CS 354Review for Final

Mark KilgardUniversity of TexasMay 3, 2012

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Today’s material

In-class quiz On surfaces & programmable tessellation

lecture Lecture topic

What to study for the final

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Daily Quiz1. How many control points

specify a bi-cubic Bezier path?

2. How many control points specify a Bezier triangle patch?

3. Programmable tessellation relies on two programmable shader stages. What are the names of these two stages? (Give either the OpenGL or Direct3D names.)

4. Multiple choice: Subdivision surfaces repeatedly apply these two steps

a) refinement & smoothing

b) divide & conquer

c) sort & merge

d) push & pop

e) push & pull

On a sheet of paper• Write your EID, name, and date• Write #1, #2, #3, #4 followed by its answer

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Last time, this time

Last lecture, we discussed Surfaces, programmable tessellation,

non-photorealistic graphics This lecture

Review for the final Course-Instructor survey

Last 10 minutes of class

Projects Project 4 on ray tracing on Piazza

Due Thursday May 3, 2012 (extended) As announced, no slip days Arrange your demo with TA for Friday, May 4

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Grading

Testing 60% Daily quizzes 10%

3-4 questions, easy

Homework 10% Occasional, mostly

math questions Project 0’s simple

rendering considered homework

Exams 40% Mid-term 15% Final 25%

Software projects 40% Object loader GLSL Shading Motion capture

animation Ray Tracer

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What will you learnWhat you should have learned

Fundamentals of computer graphics Transformations and viewing Rasterization and ray tracing Lighting and shading Graphics hardware technology Mathematics for computer graphics

Practical graphics programming OpenGL programming Shader programming Performance analysis

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When and whereis the final exam?

Thursday, May 10, 2:00-5:00 pm Three hours Afternoon

Burdine 116 Same room as

weekly lecture

you are here

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Exam Instructions & Policies

Open textbook Open notes, including lecture slides Calculators allowed/encouraged Prohibitions

No smart phones No computers No Internet access

Show your work to justify your answer and provide a basis for partial credit

Sit with an empty seat between each student

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Final exam generalities

Cumulative Covers material in both halves of the course Revisit your mid-term & homework assignments Knowledge from projects Textbook readings 1/3 first half, 2/3 second half

Exam structure Math questions Concept questions

Some multiple choice, some fill-in-the-blank, some true-or-false

No essay questions; some “explain briefly” questions About 80% longer than mid-term

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Final exam scope

Every lecture is covered by at least one exam question

Concepts developed in the projects is fair game

Variations of quiz questions, sometimes lifted exactly See Piazza for all the quiz answers

Search for #quiz

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Bezier Curve Formulation

Given control points, specify parametric formula for Bezier curve Quadratic (3 control points)

P0*(1-t)2 + 2*P1*(1-t)*t + P2*t2

Cubic (4 control points) P0*(1-t)3 + P1*3*t*(1-t)2 + P2*3*t2*(1-t)+P3*t3

Be prepared to evaluate the Bezier curve for various values of “t”

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Bezier Curves

Geometric intuition Given four cubic Bezier control points

P0, P1, P2, P3

What is the geometric interpretation of the vector from P0 to P1 and P2 to P3?

Be able to show these directions are the tangents of the curve at the end-points

Differentiate and evaluate at t=0 and t=1

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Bezier CurvesDe Casteljau's algorithm

Given a cubic Bezier curve C(t) Defined by control points P0, P1, P2, P3 How to evaluate C(t) with De Casteljau’s approach?

Make new points with linear interpolation AB = lerp(P0,P1,t) BC = lerp(P1,P2,t) CD = lerp(P2,P3,t) ABC = lerp(AB,BC,t) BCD = lerp(BC,CD,t) ABCD = lerp(ABC,BCD,t) lerp(a,b,t) = a+t*(b-a) = (1-t)*a + t*b

ABCD evaluates to C(t)

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De Casteljau's algorithmVisualized

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Bezier Subdivision

Original Bezier curve C(t) can be split into two sub-curves C1(t) and C2(t)

Assuming C(t)’s control points are P0, P1, P2, P3

C1(t)’s control points are P0,AB,ABC,ABCD

C2(t)’s control points are ABCD,BCD,CD,P3

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Bezier Surface Evaluation

Given bicubic Bezier path form Evaluate P(u,v) Also compute gradients dP/du and dP/dv Compute surface normal

Assume simple enough control points to make this tractable

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Ray Tracing MathRay-Sphere Intersection

Given ray with origin O and direction D ray(t) = O + D*t

Given sphere at C with radius r and P is on the sphere (P-C)•(P-C)=r2

Compute the values of t where the ray intersects the sphere Involves solving the quadratic formula

Interpretation of the intersection What does zero solutions mean? What does two solutions mean? What does one solution mean? What do parametric values of t<0 mean?

Compute actual point(s) of intersection of ray and sphere

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Ray Tracing MathRay-Sphere Shading

Given an intersection point on a sphere Determine the surface normal

Easy because normal N just normalized P-C Given lighting position L

Compute diffuse coefficient max(0,normalize(L-P)•N)

Why the max with zero? Given eye position E

Compute half-angle vector H=normalize(E-P+

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Ray Types

Eye (or view) rays From eye to surfaces

Light (or shadow) rays Surface in shadow of light if ray obstructed Surface illuminated by light if ray unobstructed

Reflected rays Specular bounce

Refracted rays Ray travels through interface Be able to compute refracted vector given incoming

ray direction, surface normal, and index of refraction Be able to identify types of rays in a diagram

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Rendering Equation

Be able to identify and briefly explain the constituent parts of the Rendering Equation

Explain two different forms What does each form integrate over?

Hemisphere of incoming directions at surface point All possible surface points in the scene

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Rendering Equation

Theory for light-surface interactions

dtLtftLtL ireo )(),,,(),,,,(),,,(),,,( nxxxx

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Rendering Equation Parts

Lo = outgoing light from x in direction ω x = point on a surface ω = normalized outgoing light vector λ = wavelength of light t = time Le = emitted light at x going towards from ω n = surface normal at x

dtLtftLtL ireo )(),,,(),,,,(),,,(),,,( nxxxx

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Rendering Equation Integral

∫ = integrate over a region Ω = region of a hemisphere

Together: “integrate overall the incoming directions for a hemisphere at a point x”

ωˊ = normalized incoming light vector Li = incoming light at x coming from ωˊ n = surface normal at x (-ωˊ • n) = cosine of angle between incoming

light and surface normal dωˊ = differential of incoming angle

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Bidirectional Reflectance Distribution Function

Ratio of differential outgoing (reflected) radiance to differential incoming irradiance

Physically based BRDF properties Must be non-negative Must be reciprocal

Swap incoming & relected directions generates same ratio Conserves energy

Integrating over entire hemisphere must be ≤ 1

dtdL

tdL

tdE

tdLtf

i

r

i

rr )(),,,(

),,,(

),,,(

),,,(),,,,(

nx

x

x

xx

L = radiance

E = irradiance

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Cosine Weighting for Irradiance

At shallow angles, incoming light spreads over a wider area of the surface Thus spreading the energy Thus dimming the received light

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o

Two Versions ofRendering Equation

dtLtf

tLtL

ir

e

o )(),,,(),,,,(

),,,(),,,( nxx

xx

yyxyx

xx

yx

y

yx dGtLtf

tLtL

r

e

o ),(),,,(),,,,(

),,,(),,,(

Integrate over hemisphere Integrate over all surface points

Occlusion (G) is zero

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Global Illumination Terms

Be able to explain Irradiance vs. radiance Form factor

Percent of energy from patch A that reaches (is transferred to) patch B

Radiosity Caustics

Photon mapping can simulate these Why can’t normal ray tracing simulate these?

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Typography

Typographic units Em, Pica, Point

Character encoding Unicode

How does Unicode relate to the 7-bit and 8-bit ASCII character sets?

Why is kerning done? Why font hinting? Why ClearType? Identify serif vs. non-serif type faces

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Resolution-independent 2D

What is a path? What is a contour? How are curved segments indicated?

Quadratic and cubic Bezier curves + elliptical arcs

How do you know if a point is within a path? Non-zero rule, even-odd rule

How are paths shaded (painted?) Gradients, solid color, images

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Blend Modes

Given f(Ac,Bc) and X, Y, Z parameters, build up a Porter-Duff blend equation

Evaluate that blend equation for a given source and destination RGBA value

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Graphics Pipeline Stages

Know their order Mid-term asked to order stages Worth revisiting

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Coordinate SpaceTransformations

Be able to recognize / specify the following coordinate space transformations Translate Rotate Scale (uniform, non-uniform) Orthographic view Perspective view

Be able to concatenate transformations Be able to transform normals and positions

by concatenated transformations

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Edge Equation Setupfor Rasterization

Given three points in window space, generate the three edge equations Front facing or back facing?

Test if pixel locations are inside/outside the triangle

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Plane Equation Setupfor Interpolation

Given window-space vertices for a triangle, be able to setup plane equation for interpolating Z

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Mipmap Generation

Extend to 3D textures How many bytes would a 256x128x128

texture take up? Width, height, and depth dimensions half with

successively smaller mipmap levels

Be able to generate texels for a small mipmap pyramid Say 8x2

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GLSL Shading Language

Be able to map implement simple GLSL standard library routines in GLSL as simple math operations and sqrt Example: length

sqrt(v.x*v.x+v.y*v.y+v.z*v.z) Others: dot, mul, normalize, reflect, refract,

distance, determinant Know the standard GLSL types

So you can not which types are not actual GLSL built-in types

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Scene Graph

Be able to identify when a bounding box is trivially outside a view frustum Are all its vertices on the “wrong” side of a

view frustum clip plane? Then trivially culled

Briefly distinguish View frustum culling Occlusion culling Portal culling

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Acceleration Structures

Name common acceleration structures Weight advantages and disadvantages

Explain ray traversal given a heirarchical acceleration structure

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Shadows

Name dominant shadowing algorithms for interactive rendering Understand their artifacts and limitations Order the shadow algorithm steps properly

Given 2D diagrams, be able to tell if points are “inside” or “outside” a shadow volume

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Color

Perform color space conversions RGB to YIQ

What is a color gamut?

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Particle system

Given a particle with initial velocity shooting it upward and gravity, when will it hit the ground? Involves solving a quadratic equation

Algorithmic complexity of N particles mutually attracted by inverse square law (gravity) How could this be reduced?

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Compression

Why does hardware’s DXTC/S3TC formats have a fixed compression ratio? What is the ratio?

Why is sampling chroma values at a lower frequency than luminance values a common compression technique?

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Forces Driving Improvements in Computer Graphics

Human desire for VisualIntuition and Entertainment

Embarrassing

Parallelism ofGraphics

Increasing

Semiconductor

Density

Particularly thehardware-amenable,

latency tolerantnature of rasterization

Particularlyinteractive video games

ComputerGraphics

Moore’sLaw

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Course-Instructor Survey

Instructor’s Name Mark Kilgard

Course Abbreviation and Number CS 354

Course Unique Number 53035

Semester and Year Spring 2012