Fatt x I x (N • L)

LightingLighting

• 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!

– Fatt x I x Kd x (N • L)

N

VI length = cos(t)

Radiance intensity: intensity/solid angle

NV

Sample question (2002)Sample question (2002)

• If you place a candle in the middle of a hollow sphere, what happens to the total amount of light falling on the inside surface of the sphere as the sphere gets bigger? Defend your answer in one or two sentences.

LightingLighting

• 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.

LightingLighting

• 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

– Lspec = Ks x cosn(a), a= angle between viewer and reflected ray

– how do you compute the reflected ray vector?

• (assume normalized vectors)

N LR

V

LightingLighting

• 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 H L

ShadingShading

• 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!

ShadingShading

• Angle interpolation– interpolate normal angles according to the

implicit surface– compute shading at each point of the

implicit surface– CORRECT! But very expensive

ShadingShading

• 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 ShadingLighting and Shading

• Know the OpenGL 1.1, 1.2 light equations (what terms mean what)

Exotic uses of texturesExotic uses of textures

• Environment/reflection mapping• Alphas for selecting between

textures/shading parameters• Bump mapping• Displacement mapping• Object placement• 3d textures

Good Luck!Good Luck!

Good Luck on the Final!

More review questions at:http://graphics.stanford.edu/courses/cs248-99/final_review