Warpin g and Morphin g ofGraphical Ob jects

Jonas Gomes

IMPA, Rio de Janeiro

Abstraction Pipeline

• define a graphical object

• define warpin g and morphin g

Graphical Ob jects

Drawin gs Volume data Ima ges

Graphical Ob jects

SurfaceSound

Graphical Ob ject

• Shape• Geometry• Topology

• Attributes• Properties• Color, temperature, texture, ...

Graphical Ob ject

• Same shape (rectan gle)

• Different attributes (texture)

Graphical Ob ject

• Physical attributes

Definition of aGraphical Ob ject

• Shape

• Attributes

• Dimension of the GO

Image

• Shape is a rectan gle

• Attribute is color

• Dimension = 2

Audio

• Shape is an interval

• Attribute is pressure

• Dimension = 1

Solid (volumetric ob ject)

• Shape is an spacialdomain

• Attributes: density, ...

• Dimension = n

Curves (Drawin gs)

• One-dimensional graphicalobjects of the plane

Surfaces

• Two-dimensional graphicalobjects of the space

Two-Dimensional Solids

• 2D graphical objects of the plane

• Binary ima ge• Shape is the focus

Animation

• Variation of a graphical objectalon g the time

time

Transformation of GraphicalObjects

• Transformin g shape

• Transformin g attributes

Transformin g Shape

• transformation of the ima ge shape

Transformin g Attributes

• Texture transformation

Transformin gShape and Attributes

Color andGeometry

Color,GeometryandTopolo gy

Classes of Transformation

• Distance between points

• Isometry

• Contraction

• Expansion

Classes of Transformation

isometry expansion contraction

• Change of frequencies

Mixed transformation

Our goal:Continuous Deformation

Continuous Deformation

• Twist : Rotation an gle increaseswith hei ght

Continuous twist

• Parameter space ( z axis )

Families of Transformations

Parameter Space

Graphical Object

v

p

T(p,v) =

From families to animation

f

f(t)t

Warpin g and Morphin g

• Warpin g• continuous family of transformations

of a graphical object

• Morphin g (metamorphosis)• warpin g between two graphical

objects

Warpin g and Morphin g

• Warpin g• Source object• No target object

• Morphin g• Source object• Target object

Warpin g

• Continuous shape transformation• Warp the object shape• Change geometry and topology

• Continuous attribute transformation• Change the attributes

Morphin g

• Continuous shape blendin g

• Continuous attribute blendin g

Shape Warp

Source

Shape Blendin g

Source Target

Linear Blendin g

• A and B objects in a vector space

c(t) = (1 - t) A + t B

c(0) = A, c (1) = B

• Functions (attributes)

• Subsets of space (shape)

Bilinear Blendin g

• Trilinear blendin g

Affine blendin g

• Linear in barycentric coordinates

Attribute Blendin g

• Linear Color Interpolation(cross dissolve)

Shape Warp + AttributeBlendin g

• Adaptive Color Interpolation

Some Guidelines for a goodmorphin g

• Feature preservation

• Smoothness preservation

• Avoid linearities• use adaptive methods

Ali gnin g features

+ +

= =

Comparison

• Feature ali gnment

• Adaptive color blendin g

Description of Graphical Ob jects

• Function description• shape• attributes

• Description methods• implicit• parametric• algorithmic (virtual machine)

Unif ying the problem

• Function description• graphical objects

■ shape■ attributes

• Transformations■ warping■ morphing

Function description

• Spatial (time) domain

Variation of physical ma gnitudes

Spatial (time) domain

• Audio: (time, amplitude)

Function description

• Frequency domain

Occurrence of frequencies on the function

frequency

amplitude

Frequenc y domainDescription

Function description

• time-frequency domain

• Frequencies on a period of time

• Uncertainty principle

time

frequency

Time-frequenc y domainDescription

Function description

• Time-scale domain

For each scale, describes frequencieson a certain period of time

Time-scale description

scale increases

Conversion betweendescriptions

• Function Transforms• Fourier transform• Window Fourier transform• Wavelets• etc.

• Invertibility

Some Guidelines for agood morphin g

• Avoid linearities (adaptiveness)

• Bilinear map of chessboard

• Avoid linearities

• Projective mappin g of chessboard

Some Guidelines for agood morphin g

Yet another example:

Better advice

• Keep linear morphin g as an option

• Effectiveness of linear morphin g• Depends on the objects• Depends on the description of the objects

■ Audio■ Animation

Linear warp of audio

• time domain

Linear warp of audio

• frequency domain

Linear warp of motion

• Animation: one-parameter family

Motion paths• Spatial variation of samples

• Motion warpin g

Motion Warpin g

• Andrew Witkin and Zoran PopovicMotion Warping.SIGGRAPH ‘95 Proceedin gs

Motion of articulated bod y

• Hierarchy of joints• j1, j2, ..., jn

• Joint motion• translation• rotation angles

Joint motion

• Joint jm

• Joint an gles

x

y

z

jm

Human bod y motion

• periodic motion of joints• Fourier description

Linear motion blendin g

Linear motion blendin g

• interpolation• 0 < s < 1

• extrapolation• s < 0• s > 1

References

• Unuma, M.; Anjyo, K. ; Takeuchi, R.:Fourier Principles for Emotion-based Human Fi gure Animation .SIGGRAPH ‘95 Proceedin gs.

Where to go?

• Graphical objects

• Transformations

Representation of Graphical Ob jectsand Transformations

• Representation of functions

FunctionDescription

FunctionRepresentation

Continuous Discrete

Represent shape

Represent attributes

• Quantization

• Computation

• Perception

Represent transformations

• Warpin g andMorphin g

• Motion paths

• Time discretization

Unif ying the problem

• Graphical objects

• Transformations

• Study of functions• function description• function representation

Representation and Reconstruction

Continuous

Discrete

Representation Reconstruction

What does Reconstruction bu y?

• User interaction• visualization• audio playing

• Operations• warping• morphing

• Specification

Representation andSpecification

UserSpecification

FunctionRepresentation

Continuousfunction

• Specification of graphical objects

• Specification of transformations

Function Representation

• function decomposition

• Dictionary

Representation and Reconstruction

• Reconstruction Interpolation

Point Samplin g

Point Samplin g

• dictionary functions: Dirac delta

• Dirac delta decomposition

Representation and Reconstruction

• Exact representation• recover the geometry• recover the attributes

• Non-exact representation• different flavors

■ approx. geometry + same topology■ approx. geometry + different topology

Shape Reconstruction

• geometry and topolo gy

Shape Reconstruction

• linear and cubic reconstruction

Polygonal B-Rep

• Point samplin g

• Piecewise linear reconstruction

• Curves and surfaces

Matrix Representation

• Uniform decomposition ( grid)

• images

• volume data

Rasterization

• Computation of the matrixrepresentation• grid definition• attributes computation• quantization

• Spatial resolution

• Attribute resolution

Matrix Representations

Problems of Function Representation

• Define the function space

• Define the dictionary

• Construct the dictionary

• Compute the coefficients

Problems of Function Reconstruction

• Define reconstruction basis

• Infinity elements on the basis

• Basis function extends to infinity

• Representation not exact

Point Samplin g

Point samplin g: Aliasin g

Aliasin g Artifacts

Aliasin g and Reconstruction

Avera ge Samplin g

• dictionary functions

• average decomposition

Avera ge samplin g

• Low-pass filterin g• eliminate high frequencies

• Point samplin g

Avera ge Methods

• Box avera ge

Avera ge Methods

• Trian gle avera ge

Higher Order Avera ges

• quadratic

• cubic

• Gaussian, ...

Other Representations

• Different descriptions

• Time-frequency

• Scale-frequency

• Wavelets

Transformin g Graphical Ob jects• Resamplin g

Object Resamplin g

Some Guidelines for agood Morphin g

• Topolo gy preservation

Some Guidelines for agood Morphin g

• Monotonicity

Bilinear warpin g

Projective warpin g

Some Guidelines for a good Mor phin g

• Use of transformation grou ps

f

f(t)t

• Slow-in and slow-out

Some Guidelines for a good Morphin g

Warpin g with leaka ge Warpin g the fore groundto avoid leaka ge

• Avoid leaka ge

Some Guidelines for a good Mor phin g