Level%SetMethods(&(Dynamic( ImplicitSurfaces(faculty.missouri.edu/.../lecture-notes/3-level-set.pdf · • S. Osher(and(R.(Fedkiw,(Level(SetMethods(and(Dynamic(• J.A.(Sethian,(Level(SetMethods(and(FastMarching(•

Level-‐Set Methods & Dynamic Implicit Surfaces

Overview

•  Implicit Surfaces •  Level Set Methods

Overview


Overview

•  Implicit Surfaces –  Implicit func=ons

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

–  Signed distance func=ons •  Distance func=ons •  Signed distance func=ons •  Geometry and calculus tool box

Introduc=on

•  Tracking boundaries is an important problem in many areas of computer science –  Image Processing – Machine Vision – Biometrics – Human Computer Interac=on – Entertainment

Introduc=on

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

– How do you track the boundary ahead in =me?

Explicit Representa=on

•  Marker/String Methods •  A standard approach to modeling a moving boundary (front/contour/interface) is to discre=ze it into a set of marker par=cles whose posi=ons are known at any =me.

•  To reconstruct the front the par=cles are linked by line segments in 2D and triangles in 3D.


•  Marker/String Method – The idea is to advance the par=cles in the direc=on of the arrows, recalculate the arrows, advance the par=cles, and repeat

– More par=cles should mean greater accuracy


•  Marker/String Method Issues – The markers may poten=ally cross over one another and require untangling

– The boundary may split or merge


•  Marker/String Method Issues – The front may expand or contract requiring the addi=on or removal of markers

Implicit Vs. Explicit

•  Tracking the interface through points (markers) on its surface is a Lagrangian formula=on.

•  Capturing an interface through the evolu=on of the implicit surface is a Eulerian formula=on.

Implicit Vs. Explicit

•  Implicit representa=ons overcome the previous problems at the cost of memory efficiency, some new limita=ons.

•  In the case of the water simula=on Foster and Fedkiw found a hybrid approach combining Eulerian and Lagranian formula=ons works the best.

!!!

Implicit Func=ons •  Points

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

-‐2 -‐1 0 1 2

Outside Ω+ Outside Ω+ Inside Ω-‐

Ω∂Interface ( )( ) ( ){ }1,1

,11,1,1

−=Ω∂

∞−∞−=Ω

−=Ω+

−

∪



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

-‐2 -‐1 0 1 2

Outside Ω+ Outside Ω+ Inside Ω-‐

Ω∂Interface

( )( )( )( ) 0:

0:0:12

=Ω∂

>Ω

<Ω

−=

+

−

xxxxx

ϕ

ϕ

ϕ

ϕImplicit Representa=on


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

–  In the previous case the interior is always nega=ve and the exterior is always posi=ve.

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

Implicit Func=ons

•  Curves –  In 2 spa=al dimensions a closed curve separates Ω into two sub-‐

regions.

Outside Ω+

φ > 0

Inside Ω-‐

φ < 0

Ω∂Interface ( ) 0122 =−+= yxx!ϕ

Inside Ω-‐

φ < 0

Implicit Func=ons

•  Surfaces –  In 3+ spa=al dimensions higher dimensional surfaces are used to

represent the zero level set of φ.

Outside Ω+

φ > 0

Ω∂Interface ( ) 01222 =−++= zyxx!ϕ

Discrete Representa=ons of Implicit Func=ons

•  Complicated 2D curves and 3D surfaces do not always have analy=cal descrip=ons.

•  The implicit func=on φ need to be stored with a discre=za=on, i .e. in the impl ic it representa=on, we will know the values of the implicit func=on φ at only a finite number of data points and need to use interpola=on to find the values of φ else where.

Discrete Representa=ons of Implicit Func=ons

•  The loca=on of the interface, the φ(x) = 0 zero iso-‐contour need to be interpolated from the known values of φ at the data points.

•  This can be done using contour plo`ng algorithms such as Marching Cubes algorithm.

•  The set of data points where the implicit func=on φ is defined is called a grid.

•  Uniform Cartesian grids are mostly used. •  Other grids include unstructured, adap=ve, curvilinear, etc.

Iso-‐contour extrac=on

φ(x,y)=0 φ(x,y)>0 φ(x,y)<0


φ(x,y)=0 φ(x,y)>0 φ(x,y)<0


φ(x,y)=0 φ(x,y)>0 φ(x,y)<0


φ(x,y)=0 φ(x,y)>0 φ(x,y)<0

Marching Squares

φ(x,y)=0 φ(x,y)>0 φ(x,y)<0

Four unique cases (a7er considering symmetry)

Marching Squares

Principle of Occam’s razor: If there are mul@ple possible explana@ons of a phenomenon that are consistent with the data, choose the simplest one.

Marching Squares

Linear interpola=on f(x,y)=0 f(x,y)>0 f(x,y)<0 fi, j+1 > 0

fi, j < 0

fi+1, j+1 > 0

fi+1, j > 0

Marching Squares

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

Marching Squares

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

Δx / h = (-‐a) / (b-‐a)

Δx = ah / (a-‐b)

Marching Squares

Marching Squares

Marching Squares-‐ambiguity

More informa@on are needed to resolve ambiguity


φ(x,y)=0 φ(x,y)>0 φ(x,y)<0

Marching Cubes Algorithm

Lorensen & Cline: Marching Cubes: A high resolu@on 3D surface construc@on algorithm. In SIGGRAPH 1987.

Implicit Func=ons: Discrete Representa=ons

•  Points – Another way to represent φ(x) is to discre=ze Ω into grid cells and put the value of φ(x) in each cell.

… 8 3 0 -1 0 3 8 …

0 1 2 3 -‐1 -‐2 -‐3

( ) 12 −= xxϕ


– Or define φ(x) as a height.

0 1 2 3 -‐1 -‐2 -‐3

( ) 12 −= xxϕ

0

Implicit Surface & Isosurface •  When c is 0, we say that f implicitly defines a locus, called an

implicit surface, .

•  The implicit surface is some=mes called the zero set (or zero surface) of f.

•  The isosurface (also called a level set or level surface) is , , where c is the iso-‐value of the surface. }0)(|{ 3 =∈ pp fR

})(|{ 3 cfR =∈ pp

Boolean Opera=ons of Implicit Func=ons

•  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 intersec=on 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 subtrac=on of the interior regions of φ1(x) by the interior regions of φ2(x).

Construc=ve Solid Geometry

Geometry Toolbox

•  The gradient of the implicit func=on •  The normal of the implicit func=on •  Mean curvature of the implicit func=on •  Numerical approxima=ons

Gradient/Normal

Gradient/Normal

Curvature

Curvature

Calculus Toolbox

•  Characteris=c func=on •  Heaviside func=on •  Delta func=on •  Volume integral •  Surface integral •  Numerical approxima=on

Signed Distance Func=ons

•  Signed distance func=ons are a subset of implicit func=on defined to be posi=ve on the exterior, nega=ve on the interior with

•  A distance func@on is defined as:

( ) ( )Ω∂∈

−=

I

I

xwherexxxd

!

!!! min

|1)(| =∇ x!φ


•  A signed distance func@on adds the appropriate signing of the interior vs. exterior.

⎪⎩

⎪⎨

⎧

Ω∈−

Ω∂∈=

Ω∈

=−

+

xxdxxd

xxdx

!!

!!!!

!

)(0)(

)()(φ

|1)(| =∇ x!φ


•  1D Example: –  Implicit func=on – Signed distance func=on



( ) 12 −= xxϕ( ) 1|| −= xxϕ

( ) 1, 22 −+= yxyxϕ( ) 1, 22 −+= yxyxϕ

( ) 1,, 222 −++= zyxzyxϕ

( ) 1,, 222 −++= zyxzyxϕ

Boolean Opera=ons of Signed Distance Func=ons

•  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 intersec=on 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 subtrac=on of the interior regions of φ1(x) by the interior regions of φ2(x).

Geometry and Calculus Toolboxes of the Signed Distance Func=ons

Since •  The Delta func=on is: •  The Surface integral is:

|1)(| =∇ x!φ

))(()( xx !!φδδ =

)())(()( xdxxf !!!φδ∫Ω

Geometry and Calculus Toolboxes of the Signed Distance Func=ons

Since •  The normal of the signed distance func=on is:

•  Mean curvature of the the signed distance func=on is:

Where is the Laplacian of

|1)(| =∇ x!φ

φ∇=N!

φΔ=k

zzyyxx φφφφ ++=ΔφΔ φ


•  In two spa=al dimensions the signed distance func=on for a circle with a radius r and a center (x0, y0) is given as:

( ) ( ) ( ) ryyxxx −−+−= 20

20

!φ

Overview


Overview

•  Level set methods – Mo=on in an externally generated velocity field

•  Convec=on •  Upwind differencing •  Hamilton-‐Jacobi ENO/WENO, TVD Runge-‐Kula

– Mo=on involves mean curvature •  Equa=ons of mo=on •  Numerical discre=za=on •  Convec=on-‐diffusion equa=ons

Overview

– Hamilton-‐Jacobi Equa=on •  Connec=on with Conserva=on Laws •  Numerical discre=za=on

– Mo=on in the Normal Direc=on •  The Basic Equa=on •  Numerical discre=za=on •  Adding a Curvature-‐Dependent Term •  Adding an External Velocity Field

Overview




3. Mo=on in an Externally Generated Velocity Field

1. Convec=on •  Assume the velocity of each point on the implicit surface is given as ,

•  i.e. is known for every point with •  Given this velocity field

)(xV !!

)(xV !!

0)( =x!φx!

)(xV !!

Mo=on in an Externally Generated Velocity Field

•  Level set methods add dynamics to implicit surfaces

•  Stanley Osher and James Sethian implemented the first approach to numerically solving a =me-‐dependent equa=on for a moving implicit surface


•  In the Eulerian approach the implicit func=on φ is used to both represent the interface and evolve the interface.

•  Given an external velocity field the level set equa=on is given as

0=∇⋅+ ϕϕ Vt

!

Temporal par=al deriva=ve Advec=on term


•  Using the forward difference approach for the par=al deriva=ve with respect to =me

01

=∇⋅+Δ

−+nn

nn

Vt

ϕϕφ !


•  At this point it looks like the next step would be to take the gradient of φ by using central differences.

•  But first we should expand the equa=on and look at it one dimension at a =me

01

=+++Δ

−+nz

nny

nnx

nnn

wvut

ϕϕϕϕφ


•  If we look at the u component of the vector field we might see something like to the right for direc=on.

•  If ui,j is poin=ng to the right we need to look to the cell to the right of φi,j the same applies to values to the lem.

● ● ● ● ●

● ● ● ● ●

● ● ● ● ●

● ● ● ● ●

● ● ● ● ●

xxii

Δ

−=

∂

∂ + ϕϕϕ 1

xxii

Δ

−=

∂

∂ −1ϕϕϕ

Forward Difference

Backward Difference


•  For a 1-‐d case the level set equa=on would be of the form

01

=+Δ

−+nx

ni

nn

ut

ϕϕφ


•  The update equa=on would be

•  where

nx

nnn tu φϕφ Δ+=+1

⎪⎪⎩

⎪⎪⎨

⎧

<Δ

−

>Δ

−

=−

+

0

0

1

1

iii

ii

x

ux

ux

i

ϕϕ

ϕϕ

φ

Overview




4. Mo=on involves mean curvature

•  Equa=ons of mo=on •  Numerical discre=za=on •  Convec=on-‐diffusion equa=ons

Overview



Hamilton-‐Jacobi Equa=on

•  Connec=on with Conserva=on Laws •  Numerical discre=za=on

Overview



Mo=on in the Normal Direc=on

•  The Basic Equa=on •  Numerical discre=za=on •  Adding a Curvature-‐Dependent Term •  Adding an External Velocity Field

References •  S. Osher and R. Fedkiw, Level Set Methods and Dynamic

Implicit 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 Par=al Differen=al Equa=ons and Image

Analysis, Guillermo Sapiro, Cambridge University Press, 2001.

•  J. A. Sethian, “Level Set Methods: An Act of Violence”, American Scien=st, May-‐June, 1997.

•  N. Foster and R. Fedkiw, "Prac=cal Anima=on of Liquids", SIGGRAPH 2001, pp.15-‐22, 2001.

Documents

Level%SetMethods(&(Dynamic( ImplicitSurfaces(faculty.missouri.edu/.../lecture-notes/3-level-set.pdf · • S. Osher(and(R.(Fedkiw,(Level(SetMethods(and(Dynamic(• J.A.(Sethian,(Level(SetMethods(and(FastMarching(•