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Chap 4Interpolation-Based Animation
Animation (U), Chap 4, Interpolation-based Animation
1CS, NCTU, J. H.Chuang
CS, NCTU, J. H.Chuang2
Outline
Key-Frame Systems Animation Languages Deforming Objects 3D Shape Interpolation 2D Morphing
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang3
Key-Frame Systems Hand-drawn animation
Key Frames - defined and drawn by master animators
Intermediate frames – drawn by assistant animators Computer animation
Key Frames - be generalized to apply to any VARIABLE whose value is set at specific key frames
Intermediate frames – values are interpolated according to some prescribed procedure
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang4
Key-Frame Systems
Animation (U), Chap 4, Interpolation-based Animation
Specify interpolation of key values and tangents at segment boundaries
CS, NCTU, J. H.Chuang5
Key-Frame Systems What is the key?
Difficult to interpolate hand-drawn images
Different approach in computer animation Each key frame is described by a set of parameters Sequence of key frames = points in high-dimensional
space Compute in-between by interpolating these points
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang6
Key-Frame Systems What is a key?
For a bouncing ball 3D Positions Orientation? Squishedness?
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang7
Key-Frame Systems What is a key?
For Shrek? 3D Position and orientation Joint angles of the skeleton Facial features Hair/fur? Clothing? Clouds?
Scene components? Camera Lights
Shrek (PDI/DreamWorks, 2001)
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang8
Key-Frame Systems Key-framing Procedures
Specify the key frames rigid transformation, forward/inverse kinematics
Specify the type of interpolation linear, cubic, parametric curves
Specify the speed profile of the interpolation constant velocity, ease-in/out, etc.
Computer generates the in-between frames
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang9
Key-Frame Systems Pros and Cons
Good control over motion Eliminates much of the labor in traditional
animation, but still very labor-intensive Impractical for complex scenes
water, smoke grass in the wind crowds
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang10
Key-Frame Systems Basic operation: Interpolating Curves
Point-to-point basis: straightforward
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang11
Key-Frame Systems Basic operation: Interpolating Curves
Point-to-point correspondence is not known Curve-to-curve correspondence is given
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang12
Key-Frame Systems Basic operation: Interpolating Curves
Curve-to-curve correspondence is given What happen at intermediate points along the
curve is left undefined
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang13
Key-Frame Systems Basic operation: Interpolating Curves
If both curve are Bezier curves interpolating control points, or Generate curve points on both curves, followed by
point-to-point based interpolation Moving Point Constraints approach [Reeve
’81] Allows users to specify more information about
the point correspondence along the curves and the speed of interpolating these points
Uses “patch technology”Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang14
Key-Frame Systems Basic operation: Interpolating Curves
Moving Point Constraints approach [Reeve ’81] Defines a segment of the curve to interpolate,
bounded on top and bottom by interpolation constraints
Interpolation of the very top and very bottom of the curve
Define an intermediate curve based on the constraints – C(t)
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang15
Key-Frame Systems Basic operation: Interpolating Curves
Animation (U), Chap 4, Interpolation-based Animation
Moving points
Moving points
Moving points
Moving points
Moving points
CS, NCTU, J. H.Chuang16
Animation Languages
What is animation languages? A set of structured commands that can be used
to encode information necessary to produce animations Script-based
Text instructions Flowchart-like diagrams encode relationships
between objects and procedures
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang17
Animation Languages
Artist-oriented animation languages Full-featured programming languages for
animation Graphical languages – dataflow network Actor-based animation languages
Actor: a graphical object with its associated data and procedures, including geometric description, display attributes, and motion control.
Communication between actors: message passing
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang18
Deforming objects
Deforming and morphing an object is a visually powerful animation technique Flexible body animation makes the objects much
more expressive and alive How?
Physically based simulation Less control by animators Computationally expensive
By animator’s direct manipulation Key and interpolation
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang19
Deforming objectsPicking and Pulling (Editing)
Displace one or more of object’s vertices Others are propagated with attenuated distances
specified by a function of distance between the seed vertex and the vertex to be displaced
Minimum number of edges connecting these two vertices
Minimum distance traveled over the surface between these two vertices
Geodesic distance
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang20
Deforming objectsPicking and Pulling (Editing)
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang21
Deforming objectsPicking and Pulling (Editing)
Animation (U), Chap 4, Interpolation-based Animation
0 1
1
0 1
1
)( 1
1
kn
i
kn
i
iS k
k
i: minimum of connecting edgesn: maximum range of effectk: user-selected scale factor
CS, NCTU, J. H.Chuang22
Deforming objectsPicking and Pulling (Editing)
Animation (U), Chap 4, Interpolation-based Animation
K=0: linear attenuationK<0: more elasticK>0: more rigid displacement
CS, NCTU, J. H.Chuang23
Deforming objectsDeforming an Embedding Space
Deforming an Embedding Space Establish a local coordinate system that encases
the area of the object to be distorted Transform vertices to local coordinates
The local coordinate system is deformed by users in some way – easier or more intuitive
The local coordinate of the vertices are used to map their positions in global space
Example: Free-Form deformation (FFD)
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang24
Deforming objectsDeforming an Embedding Space
Deforming an Embedding Space Is easier or more intuitive than to manipulate
vertices of the object Restricted to possible distortions of the local
coordinate system Mapping should be continuous
More powerful than affine transformations
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang25
Deforming objects2D Grid Deformation
Local coordinate system a 2D grid in which an object is placed, aligning
with the global axes Local-to-global mapping
translation and scaling Deformation
Moving grid points to distort the local space Object’s vertices are relocated in the distorted grid
by bilinear interpolation relative to the grid cell
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang26
Deforming objectsDeforming an Embedding Space
Animation (U), Chap 4, Interpolation-based Animation
A: Global coordinate: (25.6, 14.7), Local coordinate : (5.6, 2.7)
CS, NCTU, J. H.Chuang27
Deforming objectsDeforming an Embedding Space
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang28
Deforming objectsDeforming an Embedding Space
Animation (U), Chap 4, Interpolation-based Animation
P=(0.6)(0.7)P00+(0.6)(1.0-0.7)P01+(1.0-0.6)(0.7)P10
+(1.0-0.6)(1.0-0.7)P11
CS, NCTU, J. H.Chuang29
Deforming objectsPolyline Deformation
Similar to grid approach object vertices are mapped to the polyline Polyline is modified Object vertices are mapped to the same relative
location on the polyline Polyline system
Polyline Boundary lines
Bisectors Perpendicular lines at end points
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang30
Deforming objectsPolyline Deformation
Animation (U), Chap 4, Interpolation-based Animation
For a given object vertex, we record1. The closest line segment (L2)2. The distance (d) to L2
3. The ratio r
CS, NCTU, J. H.Chuang31
Deforming objectsPolyline Deformation
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang32
Deforming objectsFreeform Deformation (FFD)
3D extension of 2D grid deformation A localized coordinate grid is superimposed over
an object For each object vertex, coordinate s relative to
local grid are determined The grid is manipulated by the user Each object vertex is mapped back into the
modified grid Cubic interpolation is typically used with FFD
Bezier interpolation in Sederberg’s paperAnimation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang33
Deforming objectsFreeform Deformation (FFD)
Animation (U), Chap 4, Interpolation-based Animation
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CS, NCTU, J. H.Chuang34
Deforming objectsFreeform Deformation (FFD)
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang35
Deforming objectsFreeform Deformation (FFD)
To facilitate the modification of local coordinate system, a grid of control points is created
Animation (U), Chap 4, Interpolation-based Animation
nkmjli
Un
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0,0,0for
0
CS, NCTU, J. H.Chuang36
Deforming objectsFreeform Deformation (FFD)
As the control points moved, the point P(s,t,u) moves
Animation (U), Chap 4, Interpolation-based Animation
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CS, NCTU, J. H.Chuang37
Deforming objectsFreeform Deformation (FFD)
Multiple FFD control grids can be joined with continuity constraints across the boundaries
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang38
Deforming objectsFreeform Deformation (FFD)
Other FFD control grids
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang39
Deforming objectsFreeform Deformation (FFD)
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang40
Deforming objectsFreeform Deformation (FFD)
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang41
Deforming objectsFreeform Deformation (FFD)
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang42
Deforming objectsComposite FFD – Sequential
An object is modeled by progressing through a sequence of FFDs, each of which imparts a particular feature to the object Various detail elements can be added to an object in
stages as opposed to trying to create on complex FFD designed to do everything at once
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang43
Deforming objectsComposite FFD – Sequential
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang44
Deforming objectsComposite FFD – Hierarchical
Allows the user to work at various levels of detail Finer-resolution FFDs are embedded inside FFDs
higher in hierarchy As a coarser-level FFD is used to modify object’s
vertices, it also modifies the control points of any children FFDs that are within space affted by the deformation
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang45
Deforming objectsComposite FFD – Hierarchical
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang46
Deforming objectsAnimated FFD
FFDs can be used to control the object’s animation By deformation tools By animating the FFD control points
Deformation tools – a composition of An user-defined initial lattice A final lattice – modified from initial lattice by the user Object’s animation can be driven by
Moving the tool Moving the object
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang47
Deforming objectsAnimated FFD
Animation (U), Chap 4, Interpolation-based Animation
Deformation tool applied to an object
CS, NCTU, J. H.Chuang48
Deforming objectsAnimated FFD
Animation (U), Chap 4, Interpolation-based Animation
Deformation by moving the deformation tool relative to an object
CS, NCTU, J. H.Chuang49
Deforming objectsAnimated FFD
Animation (U), Chap 4, Interpolation-based Animation
Deformation by movingthe object through FFD space
In logical FFD space In distorted FFD space
CS, NCTU, J. H.Chuang50
Deforming objectsAnimated FFD
Animating the FFD control points using, e.g., key-frame animation or by the result of physically based simulation
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang51
Deforming objectsProblems of Editing and FFD
Details are hard to be preserved
Animation (U), Chap 4, Interpolation-based Animation
Figure 3: Detail preservation is exhibited using Green Coordinates (on the right), where the details adhere to the surface deformation and rotate accordingly. In the middle, the MVC result is depicted where the details maintain their original orientation and therefore shear. From [Green Coordinates SIGGRAPH08]
Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang
3D Shape Interpolation
52
Interactive surface decomposition for polyhedral morphing Arthur Gregory et al. The Visual Computer 15(9), 1999
CS, NCTU, J. H.Chuang53
3D Shape Interpolation
Surface-based Vertex-to-vertex correspondence Interpolation between corresponding vertices Limitations on topological consistence
volume-based Problems
Surface representation -> volumetric representation More computationally expensive
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang54
3D Shape Interpolation
Topology of surface or object Surface topology
Connectivity Manifold vs. non-manifold
Object topology Genus - hole
Topologically equivalent A doughnut and a teacup is topologically equivalent
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang55
3D Shape Interpolation
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang56
3D Shape InterpolationFor Meshes With Same Topology
Animation (U), Chap 4, Interpolation-based Animation
Vertex-to-vertex correspondence problem Genus 0
Spherical parameterization Merging Find the vertex-to-vertex corresponding
Genus >= 0 Consistent Dissection
Parameterize patches in correspondence to planar domains Merging or re-meshing Derive vertex-to-vertex correspondence
Vertex-to-vertex interpolation problem
Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang
3D Shape Interpolation Spherical Parameterization
57
Spherical parameterization in [Zwicker and Gotsman 2004]
Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang
3D Shape Interpolation Spherical Parameterization
58
CS, NCTU, J. H.Chuang59
3D Shape InterpolationFor Meshes With Same Topology
Animation (U), Chap 4, Interpolation-based Animation
Vertex-to-vertex correspondence problem Genus 0
Spherical parameterization Merging Find the vertex-to-vertex corresponding
Genus >= 0 Consistent Dissection
Parameterize patches in correspondence to planar domains Merging or re-meshing Derive vertex-to-vertex correspondence
Vertex-to-vertex interpolation problem
CS, NCTU, J. H.Chuang60
3D Shape InterpolationFor Meshes With Same Topology
Animation (U), Chap 4, Interpolation-based Animation
Genus 0
Genus 1
Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang
3D Shape InterpolationUser Guided Common Dissection
61
Interactive surface decomposition for polyhedral Morphing, by Arthur Gregory et al. The Visual Computer 15(9), 1999
Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang
3D Shape InterpolationUser Guided Common Dissection
62
Input polyhedral with user-specified correspondences
User interface for igloo-house morph showing completed feature net (red) with morphing patches
Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang
3D Shape InterpolationPatch Parameterization
63
CS, NCTU, J. H.Chuang
3D Shape Interpolation Parameterization Overlaying
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang
3D Shape InterpolationPatch Parameterization
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang
3D Shape InterpolationPatch Parameterization
Animation (U), Chap 4, Interpolation-based Animation
Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang
3D Shape InterpolationExample
67
Interactive surface decomposition for polyhedral morphingArthur Gregory et al. The Visual Computer 15(9), 1999
CS, NCTU, J. H.Chuang
3D Shape Interpolation
Animation (U), Chap 4, Interpolation-based Animation
Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang
3D Shape Interpolation Automatic Consistent Dissection
69
CS, NCTU, J. H.Chuang
3D Shape InterpolationPatch Parameterization and Re-meshing
Animation (U), Chap 4, Interpolation-based Animation
Modify the sampling and connectivity of a geometry
Convert a irregular mesh to a (semi-)regular mesh
Parametrize Re-meshing
CS, NCTU, J. H.Chuang
3D Shape Interpolation Patch Parameterization and Re-meshing
Example
Animation (U), Chap 4, Interpolation-based Animation
4 basic points
Additional points
User-specified feature points
71
3D Shape Interpolation Patch Parameterization and Re-meshing
Example
Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang
Consistent common dissection of pig and triceratops models
Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang
3D Shape Interpolation Patch Parameterization and Re-meshing
Example
M0 : 164 faces
M2 : 2,624 faces
M4 : 41,984 faces
M0 : 164 faces
M2 : 2,624 faces
M4 : 41,984 faces
Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang
3D Shape Interpolation Patch Parameterization and Re-meshing
Example Another example w.r.t corresponding features
Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang
3D Shape InterpolationExample for More than 2 Meshes
75
Applied to more than two meshes
3D Shape InterpolationExample for More than 2 Meshes
Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang
Image Morphing
Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang
Morph from the source image to the destination image
Specifying corresponding elements in the two images Coordinate grid approach Feature-based approach
CS, NCTU, J. H.Chuang
Image MorphingCoordinate grid approach
User-defined curvilinear grid over each image Make sure corresponding elements in the images
are in the corresponding cells Locate the same number of grid intersection point
on both images Connecting curves are generated using intersection
points as control points for a spline curve, e.g., Catmull-Rom spline
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang
Image MorphingCoordinate grid approach
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang
Image MorphingCoordinate grid approach
Morphing from source to destination Generate an intermediate grid
Linearly – two adjacent key frames Higher-order interpolation – more than two adjacent
key frames Warp the source pixels and destination pixels to
the intermediate grids Perform a cross dissolve in pixel-by-pixel basis to
generate the final images
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang
Image MorphingCoordinate grid approach
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang
Image MorphingCoordinate grid approach
Two-pass warping from source to intermediate Source grid to an auxiliary grid (for x direction) auxiliary grid to the intermediate grid (for y direction)
Source grid to an auxiliary grid (for x direction) Based on scan line
For each pixel in the auxiliary grid Find the range of pixel coordinates in the source image Use fractional coverage to effect anti-aliasing
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang
Image MorphingCoordinate grid approach
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang
Image MorphingCoordinate grid approach
Once both images have been warped to the intermediate grid, cross-dissolve on a pixel-by-pixel basis is applied.
Animation (U), Chap 4, Interpolation-based Animation
concerns aestheticon of value thecontrollocally
appealing vosually moreoften is blendlinear -non
linear
becan where
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CS, NCTU, J. H.Chuang
Image MorphingCoordinate grid approach
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang
Image MorphingFeature-based approach
Establish correspondence using feature lines Feature lines are drawn to identify features in
correspondence Feature lines are interpolated to form an
intermediate feature line sets Based on interpolating endpoints or interpolating center
points and orientation
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang
Image MorphingFeature-based approach
On the intermediate image Establish a mapping for each pixel in the intermediate
image to each interpolated feature line Find a relative weight indicating the amount of
influence that feature line should have on the pixel On the source image
Use the mapping to locate source image pixel that corresponds to the intermediate image pixel
Use the relative weight to average the source image locations generated by multiple feature lines into a final source image location
Use the final location to determine the color of intermediate image pixelAnimation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang
Image MorphingFeature-based approach
Feature line coordinate for a feature line defined by P1 and P2 on the intermediate image Define a local coordinate system (U, V) For a pixel P, its coordinate (u, v) is
Animation (U), Chap 4, Interpolation-based Animation
12
121
12
121
)()(
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PP
PPPPu
PP
PPPPv
CS, NCTU, J. H.Chuang
Image MorphingFeature-based approach
Feature line Q1 and Q2 on source image that corresponds to feature line defined by P1 and P2 on the intermediate image Local coordinate system (S, T) How to find pixel Q that
corresponds to P?
Animation (U), Chap 4, Interpolation-based Animation
vTuSQQ 1
CS, NCTU, J. H.Chuang
Image MorphingFeature-based approach
How to find pixel Q that
corresponds to P? Coordinates of Q are floating numbers Pixels corresponding to P are in an area Requires some kind of filtering
Nearest neighbor Linear Interpolation Quadrilateral formed by mapping corners of P
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang
Image MorphingFeature-based approach
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang
Image MorphingFeature-based approach
Multiple feature lines In addition to mapping, each pixel P is associated
with a weight based on P’s position relative to the a feature line in the intermediate image
Animation (U), Chap 4, Interpolation-based Animation
mapping. theofcharacter
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CS, NCTU, J. H.Chuang
Image MorphingFeature-based approach
(a, b, p) parameters If a ~ 0, the mapping is a rigid transformation When a increases, makes the effect of lines over
the image smoother. Increasing p increases the effect of longer line. Increasing b makes the effect of a line fall off more
rapidly (a, b, p) parameters can be global or on a
feature-line-by-feature-line basis.
Animation (U), Chap 4, Interpolation-based Animation
CS, NCTU, J. H.Chuang
Image MorphingFeature-based approach
How to scale the displacement by using weights? For a given pixel in the intermediate image
The displacement indicated by each feature line pair is scaled by its weight
Final displacement is the weighted sum of all displacements for each feature line pair
This gives the displacement from the intermediate pixel to its corresponding position in the source image.
Animation (U), Chap 4, Interpolation-based Animation