Computer Animation Keyframe and Interpolation

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Computer Animation Keyframe and Interpolation
CMPSEM026
Abdennour El Rhalibi
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Animation Techniques

• Key Frame Animation
• Position
• Interpolation Linear, quadratic, cubic
• Hermite Interpolation.
• Catmull-Rom Interpolation
• Bezier Interpolation.
• TCB Interpolation.
• Rotation
• Interpolation of Euler angles.
• Interpolation of Quaternions.
• Spherical Linear Interpolation.
• Spherical TCB Interpolation.
• Path Animation
• Uniform speed parameterization.
• Tangent guided rotation.

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Keyframing

• 2D/Hand-Drawn/Cel Animation
• Highly skilled animator draws the important, or
  key frames
• Less skilled (lower paid) animator draws the
  inbetween frames
• 3D/Computer Generated Animation
• Animator specifies the important key frames
• Computer generates the in-betweens automatically
  using interpolation

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Interpolation and Basic Techniques

• Abstract out the important features of the
  animation ?
• A very subjective thing to do ? the appropriate
  feature for one kind of animation will not be the
  appropriate feature for another kind of
  animation.
• Our focus ? various techniques in which the
  animator is responsible for specifying most of
  the information (low level of abstraction).

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Level of abstraction
High
Weak relationship (Model-based techniques)
Information provided by the animator
Low
the resulting motion
Strong relationship
We are here!
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Interpolation and Basic Techniques

• Interpolation
• Motion along a curve (arc length)
• Interpolation of rotations (quaternions)
• Path following

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Interpolation

• Generating in between values (connecting the
  dots)
• Examples are linear and cubic polynomial
  interpolation
• We can interpolate any changeable value!!!
• position
• orientation
• Color (sunrise)
• Intensity (dimmed lights)
• camera focal length (zoom)

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Interpolation

• At the foundation of almost all animation is the
  interpolation of values.
• The simplest case is interpolating the position
  of a point in space.
• Even this is non-trivial to do correctly and
  requires some discussion of several issues
• the appropriate parameterization of position,
• the appropriate interpolating function,
• and maintaining the desired control of the
  interpolation over time.

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Interpolation

• Often, an animator has a list of values
  associated with a given parameter at specific
  frames (called key frames or keys) of the
  animation.
• The question to be answered is how best to
  generate the values of the parameter for the
  frames between the key frames.
• The parameter to be interpolated may be
• a coordinate of the position of an object,
• a joint angle of an appendage of a robot,
• the transparency attribute of an object,
• any other parameter

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Interpolation

• Our focus ?How to choose the most appropriate
  interpolation technique and apply it in the
  production of an animated sequence.
• One of the first decisions to make is whether the
  given values represent actual values that the
  parameter should have at the key frames
  (interpolation), or whether they are meant merely
  to control the interpolating function and do not
  represent actual values the parameter will assume
  (approximation).

interpolation
approximation
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Interpolation

• Other issues that influence which interpolation
  technique to use include how smooth the resulting
  function needs to be (i.e. continuity),

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Interpolation

• how much computation you can afford to do (order
  of interpolating polynomial, and whether local or
  global control of the interpolating function is
  required.

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• Spline curves control points
• Spline
• Created by mathematical technique that fits a
  curve to a set of given points called Control
  Points Curve fitting
• Artist places the points, and program creates a
  curve based on the points
• Polyline
• series of straight lines through the points
• B-splines
• blend position of control points without
• passing through any of them
• Non-uniform rational B-splines (NURB)

Ex
Ex
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• Bezier curve
• Named after French engineer Pierre Bezier
• Direct control over the location of the
• curve with anchor points
• Over the curvature between the points
• with two off-the curve control points
• Difficult off-path control points
• Catmull-Rom Spline
• Interpolating spline
• Restricts the curvature choice but
• places all points on the curve
• Make drawing and editing more
• straightforward

Often used in animation path movement
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Curves
Hermite
Bezier
Catmull-Rom
Blended parabolas
B-splines, NURBS
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Hermite
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Bezier
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B-Spline / NURBS
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Often used in animation path movement
Catmull-Rom
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Often used in animation path movement
Blended Parabolas
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Interpolation

• Lets assume an interpolating technique has been
  chosen and that a function P(t) has been selected
  which, for a given value of t, will produce a
  value, i.e., p P(t).
• If position is being interpolated then three
  functions are used in the following manner. The
  x, y and z coordinates for the positions at the
  key frames are specified.

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Interpolation

• Key frames are associated with specific values of
  the time (t) parameter.
• The x, y and z coordinates are considered
  independently.
• For example, the points (x,t) are used as control
  points for the interpolating curve so that
  XPx(t)
• Similarly, YPy(t) and ZPz(t) are formed so that
  at any time, t, a position (x,y,z) can be
  produced.

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Interpolation

• Varying the parameter of interpolation, in this
  case t, by a constant amount, does not mean that
  the resulting values, in this case Euclidean
  position, will vary by a constant amount.
• This means that just because t changes at a
  constant rate, the interpolated value will not
  necessarily, in fact seldom, have a constant
  speed.

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Interpolation ?Parametric Motion

• This animation shows a ball moving along a
  parametrically defined cubic curveP(t) at3
  bt2 ct d by uniformly varying the parameter
  (i.e. t 0.0, 0.1, 0.2, etc.).
• Note that it travels further between frames at
  the start of the curve than at the end of the
  curve Equi-distant values in parametric space do
  not result in equi-distant points in Euclidean
  space.

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Interpolation

• In order to ensure a constant speed for the
  interpolated value, the interpolating function
  has to be parameterized by arc length, i.e.,
  distance along the curve of interpolation.
• Some type of reparameterization by arc length
  should be performed for most applications.
• Usually this reparameterization can be
  approximated without adding undue overhead or
  complexity to the interpolating function.

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Linear InterpolationThe Problem

• In Animation we often need to determine the value
  of some quantity in between two places where its
  value is known. For example,
• We are rendering a line. One endpoint is white
  the other is blue. What color should the rest of
  the line be?
• We are animating a spinning wheel. We know its
  angular position at two key frames what is the
  angle between the two frames?
• We are rendering a partially transparent polygon.
  We know the transparency at two of its vertices
  what is the transparency between these vertices?
• We are drawing a smooth curve. We know its slope
  at two different points what is the slope
  between the two points?
• We are texturing a polygon (painting an image on
  it). What color is the polygon between two pixels
  in the image (texels)?
• We are doing lighting calculations for a surface.
  We know the effect of lighting at two points
  what is the effect between the two points?

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Linear InterpolationLirping

• Approximating a quantity between places where the
  quantity is known, is called interpolation.
• Approximating the quantity in other places is
  called extrapolation.
• The simplest interpolation method is linear
  interpolation.
• Linear interpolation assumes that the graph of
  the quantity between the two known values is a
  straight line.
• Of course, this assumption is often false, but
  remember that we are only trying to approximate.
• We abbreviate linearly interpolate as lirp
  some people spell it lerp.

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Linear InterpolationSimple Case

• When we lirp, we are given two values of a
  quantity we determine its value somewhere
  between these.
• To do this, we assume that the graph of the
  quantity is a line segment (between the two given
  values).
• Here is a simple case
• When t 0, the quantity is equal to a.
• When t 1, the quantity is equal to b.
• Given a value of t between 0 and 1, the
  interpolated value of the quantity is
• Check that this satisfies all the requirements.
• Note We only do this for t between 0 and 1,
  inclusive.

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Linear InterpolationExample 1

• An object lies at position y 2 at time t 0
  and position y 3.1 at time t 1. Using linear
  interpolation, approximate the position of the
  object at time t 0.6.

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Linear InterpolationGeneral Case

• What if we are not given values of the quantity
  at 0 and 1?
• Say the value at s1 is a and the value at s2 is
  b, and we want to find the interpolated value at
  s.
• We setand proceed as before

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Linear InterpolationExample 2

• The brightness of a moving light source is 1.0
  when it lies at position x 1.1 and 0.5 when it
  lies at position x 1.5. Using linear
  interpolation, approximate the brightness when it
  lies at position x 1.4.

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Linear InterpolationLirping Colors Points

• Much of the interpolation done in CG involves
  several numbers at once. An important example is
  colors, which are typically specified as three
  numbers R, G, and B.
• To lirp between two colors, we actually lirp
  three times between the two R values, between
  the two G values, and between the two B values.
• The three lirping calculations are carried out
  independently of each other.
• This method actually ignores some important
  characteristics of human vision however, when
  the two colors are very similar, it is a fine
  method to use.
• Lirping between 2-D and 3-D points is done
  similarly.

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Linear InterpolationExample 3

• The position of an object is (0, 1) at time t 0
  and (2, 10) at time t 1. Using linear
  interpolation, approximate the position of the
  object at time t 0.1.

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Position Interpolation Problem generate a path
through points at designated times with smooth
motion
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Answers

• Example 1
• a 2, b 3.1, and the relevant value of t is
  0.6.
• Approximation (1 0.6) ? 2 0.6 ? 3.1 2.66.
• Example 2
• We start at 1.1 and end at 1.5. We want to
  approximate at 1.4.
• So t (1.4 1.1)/(1.5 1.1) 0.75.
• Now, a 1.0, b 0.5, and t 0.75.
• Approximation (1 0.75) ? 1.0 0.75 ? 0.5
  0.625.
• Example 3
• We lirp the two coordinates separately.
• 1st coord. a 0, b 2, t 0.1. (1 0.1) ? 0
  0.1 ? 2 0.2.
• 2nd coord. a 1, b 10, t 0.1. (1 0.1) ? 1
  0.1 ? 10 1.9.
• We finish by putting the two coordinates
  together (0.2, 1.9).

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Interpolation vs. Approximation

• Have a set of data points (key frames)
• Usually in 3D
• Want to fill in whats in between
• Create continuous curve P(u)
• Interpolation
• Curve goes through data
• Approximation
• Curve goes near the points (data) used as weights
  or control points

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Interpolation Qualities

• Want smooth curves
• Local control
• Local data changes have local effect
• Continuous with respect to the data
• No wiggling if data changes slightly
• Low computational effort

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Continuity

• Describe curves in terms of their continuity in a
  segment and between segments
• Parametric continuity Cx
• Only P is continuous C0
• Positional continuity
• P and first derivative are continuous C1
• Tangential continuity
• P first second C2
• Curvature continuity

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Curves
Explicit form y f(x)
Implicit form 0 f(x,y)
x f(u)
Parametric form
y g(u)
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Parametric Curves

• You are probably familiar with curves like y x²
  or f(x)x².
• A parametric curve is similar to those, but the
  x- and y-values are calculated in separate
  functions.
• To do this we need another variable, say a.
• So instead of f(x), we say X(a) and Y(a), where
  X(a) gives you an x-value for each value of a,
  and Y(a) gives you an corresponding y-value for
  the same value of a.
• Here is an example
• X(a) a/2 Y(a) a5
• If we substitute a with a number we can get a
  point that lies on the curve/line a 6
• X(6) 6/2 3 x
• Y(6) 65 11 y
• We now know that (3,11) is a point on the curve.
• We can easily make a curve in 3D just by defining
  Z(a).

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Simple line

• The simplest form of interpolation is a straight
  line from a control point A, to a control point
  B.
• I will use Ax to denote the x-coordinate of
  control point A, and Ay to denote the
  y-coordinate.
• The same goes for B.
• I am introducing another variable, b, just to
  make it a bit more simple to read the functions.
• However, the variable b is only a in disguise, b
  is always equal to 1-a.
• So if a 0.2, then b 1-0.2 0.8. So when you
  see the variable b, think (1-a).
• This parametric curve describes a line that goes
  from point A, to point B when the variable a goes
  from 1.0 to 0.0
• X(a) Axa Bxb
• Y(a) Aya Byb P(a) Axa Bx(1-a)
• Z(a) Aza Bzb

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Simple line

• This parametric curve describes a line that goes
  from point A, to point B when the variable a goes
  from 1.0 to 0.0
• X(a) Axa Bxb
• Y(a) Aya Byb
• Z(a) Aza Bzb
• If you set a 0.5
• (and thus also b 0.5, since b 1.0-a 1.0-0.5
  0.5)
• then X(a), Y(a) and Z(a) will give you the 3D
  coordinate of the point on the middle of the line
  from point A to point B.
• Note that the reason this works is because a b
  is always equal to one, and when you multiply
  something with one, it will stay unchanged.
• This makes the curve behave predictably and lets
  you place the control points anywhere in the
  coordinate system, since when a 1.0 then b
  0.0
• thus making the point equal to one of the
  control points, and completely ignoring the other
  one.

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Linear interpolation

• Curve continuity ?
• Locality ?
• Computational effort ?

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Linear interpolation

• Curve continuity ?
• -- Low (C0)
• Locality ?
• Computational effort ?
• P(a) Axa Bx(1-a), a ?0..1

Important Criteria Continuity is missing for
Linear Interpolation
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Quadratic

• Ok, now we have done lines, but what about
  curves?
• (ab), where b (1.0-a) is the key.
• We know that a polynomial function is a curve, so
  why not try to make (ab) a polynomial function?
• Let's try
• (ab)² a² 2ab b²
• Knowing that b1-a we see that a² 2ab b² is
  still equal to one,
• since (ab) 1, and 1² 1.
• We now need three control points, A, B and C,
  where A and C are the end points of the curve,
  and B decides how much and in which direction it
  curves.
• Except for that, it is the same as with the
  parametric line.
• X(a) Axa² Bx2ab Cxb²
• Y(a) Aya² By2ab Cyb²
• Z(a) Aza² Bz2ab Czb²
• P(a) Axa² Bx2a(1-a) Cx(1-a)²

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Quadratic

• This parametric curve describes a curve that goes
  from point A, to point C in the direction of C
  when the variable a goes from 1.0 to 0.0
• X(a) Axa² Bx2ab Cxb²
• Y(a) Aya² By2ab Cyb²
• Z(a) Aza² Bz2ab Czb²
• If you set a 0.5, then a² 0.25, 2ab 0.5
  and
• b² 0.25 giving the middle point, B, the biggest
  impact and making point A and C pull equally to
  their sides.
• If you set a 1.0, then a² 1.0, ab 0.0 and
  b² 0.0 meaning that result is the control point
  A itself,
• Just the same as setting a 0.0 will return the
  coordinates of control point C.

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Polynomial interpolation

• Curve continuity ?
• Locality ?
• Computational effort ?

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Example Polynomial interpolation

• Curve continuity ?
• Locality ?
• --
• Computational effort ?
• P(a) Axa² Bx2a(1-a) Cx(1-a)²

Important Criteria of Locality is missing for
Polynomial Interpolation
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Cubic

• We can also increase the number of control points
  using (ab)³ instead of (ab)², giving you more
  control over how to bend the curve.
• (ab)³ a³ 3a²b 3ab² b³
• Now we need four control points A, B, C and D,
  where A and D are the end points.
• X(a) Axa³ Bx3a²b Cx3ab² Dxb³
• Y(a) Aya³ By3a²b Cy3ab² Dyb³
• Z(a) Aza³ Bz3a²b Cz3ab² Dzb³
• It still works the same way as the previous ones,
  as a goes from 1.0 to 0.0 (and thus b from 0.0 to
  1.0) the functions will return coordinates on a
  smooth line from control point A to control point
  D, curving towards B and C on the way.

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Cubic

• X(a) Axa³ Bx3a²b Cx3ab² Dxb³
• Y(a) Aya³ By3a²b Cy3ab² Dyb³
• Z(a) Aza³ Bz3a²b Cz3ab² Dzb³
• You can easily use (ab) to the power of n and
  get n1 control points (where n is any integer
  equal to, or greater than one).
• But the more control points you have, the more
  computational effort is needed.
• Personally I like the cubic ones best. You can
  easily make circles with them, and you can
  control the direction of the curve independently
  at each control point.

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A general solution

• Use SEVERAL polynomials
• Complete curve consists of several pieces
• All pieces are of low order
• Third order is the most common (cubic)
• Pieces join smoothly
• This is the idea of spline curves (splines)

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Spline curves

• Many different types
• Some are better for interpolation
• Others for geometric modeling
• Main Types
• Hermite interpolation
• Catmull-Rom Splines
• Bezier curves
• Other types
• Splines with tension (Tension Continuity Bias
  TCB)
• B-splines
• NURBS

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Comparison of Basic Cubic Splines