May 3, 2012; CS 354 Computer Graphics; University of Texas at Austin

1. CS 354
Review for Final
Mark Kilgard
University of Texas
May 3, 2012

2. CS 354 2
Today’s material
 In-class quiz
 On surfaces & programmable tessellation
lecture
 Lecture topic
 What to study for the final

3. CS 354 3
On a sheet of paper
Daily Quiz • Write your EID, name, and date
• Write #1, #2, #3, #4 followed by its answer
1. How many control points
specify a bi-cubic Bezier path?
 Multiple choice: Subdivision
surfaces repeatedly apply
these two steps
a) refinement & smoothing
3. How many control points
specify a Bezier triangle patch? b) divide & conquer
c) sort & merge
4. Programmable tessellation d) push & pop
relies on two programmable
shader stages. What are the e) push & pull
names of these two stages?
(Give either the OpenGL or
Direct3D names.)

4. CS 354 4
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

5. CS 354 5
Grading
 Testing 60%  Software projects 40%
 Daily quizzes 10%  Object loader
 3-4 questions, easy  GLSL Shading
 Homework 10%  Motion capture
 Occasional, mostly animation
math questions
 Ray Tracer
 Project 0’s simple
rendering considered
homework
 Exams 40%
 Mid-term 15%
 Final 25%

6. CS 354 6
What will you learn
What 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

7. CS 354 7
When and where
is the final exam?
 Thursday, May 10, 2:00-5:00 pm
 Three hours
 Afternoon
 Burdine 116
 Same room as
weekly lecture
you are here

8. CS 354 8
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

9. CS 354 9
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

10. CS 354 10
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

11. CS 354 11
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”

12. CS 354 12
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

13. CS 354 13
Bezier Curves
De 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)

14. CS 354 14
De Casteljau's algorithm
Visualized

15. CS 354 15
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

16. CS 354 16
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

17. CS 354 17
Ray Tracing Math
Ray-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

18. CS 354 18
Ray Tracing Math
Ray-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+

19. CS 354 19
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

20. CS 354 20
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

21. CS 354 21
Rendering Equation
 Theory for light-surface interactions
Lo (x, ω , λ , t ) = Le (x, ω , λ , t ) + ∫ f r (x, ω ′, ω , λ , t ) Li (x, ω ′, λ , t ) (−ω ′ • n) dω ′
Ω

22. CS 354 22
Rendering Equation Parts
Lo (x, ω , λ , t ) = Le (x, ω , λ , t ) + ∫ f r (x, ω ′, ω , λ , t ) Li (x, ω ′, λ , t ) (−ω ′ • n) dω ′
Ω
 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

23. CS 354 23
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

24. CS 354 24
Bidirectional Reflectance
Distribution Function
 Ratio of differential outgoing (reflected) radiance
to differential incoming irradiance
dLr (x, ω , λ , t ) dLr (x, ω , λ , t )
f r (x, ω ′, ω , λ , t ) = =
dEi (x, ω ′, λ , t ) dLi (x, ω ′, λ , t ) (−ω ′ • n) dω ′
 Physically based BRDF properties L = radiance
 Must be non-negative E = irradiance
 Must be reciprocal
 Swap incoming & relected directions generates same ratio
 Conserves energy
 Integrating over entire hemisphere must be ≤ 1

25. CS 354 25
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

26. CS 354 26
Two Versions of
Rendering Equation
Occlusion (G) is zero
o
Le (x, ω , λ , t ) +
Le (x, ω , λ , t ) +
Lo (x, ω , λ , t ) =
∫
Ω
f r (x, ω ′, ω , λ , t ) Li (x, ω ′, λ , t ) (−ω ′ • n) dω ′ Lo (x, ω , λ , t ) =
∫ f r (x, ωyx , ω , λ , t ) L(y, ωyx , λ , t ) G (x, y ) dy
y∈Γ
Integrate over hemisphere Integrate over all surface points

27. CS 354 27
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?

28. CS 354 28
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

29. CS 354 29
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

30. CS 354 30
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

31. CS 354 31
Graphics Pipeline Stages
 Know their order
 Mid-term asked to order stages
 Worth revisiting

32. CS 354 32
Coordinate Space
Transformations
 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

33. CS 354 33
Edge Equation Setup
for 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

34. CS 354 34
Plane Equation Setup
for Interpolation
 Given window-space vertices for a
triangle, be able to setup plane equation
for interpolating Z

35. CS 354 35
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

36. CS 354 36
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

37. CS 354 37
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

38. CS 354 38
Acceleration Structures
 Name common acceleration structures
 Weight advantages and disadvantages
 Explain ray traversal given a heirarchical
acceleration structure

39. CS 354 39
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

40. CS 354 40
Color
 Perform color space conversions
 RGB to YIQ
 What is a color gamut?

41. CS 354 41
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?

42. CS 354 42
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?

43. CS 354 43
Forces Driving Improvements in
Computer Graphics
Em
Pa barra Computer g
sin tor
ral ea
leli ssing I ncr nduc
Gra sm o
ph
ics of Graphics mic nsity
Se De
Particularly the Moore’s
hardware-amenable, Law
latency tolerant
nature of rasterization
Human desire for Visual
Intuition and Entertainment
Particularly
interactive video games

44. CS 354 44
Course-Instructor Survey
 Instructor’s Name
 Mark Kilgard
 Course Abbreviation and Number
 CS 354
 Course Unique Number
 53035
 Semester and Year
 Spring 2012