Implicit GeometryImplicit Geometry

OverviewOverview

• Implicit Geometryp c t Geo et y– Implicit functions

• Points• Curves• Surfaces• Geometry toolboxy• Calculus toolbox

– Signed distance functionsDi f i• Distance functions

• Signed distance functions• Geometry and calculus tool box

IntroductionIntroduction

• Tracking boundaries is an important problemTracking boundaries is an important problem– Image Processing– Machine Vision– Biometrics– Human Computer Interactionp– Entertainment– Computational Physics– Computational Fluid Dynamics 

IntroductionIntroduction

• Image ProcessingImage Processing– Video, Image and Volume Segmentation

IntroductionIntroduction

• Machine VisionMachine Vision

• Biometrics

IntroductionIntroduction

• Human Computer InteractionHuman Computer Interaction

• Entertainment

IntroductionIntroduction

• Physical Simulations like WaterPhysical Simulations like Water

IntroductionIntroduction

• The ProblemThe Problem– How is it possible to efficiently and accuratelyrepresent the boundary?represent the boundary?

– How do you track the boundary ahead in time?– How do you track the boundary ahead in time?

Explicit RepresentationExplicit Representation

• Marker/String MethodsMarker/String Methods

• A standard approach to modeling a moving boundary(front/contour/interface) is to discretize it into a set( / / )of marker particles whose positions are known at anytime.

• To reconstruct the front the particles are linked byline segments in 2D and triangles in 3D.

Explicit RepresentationExplicit Representation

• Marker/String MethodMarker/String Method– The idea is to advance the particles in the direction of the arrows recalculate the arrowsdirection of the arrows, recalculate the arrows, advance the particles, and repeat

– More particles should mean greater accuracy– More particles should mean greater accuracy

Explicit RepresentationExplicit Representation

• Marker/String MethodMarker/String Method Issues– The markers may potentially– The markers may potentially cross over one another and require untanglingq g g

– The boundary may split orThe boundary may split or merge

Explicit RepresentationExplicit Representation

• Marker/String MethodMarker/String Method Issues– The front may expand or– The front may expand or contract requiring the addition or removal of markers

Implicit Vs. ExplicitImplicit Vs. Explicit

• Tracking the interface through pointsTracking the interface through points(markers) on its surface is a Lagrangianformulationformulation.

• Capturing an interface through the evolutionof the implicit surface is a Eulerianof the implicit surface is a Eulerianformulation.

Explicit FunctionsExplicit Functions• Points

– Explicitly divide a 1 d region Ω into two sub regions with– Explicitly divide a 1‐d region Ω into two sub‐regions with an interface at x = ‐1 and x = 1 

Outside Ω+ Outside Ω+Inside Ω‐

‐2 ‐1 0 1 2

Interface

11

1,1

Explicit Representation

1,1

,11,

Implicit FunctionsImplicit Functions• Points

– Implicitly divide a 1 d region Ω into two sub regions with– Implicitly divide a 1‐d region Ω into two sub‐regions with an interface at x = ‐1 and x = 1 

Outside Ω+ Outside Ω+Inside Ω‐

‐2 ‐1 0 1 2Implicit Representation

Interface

0:

12

x

xx

Implicit Representation

Interface 0:

0:

xx

Implicit FunctionsImplicit Functions• Points

– An implicit representation provides a simple,numerical method to determine the interior,exterior and interface of a region Ωexterior and interface of a region Ω.

– In the previous case the interior is always negativeand the exterior is always positiveand the exterior is always positive.

– The set of points where φ(x) = 0 is known as thezero level set or zero isocontour and will alwayszero level set or zero isocontour and will alwaysgive the interface of the region.

Implicit FunctionsImplicit Functions

• in Rn subdomains are n‐dimensional whilein R , subdomains are n dimensional, while the interface has dimension n − 1. 

• We say that the interface has codimension• We say that the interface has codimensionone.

Implicit FunctionsImplicit Functions

• Curves– In 2 spatial dimensions a closed curve separates Ω into two sub‐

regions.  g

Outside Ω+

0φ > 0

Inside Ω‐

φ < 0φ < 0

Interface

0122 yxx

Implicit FunctionsImplicit Functions

Implicit FunctionsImplicit Functions

• Surfaces– In 3+ spatial dimensions higher dimensional surfaces are used to 

represent the zero level set of φ.p φ

Outside Ω+

0

Inside Ω‐

φ < 0

φ > 0

φ < 0

Interface

01222 zyxx

Geometric RepresentationGeometric Representation

• For complicated surfaces in 3D the explicit representation canp p pbe quite difficult to discretize. One needs to choose a numberof points on the two‐dimensional surface and record theirconnectivityconnectivity.

• Connectivity can change for dynamic/deformable surfaces,y g y / ,i.e., surfaces that are moving around.

• One of the nicest properties of implicit surfaces is thatconnectivity does not need to be determined.

Geometric RepresentationsGeometric Representations

• Continuous representationContinuous representation– Explicit representation

Implicit representation– Implicit representation

• Discrete representation– Triangle meshes

– Points/Particles

Triangle MeshTriangle Mesh

Explicit vs Implicit RepresentationExplicit vs Implicit Representation

Explicit:Explicit:• Pros: easier display• Cons: maintain connectivity, difficult to handleCons: maintain connectivity, difficult to handletopology change.

Implicit:• Pros: No connectivity, easier to handle topology

hchange.• Cons: need extra step to display.

Discrete Representations of Implicit FunctionsDiscrete Representations of Implicit Functions

• Complicated 2D curves and 3D surfaces do notComplicated 2D curves and 3D surfaces do notalways have analytical descriptions.

• The implicit function φ need to be stored witha discretization, i.e. in the implicita discretization, i.e. in the implicitrepresentation, we will know the values of theimplicit function φ at only a finite number ofdata points and need to use interpolation tofind the values of φ else where.

Discrete Representations of Implicit FunctionsDiscrete Representations of Implicit Functions

• The location of the interface, the φ(x) = 0 zero iso‐, φ( )contour need to be interpolated from the knownvalues of φ at the data points.h b d l l h• This can be done using contour plotting algorithmssuch as Marching Cubes algorithm.

• The set of data points where the implicit function φ• The set of data points where the implicit function φis defined is called a grid.

• Uniform Cartesian grids are mostly used.g y• Other grids include unstructured, adaptive,curvilinear, etc.

Iso‐contour extractionIso contour extraction

φ(x,y)=0φ(x,y) 0

φ(x,y)>0

φ(x y)<0φ(x,y)<0

Iso‐contour extractionIso contour extraction

φ(x,y)=0φ(x,y) 0

φ(x,y)>0

φ(x y)<0φ(x,y)<0

Iso‐contour extractionIso contour extraction

φ(x,y)=0φ(x,y) 0

φ(x,y)>0

φ(x y)<0φ(x,y)<0

Iso‐contour extractionIso contour extraction

φ(x,y)=0φ(x,y) 0

φ(x,y)>0

φ(x y)<0φ(x,y)<0

Marching SquaresMarching Squares

φ(x,y)=0φ(x,y) 0

φ(x,y)>0

φ(x y)<0φ(x,y)<0

Four unique cases (after considering symmetry)

Marching SquaresMarching Squares

Principle of Occam’s razor:Principle of Occam s razor:

If there are multiple possible explanations of aphenomenon that are consistent with the data,p ,choose the simplest one.

Marching SquaresMarching Squares

Linear interpolationLinear interpolation

f(x,y)=0

f(x y)>0f(x,y)>0

f(x,y)<0 fi, j+1 > 0 fi+1, j+1 > 0

fi, j < 0 fi+1, j > 0

Marching SquaresMarching Squares

fi, j = a < 0 fi+1, j = b > 0fi+x, j=0

Marching SquaresMarching Squares

fi, j = a < 0 fi+1, j = b > 0fi+x, j=0

x / h = (‐a) / (b‐a)

h / ( b)x = ah / (a‐b)

Marching SquaresMarching Squares

Marching SquaresMarching Squares

Marching Squares‐ambiguityMarching Squares ambiguity

More information are needed to resolve ambiguity

Iso‐contour extractionIso contour extraction

φ(x,y)=0φ(x,y) 0

φ(x,y)>0

φ(x y)<0φ(x,y)<0

Marching Cubes AlgorithmMarching Cubes Algorithm

Lorensen & Cline: Marching Cubes: A high resolution 3D surface construction algorithm. In  SIGGRAPH 1987. 

Geometry ToolboxGeometry Toolbox

• Inside/outsideInside/outside

• Boolean operation/CSG

h di f h i li i f i• The gradient of the implicit function

• The normal of the implicit function

• Mean curvature of the implicit function

• Numerical approximationsNumerical approximations

Inside/outside functionInside/outside function

Determine whether a point is inside/outside theDetermine whether a point is inside/outside the interface

• Easier for Implicit function just look at the• Easier for Implicit function, just look at the sign of 

0I id:

x

0::Outside0::Inside

x

x

• Difficult for explicit representation:

0::Boundary x

Difficult for explicit representation:

Inside/outside functionInside/outside function

Difficult for explicit representation:cu t o e p c t ep ese tat o :• A standard procedure is to cast a ray from thepoint in question to some far‐off place that isp q pknown to be outside the interface.

• Then if the ray intersects the interface an evennumber of times, the point is outside theinterface.Oth i th i t t th i t f dd• Otherwise, the ray intersects the interface an oddnumber of times, and the point is inside theinterfaceinterface.

Boolean Operations of Implicit Functionsp p

• φ(x) = min (φ (x) φ (x)) is the union of the interior regions of φ (x)• φ(x) = min (φ1(x), φ2(x)) is the union of the interior regions of φ1(x) and φ2(x). 

• φ(x) = max (φ1(x), φ2(x)) is the intersection of the interior regions of φ1(x) and φ2(x). 

• φ(x) = ‐φ1(x) is the complement of φ1(x) . 

• φ(x) = max (φ1(x), ‐φ2(x)) represents the subtraction of the interior regions of φ1(x) by the interior regions of φ2(x). 

Constructive Solid Geometry (CSG)y ( )

Gradient/Normal of Implicit FunctionGradient/Normal of Implicit Function

Mean Curvature of the InterfaceMean Curvature of the Interface

Calculus ToolboxCalculus Toolbox

• Characteristic functionCharacteristic function

• Heaviside function

l f i• Delta function

• Volume integral

• Surface integral

• Numerical approximationNumerical approximation

Signed Distance FunctionsSigned Distance Functions

• Signed distance functions are a subset of implicitSigned distance functions are a subset of implicit function defined to be positive on the exterior, negative on the interior with  |1)(| x

Distance FunctionsDistance Functions

• A distance function is defined as:A distance function is defined as:

I

xwherexxxd

min

• so:

Ixwhere

• so:

xxwherexxxd cc toboundary on thepoint closest theis |,|

xwherexd ,0

Distance functionDistance function

Signed Distance FunctionsSigned Distance Functions

• A signed distance function adds the appropriateA signed distance function adds the appropriate signing of the interior vs. exterior.

xxd )(

xxd

xxdx

)(0)(

)()(

• Signed distance function share all the properties of implicit function with new properties such as:

xxd )(

p p p

• Given a point , the closest point on the interface|1)(| x

x x Given a point    , the closest point      on the interface is

x cx xxc

Nx)(

Signed Distance FunctionsSigned Distance Functions

• 1D Example:1D Example:– Implicit function– Signed distance function 

12 xx 1|| xxg

• 2D Example:– Implicit function

||

1, 22 yxyxImplicit function– Signed distance function 

• 3D Example:

, yy 1, 22 yxyx

3D Example:– Implicit function– Signed distance function

1,, 222 zyxzyx 1222 zyxzyxSigned distance function  1,, zyxzyx

Signed Distance FunctionsSigned Distance Functions

Boolean Operations of Signed Distance Functionsp g

• φ(x) = min (φ (x) φ (x)) is the union of the interior regions of φ (x)• φ(x) = min (φ1(x), φ2(x)) is the union of the interior regions of φ1(x) and φ2(x). 

• φ(x) = max (φ1(x), φ2(x)) is the intersection of the interior regions of φ1(x) and φ2(x). 

• φ(x) = ‐φ1(x) is the complement of φ1(x) . 

• φ(x) = max (φ1(x), ‐φ2(x)) represents the subtraction of the interior regions of φ1(x) by the interior regions of φ2(x). 

Geometry and Calculus Toolboxes of h dthe Signed Distance Functions

Since |1)(| xSince 

• The Delta function is:|1)(| x

))(()( • The Surface integral is:

))(()( xx

)())(()( xdxxf

Geometry and Calculus Toolboxes of h dthe Signed Distance Functions

Since |1)(| xSince 

• The normal of the signed distance function is:|1)(| x

N

• Mean curvature of the the signed distance N

function is:k

Where         is the Laplacian of 

zzyyxx

Signed Distance FunctionsSigned Distance Functions

• In two spatial dimensions the signed distance gfunction for a circle with a radius r and a center (x0, ( 0,y0) is given as:

ryyxxx 20

20

OverviewOverview

• Implicit GeometryImplicit Geometry

• Level Set Methods

OverviewOverview

• Level set methodsLevel set methods– Motion in an externally generated velocity field

• ConvectionConvection

• Upwind differencing

• Hamilton‐Jacobi ENO/WENO, TVD Runge‐Kutta/ , g

– Motion involves mean curvature• Equations of motion

• Numerical discretization

• Convection‐diffusion equations

OverviewOverview

– Hamilton‐Jacobi EquationHamilton Jacobi Equation• Connection with Conservation Laws

• Numerical discretization

– Motion in the Normal Direction• The Basic Equation

• Numerical discretization

• Adding a Curvature‐Dependent Term

• Adding an External Velocity Field

ReferencesReferences

• S. Osher and R. Fedkiw, Level Set Methods and Dynamic I li it S f S i 2003Implicit Surfaces, Springer, 2003.

• J.A. Sethian, Level Set Methods and Fast Marching Methods, Cambridge, 1996.

• Geometric Level Set Methods in Imaging, Vision, and Graphics, Stanley Osher, Nikos Paragios, Springer, 2003.

• Geometric Partial Differential Equations and ImageGeometric Partial Differential Equations and Image Analysis, Guillermo Sapiro, Cambridge University Press, 2001.J A S thi “L l S t M th d A A t f Vi l ”• J. A. Sethian, “Level Set Methods: An Act of Violence”, American Scientist, May‐June, 1997.

• N. Foster and R. Fedkiw, "Practical Animation of Liquids", SIGGRAPH 2001 pp 15 22 2001SIGGRAPH 2001, pp.15‐22, 2001.