1 Multiple problems Applying Marching Cubes (thresholding) - - PDF document

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Level Sets

Roger Crawfis

Slides collected from: Fan Ding, Charles Dyer, Donald Tanguay and Roger Crawfis

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Motivation:

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Easy Case – Use Marching Cubes

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Input Data Noisy

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Non-uniform Exposure

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Intensity Varies

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Multiple problems

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Applying Marching Cubes (thresholding)

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Applying A Threshold

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Applying A Threshold

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Four (contour) Levels

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What To Do?

User Intervention!!!

We see it!! It’s right there!!! Well, the edges get pretty fuzzy .

Two step process:

• 1. Draw an initial curve (or surface) within the

desired region.

• 2. Expand that interface outward towards the

edge of our desired region.

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Interfaces

An interface (or front) is a boundary between two regions: “inside” and “outside.” In 2-D, an interface is a simple closed curve:

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Propagating Interfaces

How does an interface evolve over time? At a specific moment, the speed function F (L, G, I) describes the motion of the interface in the normal direction.

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Propagating Interfaces

• Local – depend on local geometric information

(e.g., curvature and normal direction)

• Global – depend on the shape and position of the front

(e.g., integrals along the front, heat diffusion)

• Independent – do not depend on the shape of the front

(e.g., an underlying fluid velocity that passively transports the front) Speed F(L,G,I) is a function of 3 types of properties:

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Motion Under Curvature

Example: Motion by curvature. Each piece moves perpendicular to the curve with speed proportional to the local curvature.

large positive motion small negative motion

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Motion Under Curvature

Curvature κ is the inverse of the radius r of the osculating circle.

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Motion Under Curvature

http://math.berkeley.edu/~sethian/Applets/java_curve_flow.html

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Functional Representation

Eulerian framework: define fixed coordinate system on the world. For every world point x, there is (at most) one value y = ft(x). Falling snow example:

) (

0 x

f y

t=

= ) (

1 x

f y

t=

=

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Functional Representation

However, many simple shapes are multivalued; they are not functions regardless of the orientation of the coordinate system.

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Parametric Representation

Spatially parameterize the curve x by s so that at time t the curve is xt(s), where 0 ≤ s ≤ S and the curve is closed: xt(0) = xt(S). Points on initial curve. Gradient (wrt time) is the speed in normal direction. Normal is perpendicular to curve, as is curvature.

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Parametric Representation

For motion under curvature, speed F depends only on local curvature κ – the equation of motion is thus: where curvature is and the normal is

2 / 3 2 2

) ( ) (

s s s ss s ss t

y x y x x y s + − = κ

[ ] [ ]

2 / 1 2 2 2 / 1 2 2

) ( ) ( ) (

s s T s s s s T s s t

y x x y y x y x x x s n + − =         + =         ∇ ∇ =

⊥ ⊥

• ),

( )) ( ( ) ( s n s F s x t

t t t

• ⋅

= ∂ ∂ κ

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Particle Methods

In order to compute, discretize the parameterization into moving particles which reconstruct the front. Known under a variety of names: marker particle techniques, string methods, nodal methods. = # mesh particles

s ∆ t ∆

s S ∆ = time step = parameterization step ) , (

n i n i y

x = location of point i∆s at time n∆t ∆s

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Problem Statement

Generally, given:

An initial front Equations that govern its evolution

How do we simulate the front’s evolution? Called an ‘initial value problem’

Given the initial position Solve for a position at a future time

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More formally:

Given some initial front Г:

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More formally: (2)

And a function F that specifies the velocity

• f the front in the normal direction:

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More formally: (3)

Solve for Г at some future time Level set methods are used to track an interface Water/air interface, for example

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What’s Wrong with the Obvious Solution?

Why is a level set method necessary? There seems to be a more intuitive way to solve this problem

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Marker/String Methods

Why not just connect some control points (in 3D, triangulate):

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Marker/String Methods (2)

And run the simulation on the points?

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Ocean Waves

Think of an air/water interface with two waves racing towards each other:

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Ocean Waves (2)

What happens to the control points when the waves collide?

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Shocks

Event known as a ‘shock’ Below formation called a ‘swallowtail’

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Shocks (2)

How to fix up the control points? Fixing swallowtails known as ‘de-looping’

Very difficult Some methods exist in 2D No robust 3D methods so far

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Changing Topology

Example: two fires merge into a single fire.

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Changing Topology

Buoys! In particle methods:

• Difficult (and expensive) to detect and change the particle chains
• Much more difficult as dimensionality increases

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Difficulties With Particle Methods

Instability Local singularities Management of particles: remove, redistribute, connect

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Level Set Formulation

Recast problem with one additional dimension – the distance from the interface. And then use Marching Cubes to extract the surface. ) , ( y x z

t=

= φ

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Level Set Methods

Contour evolution method due to J. Sethian and

• S. Osher, 1988

www.math.berkeley.edu/~sethian/level_set.html Difficulties with snake-type methods

Hard to keep track of contour if it self-intersects during its evolution Hard to deal with changes in topology

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The level set approach:

Define problem in 1 higher dimension Define level set function z = φ(x,y,t=0) where the (x,y) plane contains the contour, and z = signed Euclidean distance transform value (negative means inside closed contour, positive means outside contour)

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How to Move the Contour?

Move the level set function, φ(x,y,t), so that it rises, falls, expands, etc. Contour = cross section at z = 0

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Level Set Surface

The zero level set (in blue) at one point in time is a slice of the level set surface (in red)

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Level Set Surface

Later in time the level set surface (red) has moved and the new zero level set (blue) defines the new contour

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Level Set Surface

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Level Set Formulation

The interface always lies at the zeroth level set of the function Φ, i.e., the interface is defined by the implicit equation Φt (x, y) = 0.

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How to Move the Level Set Surface?

• 1. Define a velocity field, F, that specifies how contour

points move in time

Based on application-specific physics such as time, position, normal, curvature, image gradient magnitude

• 2. Build an initial value for the level set function,

φ(x,y,t=0), based on the initial contour position

• 3. Adjust φ over time; current contour defined by φ(x(t),

y(t), t) = 0 y Φ x Φ F t Φ Φ F t Φ

2 1 2 2

=                 ∂ ∂ +       ∂ ∂ + ∂ ∂ = ∇ ⋅ + ∂ ∂

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Speed Function

( )

εK 1 (k) F F F(K)

1

− = + =

( ) ( )

εK 1 x,y k F(K)

I

− ∗ =

( )

( )

x,y I G

• I

σ I

σ

e k x,y I G 1 1 k

∗ ∇

= ∗ ∇ + =

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Example: Shape Simplification

F = 1 – 0.1κ where κ is the curvature at each contour point

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Example: Segmentation

Digital Subtraction Angiogram F based on image gradient and contour curvature

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Example (cont.)

Initial contour specified manually

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Result – segmentation using Fast marching

No level set tuning

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Results – vein segmentation

No level set tuning With level set tuning

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Results – vein segmentation continued

Original Our result Sethian’s result (Fast marching + (Level set only) Level set tuning) 4/24/2003

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Result – segmentation using Fast marching

No level set tuning

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Results – brain segmentation continued

No level set tuning With level set tuning

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Results – brain image segmentation

# of iterations = 9000 # of iterations = 12000 Fast marching only, no level set tuning