Verbs and Adverbs: Multidimensional Motion Interpolation Using Radial Basis Functions

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Verbs and Adverbs Multidimensional Motion
Interpolation Using Radial Basis Functions
Charles Rose
Michael F. Cohen
Bobby Bodenheimer
Presented by Sean Jellish
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The Problem

• Creating believable animated humans is hard
• Results are difficult to reuse
• Modifying an animation can be almost as hard as
  creating the original motion
• Exact motion may not be known until runtime

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The Interpolation Method

• Sets of example motions are combined with an
  interpolation scheme to produce new motions
• Interpolated motions must keep the look and feel
  of the examples
• Examples are precious and hard to obtain
• Interpolation scheme must be efficient to run in
  real time

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Verbs and Adverbs

• Verb a parameterized motion
• Walking, Running, Swimming
• Adverb the parameters for the motion
• Happy, Tired, Frustrated, Confused
• Verb Graph a graph of motions with transitions
  between them

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Adverb Spaces

• An axes is defined for each adverb
• This creates a multidimensional adverb space of
  all possible variations of a particular verb
• Every example or interpolated motion can be
  placed in this adverb space based on the values
  of its parameters

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Parts of the System

• Two main parts
• Authoring system
• Allows for creating verbs from sets of examples
• Allows for combining verbs together
• Runtime system
• Determines which verb is currently in use
• Calculates figures pose at each frame

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Animated Figures

• System assumes figures are constructed from a
  hierarchy of rigid links connected by joints
• Each joint may have one or more DOFs
• Root of hierarchy has 6 additional DOFs
• The DOF functions are created by interpolating
  the example motions which are weighted by their
  adverbs (this is the hard part)

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Restrictions on Examples

• All examples of a particular verb must be
  structured similarly
• Start on same foot
• Take same number of steps
• The examples must have a consistent use of joint
  angles

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Annotation

• Each example motion is placed into the adverb
  space by giving it adverb values
• Key times must also be defined in each example
• Each example is given a set of constraints

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Example Populated Adverb Spaces
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Key Times

• Keytime an instant when an important structural
  event occurs
• Foot down
• Foot up
• Specifying keytimes enables the different example
  motions to be of different time durations

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Time Warping

• For interpolation to work, time must be warped so
  that examples of varying lengths can be compared
• Clock time gets transformed into a generic time
  based on the key times

• In this way all of the examples can be put into a
  canonical timeline and will be at the same
  structural point of motion for any given t

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Example Time Mapping
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Constraints

• Key times also specify the periods during which
  kinematic constraints should be enforced
• Specific constraint conditions are not evaluated
  until runtime when they are triggered by a key
  time being crossed
• To find the DOF changes needed to satisfy the
  constraint, solve the linear system

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Creating New Motions

• Populate adverb space with examples
• Every point p in the adverb space defines a
  motion with the specified parameters
• Combine radial basis functions of all the
  examples and add in a linear polynomial
• Polynomial provides an approximation to the space
• Radial basis functions locally adjust the
  polynomial

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Linear Approximation

• Create a best fit hyperplane through the adverb
  space that minimizes the error between an
  examples value in the plane and its actual value

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Radial Bases

• These are used to locally adjust the linear
  approximation returned by the hyperplane
• The basis functions are dilated cubic B-splines
• Dilation factor gives a support radius equal to
  twice the distance to nearest example

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Summing Up the Math
Radial basis functions with parameter p
Interpolated control point for new motion
The height of p in the approximated hyperplane
Sum of all the radial basis functions at p
Actual value of each of our examples in the
hyperspace
Weights of the radial basis functions
Value of our examples interpolated into the
approximated hyperplane
Special square matrix created to cancel out the
residuals
The residuals formed from the introduction of the
hyperplane
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Summing Up the Math

• Creates a wavy hyperplane
• The value of each example is on the hyperplane
  and there is a spline shaped mountain extending
  away from it in all directions for an amount
  equal to twice the distance to a neighbor
• A new motion will be somewhat effected by all of
  the examples but even more so by its close
  neighbors

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Verb Graphs

• A directed graph of verbs
• Nodes correspond to verbs
• Arcs correspond to transitions between verbs
• If multiple arcs leaving a node, each arc is
  given a likelihood of occurring
• Adverbs are shared across verbs even if they do
  not apply
• Static

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Transitions

• Transitions are meant to smoothly move control
  between verbs
• They map similar segments between two verbs
• Transition duration is determined by taking the
  average of the lengths of the transition
  intervals of the two verbs in generic time
• The two verbs are blended by fading the joint
  angles of the first verb out while fading those
  of the second verb in

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Transitions

• DOFs are found by interpolating joint positions
  between the verbs

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Transitioning at Runtime

• A search is made to find shortest path through
  graph from current verb to desired verb
• Upcoming transitions and verbs stored in queue
• Must remember position and orientation between
  verbs
• If queue goes empty and verb ends, a transition
  is chosen based on the transition weights

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Runtime System

• Events inserted into event queue in timestamp
  order and associated with callback function
• Three event types
• Normal
• Sync
• Optional
• Render event calculates DOFs for timestamp
• Display event displays rendered image when
  timestamp equals clock

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Runtime Processing
Only computed when the parameters to a verb or
the whole verb itself changes
Only four of these are needed at a time and they
are only computed once per verb adverb set