Warping and Morphing of Graphical Objectsw3.impa.br/~morph/sig-course/jonas.pdf · 1998. 10....
Transcript of Warping and Morphing of Graphical Objectsw3.impa.br/~morph/sig-course/jonas.pdf · 1998. 10....
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