Paper Summaries Any takers? Interpolation Projects Factoids - - PDF document

SLIDE 1

1

Interpolation

Paper Summaries

• Any takers?

Projects

• Proposals due next week.

Factoids

• Digital Video Frame Rate

– Star Wars was shot at 24 fps. – Keep film look

Plan for today

• Keyframing and Interpolation
• But first…

Motivation Films

• Two early attempts of significant use of CG

elements in film

SLIDE 2

2

Motivational Film

• TRON

– Disney (1982) – Clips shown at SIGGRAPH 82

Motivational Film

• Question from SIGGRAPH Bowl

– Q: Which special effects company was responsible for the effects of TRON? – A: ALL OF THEM

• MAGI
• Triple I
• Robert Abel & Associates
• Digital Effects

Motivational Film

• Interesting TRON Factoid

– No polygons! – All objects were defined using mathematical formula.

• Lesson Learned from TRON

– Great effects – Bad movie – Overestimated the “Geek Factor”

Motivational Film

• The Last Starfighter (1985)

– Effects by Digital Productions – Universal Pictures

Motivational Film

• Last Starfighter factoids

– First movie to do all special effects (except makeup) on a computer. – All shots of spacecraft, space, etc. were generated on a Cray computer.

• Lesson Learned from Last Starfighter

– Great effects – Bad movie – Overestimated the “Geek Factor”

Motivational Film

• Life imitating art imitating life…

– Sort of… – A prototype for A Last Starfighter video game was produced by Atari (remember them?) but was never distributed. – Note that there WAS a TRON video game (made by Bally/Midway – 1983)

SLIDE 3

3

Let’s get started

• Key framing and interpolation
• 6. Straight ahead action and Pose-to-pose action
• The two main approaches to hand drawn

animation.

– Straight ahead action

• “the animator works straight ahead from his first drawing in

the scene.”

– Pose-to-pose (keyframing)

• the animator plans his actions, figures out just what drawings

will be needed to animate the business, makes the drawings concentrating on the poses, ... and then draws the inbetweens.”

• Link

Key framing

• The simplest form of animating an object.
• Object has:

– A beginning state – An end state – A number of intermediate “key” states

• It is up to the keyframe system to determine

the states for inbetween frames

Key Framing

P1 P2 P3 P4 P5 P6

Time = 0 Time = 10 Time = 35 Time = 50 Time = 55 Time = 60

Key Framing

• 60 sec animation

– Video: 30 frames / sec = 1800 frames – We have 6 key frames – Need to create position of objects for the remaining 1794 frames.

Interpolation

• According to webster.com

– in·ter·po·late (n) - to estimate values of (a function) between two known values – Note: “state” can be defined as a function (in this case of time)

• F(t)

SLIDE 4

4

Interpolation

• Things to consider:

– Interpolation or approximation?

• Interpolation – Control points fall on the curve
• Approximation – Control points fall outside the

curve.

Interpolation

• Things to consider:

– Continuity – defines “smoothness” of curve. – Determined by how many derivatives are continuous

• 0th order – are the curves attached at the points

themselves (positional continuity)

• 1st order – is the first derivative continuous

(tangential continuity)

• 2nd order – is the 2nd derivative continuous

(curvature continuity)

Interpolation – Continuity

No positional continuity Order 1 Continuity Order 0 continuity 2nd order continuity For animation purposes, Tangential (1st Order) continuity is usually sufficient

Interpolation

• Things to consider:

– Local vs. Global control

• Does changing one point change the entire function?

– Questions?

Linear Interpolation

• The simplest solution

P(0) = P0 P(1) = P1 P(t)

1

) 1 ( ) ( P u P u u P ⋅ + ⋅ − =

1

) ( ) ( P u P P u P + ⋅ − =

Also can be expressed as

Linear Interpolation

• Pros

– Straight-forward – Easy to implement – Direct relationship between time and value

• Cons

– Lacks tangential continuity – Why is this important?

SLIDE 5

5

Remember our principles of animation?

• 8. Arcs

– “The visual path of action from one extreme to another is always described by an arc.”

This is not an arc!

Cubic Interpolation

• x, y, z given as function of t
• Approximated by cubic polynomials

x x x x

d t c t b t a t x + + + =

2 3

) (

y y y y

d t c t b t a t y + + + =

2 3

) (

z z z z

d t c t b t a t z + + + =

2 3

) (

Cubic Interpolation

[ ]

1

2 3

t t t T =

⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ =

z z z z y y y y x x x x

d c b a d c b a d c b a C

[ ]

) ( ) ( ) ( ) ( t z t y t x t Q =

C T Q ⋅ =

Cubic Interpolation

• Separate C matrix into basis + geometry

4 4 3 4 4 2 1 4 4 4 4 3 4 4 4 4 2 1

points control 4 3 2 1 4 3 2 1 4 3 2 1 matrix basis 44 43 42 41 34 33 32 31 24 23 22 21 14 13 12 11

⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ =

z z z z y y y y x x x x

G G G G G G G G G G G G m m m m m m m m m m m m m m m m C

Cubic Interpolation

• Basis & Control points distinguishes types of

curves

{

{

control basis

) ( G M T t Q ⋅ ⋅ =

Cubic Interpolation

• Hermite Curves

– specified by endpoints and tangent vectors from each endpoint

SLIDE 6

6

Cubic Interpolation

• Hermite Curves

⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − = 1 1 1 2 3 3 1 1 2 2

H

M ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ =

4 1 4 1

R R P P GH

Cubic Interpolation

• Problem with Hermite Curves

– Animator generally will not enjoy specifying tangent vectors – Solution: The Catmull-Rom spline

Cubic Interpolation

• Catmull-Rom spline

– Uses previous and next control point to determine tangent vector – Spline goes through all control points!

Cubic Interpolation

• Catmull-Rom

⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − − = 2 1 1 1 4 5 2 1 3 3 1 2 1

CR

M ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ =

+ + − 2 1 1 i i i i CR

P P P P G Note: for interpolation between Pi and Pi+1 you need 4 points

Cubic Interpolation

• Bezier Curves

– specified by endpoints and intermediate points, not on the curve, that approximate the tangent vectors

Cubic Interpolation

• Bezier Curves

⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − − = 1 3 3 3 6 3 1 3 3 1

B

M ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ =

3 2 1

P P P P GH

SLIDE 7

7

Cubic Interpolation

• Bezier Curves suffer from same problem as

Hermite Curves

– Animator generally will not enjoy specifying tangent vectors…even from control points

• Exception: Provide animator with an interactive curve editor

– DeCastelau Construction

• Algorithm to automatically define Bezier control points
• Described in Parent Appendix and Shomake Quat. paper

Cubic Interpolation

• B-Splines

– Control points are outside of the curve – Local control

• Moving one control point only affects a portion of

the entire curve.

– Individual curve segments share control points. – Knots - points connecting individual curve segments.

Cubic Interpolation

• B-Splines

Cubic Interpolation

• B-Splines - local control

Cubic Interpolation

• Segment Qi uses control points Pi-3, Pi-2, Pi-1 and Pi

Cubic Interpolation

• B-Splines

⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − − = 1 4 1 3 3 3 6 3 1 3 3 1 6 1

B

M ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ =

− − − i i i i H

P P P P G

1 2 3

SLIDE 8

8

Cubic Interpolation

• Uniform Nonrational B-Splines

– uniform - knots are equally spaced – nonrational - does not use homogeneous coordinates (more on this later) – B - basis -- curves are weighted sums of a basis function

Cubic Interpolation

• Rational Splines

– expressed using homogeneous coordinates

[ ]

) ( ) ( ) ( ) ( ) ( t W t Z t Y t X t Q = ) ( ) ( ) ( , ) ( ) ( ) ( , ) ( ) ( ) ( t W t Z t z t W t Y t y t W t X t x = = =

Cubic Interpolation

• Rational splines -- why bother?

– Invariant under rotation, scaling, translation, and perspective transforms – Can define precisely any of the conic sections. (spheres, ellipsoids, parabolids, etc)

Cubic Interpolation

• NURBS

– Non-uniform Rational B-Splines – Non-uniform - knots need not be equally spaced. – Most useful and used method for representing surface curves in CG. – Probably overkill for cubic interpolation

Cubic Interpolation

• Pros

– We now have our tangential continuity – More complex than linear but not too complex

• Cons

– Spline control does not correspond to time. – May require an interactive “curve editor” tool.

Cubic Splines

u=1/3 u=2/3 u=0.0 u=1.0 u does not correspond to t.

SLIDE 9

9

Cubic Splines

• Our problem

– Spline control does not correspond to time. – What we have

• P(u) – position w.r.t. spline control

– What we need

• P(t) – position w.r.t. time (note we have this for

control points)

Cubic Splines

• Solution

– Determine:

• u(t) – spline control w.r.t time

– Then

• P(t) = P (u(t))

Cubic Splines

• Solution

– Determine:

• U(t) – spline control w.r.t time

– Then

• P(t) = P (U(t))

– Can use interpolation on u(t) using t at control points

• Linear
• Cubic

– Questions?

Cubic Splines

P1 P2 P3 P4 P5 P6

Time = 0 Time = 10 Time = 35 u = 0.0 Time = 50 u=1.0 Time = 55 Time = 60 at t = 42 u = 0.46

Let’s Take this a Step Further

• 7. Slow In and Slow Out

– Refers to the spacing of the inbetween frames at maximum positions. – Example:

• a bouncing ball moves faster as it approaches or

leaves the ground and slower as it approaches leaves its maximum position

Let’s Take this a Step Further

• Suppose animator knows path but wants

more control over speed?

– What we have

• P(u) – position w.r.t. spline control

– Note that this is NOT

• S(u) – length traveled w.r.t. spline control.

– And we need to give animator control of

• S(t) – length traveled at a given time.

SLIDE 10

10

Let’s Take this a Step Further

• Suppose animator knows path but wants more

control over speed?

– What we have

• P(u) – position w.r.t. spline control

– What we need

• U(S) – length w.r.t. spline control

– And let animator define

• S(t) – length w.r.t. time (velocity/speed)

– Then

• P(t) = P (U (S (t)))

Arc Length

• Arc Length = distance traveled along a curve.

– Mathematically defined as:

du du dP u u s

u u

∫

=

2 1

) , (

2 1

Arc Length

• So, arc length is an integral

– Ways to solve

• Analytic – integral not always solvable
• Numerical Integration – more when we talk about

physics.

• Forward Differencing

Arc Length

• Forward Differencing

– Sample the curve at equally spaced u values. – Estimate arc length by calculating linear distance between sampled points. – For example,

• Given P(u) calculate at u=0.00, 0.05, 0.10, 0.15, …
• Define arc length s(u)

– s (0.0) = 0.0 – s (0.05) = distance between P(0.05) and P(0.0) – s (0.1) = s (0.05) + distance between P(0.10) and P(0.05) – Etc.

Arc Length

• Forward Differencing

– This will give us a mapping from P(u) to s(u) for sampled u values. – Perform interpolation (linear or cubic) between sampled values

Arc Length

s(u) u

Idx

0.720 0.40

8

0.600 0.35

7

0.500 0.30

6

0.400 0.25

5

0.320 0.20

4

0.230 0.15

3

0.150 0.10

2

0.080 0.05

1

0.000 0.00

Find u corresponding to s = 0.25 Find closest idx to s = 0.25 Find relative distance to s for next idx

222 . 23 . 32 . 23 . 25 . = − −

Linearly interpolate u ) 15 . 20 . ( 22 . 15 . ) 25 . ( − + = u

SLIDE 11

11

Animator control of speed

• Animator can define s(t)

– Length traveled w.r.t time – Speed

Basic ease-in/ease out

Ease-in/ease out summary

• Animator defines speed

– Arc length given time

Ease-in/ease out summary

• Spline control determined by forward

differencing

222 . 23 . 32 . 23 . 25 . = − −

Linearly interpolate u 161 . ) 15 . 20 . ( 22 . 15 . ) 25 . ( = − + = u Find u corresponding to s = 0.25 Find closest idx to s = 0.25 Find relative distance to s for next idx

Ease-in/ease out summary

• Position determined by spline

– P = P (0.161) – Questions?

Interpolating Quaternions

• Normalized Quaternions are defined on the unit

4D sphere

• Linear interpolation generates unequal spacing

when projected onto circle

Interpolating Quaternions

• Spherical Linear Interpolation

– where

θ θ θ θ sin ) sin( sin ) ) 1 sin(( ) , ( slerp

2 1 , 2 1

u q u q u q q + − =

) ( cos

2 1 2 1 2 1

v v s s q q

• +

=

• =

θ

SLIDE 12

12

Let’s go to assignment 1