Deformable ModelsDeformable Models

Segmentation methods until now (no Segmentation methods until now (no knowledge of shape:knowledge of shape:

ThresholdingThresholding

Edge basedEdge based

Region based Region based

Deformable modelsDeformable modelsKnowledge of the shape of the objectKnowledge of the shape of the object

Deformable ModelsDeformable Models

Initial shape (curve or surface) Initial shape (curve or surface)

Move the shape according:Move the shape according:Internal forces (curve/surface properties)Internal forces (curve/surface properties)E.g.: Curvature to keep the object smoothE.g.: Curvature to keep the object smooth

External forces (image properties)External forces (image properties)E.g.: To track the object to the boundaryE.g.: To track the object to the boundary

2D Snakes 2D Snakes Kass, Witkin and Terzopoulos 1987Kass, Witkin and Terzopoulos 1987

ExampleExample

Animation with a 2D countour Animation with a 2D countour adapting to the edgeadapting to the edge

Deformable ModelsDeformable Models

Various names for the same:Various names for the same:2D snakes, active contours, and 2D snakes, active contours, and deformable contours ...deformable contours ...

3D ballons, active surfaces, and 3D ballons, active surfaces, and deformable surfaces ...deformable surfaces ...

Two main groupsTwo main groupsParametric deformable models (based Parametric deformable models (based on parametric form of models)on parametric form of models)

Geometric deformable models (curve Geometric deformable models (curve evolution or level sets)evolution or level sets)

Parametric ContourParametric Contour

CurveCurveExample:Example:

Internal Energy Functional (internal forces)Internal Energy Functional (internal forces)

Potential Energy Functional (external Potential Energy Functional (external forces) forces)

Moves to minimize the energy functional: Moves to minimize the energy functional:

( ) ( ( ), ( )), [0,1]s X s Y s s X

1

2

( ) ( 2 ,sin( 2 ))

( ) (cos( 2 ),sin( 2 ))

s s s

s s s

X

X

0s

0.25s

0.5s

0.75s

1s

( ) ( ) ( ) ...elasticity curvatureS S S X X X

( ) ( ) ...bordersP P X X

( ) ( ) ( )E S P X X X

Internal Energy FunctionalInternal Energy Functional

Elasticity, avoids stretchingElasticity, avoids stretching

CurvatureCurvature

are often defined as constants are often defined as constants

21

0

( ) ( )elasticityS s dss

X

X

F F

21 2

20

( ) ( )curvatureS s dss

X

XF

( ), ( )s s

( ) weighting param eters

( ) weighting param eters

Potential Energy FunctionalPotential Energy Functional

Energy based on features of the Energy based on features of the imageimage

Potential field based on borders. We Potential field based on borders. We want small values close to borders, want small values close to borders, e.g.:e.g.:

1

0

( ) ( ( ))P p s dsX X

2( ) ( ) borderp x f x

( )f x

positive weighting parameter

y

fx

f

G

G

y

xf

Potential Field ExamplePotential Field Example

* ( )G f x

( )f x

Image smoothed by convolving with a Gaussian

222 2/)(

2

1),(

yxeyxG

Potential fieldPotential field

( ) ( * ( ))p x G f x ( ) * ( )p x G f x

Dynamic Force FormulationDynamic Force Formulation

Formulation as a dynamic problemFormulation as a dynamic problem

Allows definition of extra external Allows definition of extra external forcesforces

Based on Newton’s second law Based on Newton’s second law : :

( , )s tX2

2( ) ( ) ( )damp int extt

X

F X F X F X

0the mass unit, often then equal to previous sec tion

damp t

X

F damping (viscosity) coefficient 2

2inertial r e fo c

t

X

External ForcesExternal Forces

Combination of different forcesCombination of different forces

Pressure Force (ballon)Pressure Force (ballon)1 2( ) ( ) ( ) ... ( )ext N F X F X F X F X

( ) ( ) ( )where

is the inward

Normal t op pF X N X N X

X0p

0p

Problems with CavitiesProblems with Cavities

Gradient Vector FlowGradient Vector FlowDiffuse the vector field according to Diffuse the vector field according to strength of the edgesstrength of the edges

Dynamic Distance ForceDynamic Distance ForceUse of signed distancesUse of signed distances

Interactive VariationsInteractive VariationsUser interaction modeledUser interaction modeledas force termsas force terms

ImplementationImplementation

We work with in a discrete form:We work with in a discrete form:

The derivatives need to be The derivatives need to be approximated by, for example, finite approximated by, for example, finite differencesdifferences

( ( ), ( )), {1,... }X i Y i i n X

X

Problems of Active ContoursProblems of Active Contours

Initial contour needs to be close to final Initial contour needs to be close to final resultresult

Reparameterization is needed (3D is difficult)Reparameterization is needed (3D is difficult)

Topological adaptation (3D is Topological adaptation (3D is complicated)complicated)

Forces definition, parameter settingForces definition, parameter setting