Chap 4 Interpolation-Based Animation Animation (U), Chap 4, Interpolation-based Animation 1 CS,...

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

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

Key-Frame Systems

Specify interpolation of key values and tangents at segment boundaries

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

Key-Frame Systems What is a key?

For a bouncing ball 3D Positions Orientation? Squishedness?

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)

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

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

Key-Frame Systems Basic operation: Interpolating Curves

Point-to-point basis: straightforward

Point-to-point correspondence is not known Curve-to-curve correspondence is given

Curve-to-curve correspondence is given What happen at intermediate points along the

curve is left undefined

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

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)

Moving points

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 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

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

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

i: minimum of connecting edgesn: maximum range of effectk: user-selected scale factor

K=0: linear attenuationK<0: more elasticK>0: more rigid displacement

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)

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

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

A: Global coordinate: (25.6, 14.7), Local coordinate : (5.6, 2.7)

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

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

For a given object vertex, we record1. The closest line segment (L2)2. The distance (d) to L2

3. The ratio r

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

uUtTsSPP

s, t, uP

UTSPPTSs

TSUPPSUt

SUTPPUTs

s, t, uP

S, T, U

coordinate global its has )( coordinate local with vertex a So

)))/(()(

by determined )( coordinate local :A vertex

orthogonally necessaryinot - )(by defined

:at origin with system coordinate Local

To facilitate the modification of local coordinate system, a grid of control points is created

nkmjli

iPPijk

0,0,0for

As the control points moved, the point P(s,t,u) moves

ijkkkn

)1()1(

)1(),,(

Multiple FFD control grids can be joined with continuity constraints across the boundaries

Other FFD control grids

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

Deforming objectsComposite FFD – Sequential

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

Deforming objectsComposite FFD – Hierarchical

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

Deformation tool applied to an object

Deformation by moving the deformation tool relative to an object

Deformation by movingthe object through FFD space

In logical FFD space In distorted FFD space

Animating the FFD control points using, e.g., key-frame animation or by the result of physically based simulation

Deforming objectsProblems of Editing and FFD

Details are hard to be preserved

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

Interactive surface decomposition for polyhedral morphing Arthur Gregory et al. The Visual Computer 15(9), 1999

Surface-based Vertex-to-vertex correspondence Interpolation between corresponding vertices Limitations on topological consistence

volume-based Problems

Surface representation -> volumetric representation More computationally expensive

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

3D Shape InterpolationFor Meshes With Same Topology

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

3D Shape Interpolation Spherical Parameterization

Spherical parameterization in [Zwicker and Gotsman 2004]

3D Shape Interpolation Spherical Parameterization

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

Genus 0

Genus 1

3D Shape InterpolationUser Guided Common Dissection

Interactive surface decomposition for polyhedral Morphing, by Arthur Gregory et al. The Visual Computer 15(9), 1999

3D Shape InterpolationUser Guided Common Dissection

Input polyhedral with user-specified correspondences

User interface for igloo-house morph showing completed feature net (red) with morphing patches

3D Shape InterpolationPatch Parameterization

CS, NCTU, J. H.Chuang

3D Shape Interpolation Parameterization Overlaying

3D Shape InterpolationExample

Interactive surface decomposition for polyhedral morphingArthur Gregory et al. The Visual Computer 15(9), 1999

3D Shape Interpolation Automatic Consistent Dissection

3D Shape InterpolationPatch Parameterization and Re-meshing

Modify the sampling and connectivity of a geometry

Convert a irregular mesh to a (semi-)regular mesh

Parametrize Re-meshing

3D Shape Interpolation Patch Parameterization and Re-meshing

Example

4 basic points

Additional points

User-specified feature points

Example

Consistent common dissection of pig and triceratops models

Example

M0 : 164 faces

M2 : 2,624 faces

M4 : 41,984 faces

M0 : 164 faces

M2 : 2,624 faces

M4 : 41,984 faces

Example Another example w.r.t corresponding features

3D Shape InterpolationExample for More than 2 Meshes

Applied to more than two meshes

3D Shape InterpolationExample for More than 2 Meshes

Image Morphing

Morph from the source image to the destination image

Specifying corresponding elements in the two images Coordinate grid approach Feature-based approach

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

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

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

Once both images have been warped to the intermediate grid, cross-dissolve on a pixel-by-pixel basis is applied.

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jiCjiCjiC ds

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

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

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

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?

vTuSQQ 1

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

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

mapping. theofcharacter

overall thecontrol parameters supplied-uesr :)(

. andby formed line feature thefrom of distance :

PPPdist

(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.

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.

Chap 4 Interpolation-Based Animation Animation (U), Chap 4, Interpolation-based Animation 1 CS,...

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