level set methods in medical image analysis · ized models for applications in visualization and...

Level Set Methods in Medical Image Analysis

� Isosurfaces and Level Sets

� Representing Surfaces with Volumes

� Formulation of Interface Propagation

◦ Boundary Value Problem

◦ Initial Value Problem

� Numerical Implementation

� Medical Image Segmentation Using Level Sets

ITCS 6010:Biomedical Imaging and Visualization 1 Evolving Interfaces:Level Set Methods

Isosurfaces and Level Sets

� Isosurfaces can be used in modeling as an alternative to parameter-ized models for applications in visualization and graphics

� Level set methods are a class of methods that rely on PDEs tomodel deforming isosurfaces, with applications to image process-ing, vision, graphics.

� Level sets (moving interfaces) present an elegant means to manip-ulate the shape of isosurfaces in prescribed ways.

ITCS 6010:Biomedical Imaging and Visualization 2 Evolving Interfaces:Level Set Methods

Representing Surfaces with Volumes

� Level sets are based on implicit representation of a surface:

φ : Ux,y,z 7→ R

with U ∈ R3, and the surface S

S = {~x|φ(~x) = k}

where φ is the embedding function, S is an isosurface.

� Normal to isosurface, ~n is given by

~n(~x) =∇φ(~x)

|∇φ(~x|

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Surface Deformation

� In geometric modeling, a surface ~S is normally represented as a 2parameter object in 3D space:

~S : Vr × Vs 7→ R3x,y,z

where V × V ∈ R2.

� Deformable surfaces vary over time, i.e. S(r, s, t), are second ordercontinuous, oriented ( ~N(r, s, t)),

� Local deformations are defined by an evolution equation, relating tolocal and global shape properties, and other force functions:

∂~S

∂t= ~G(~S, ~Sr, ~Ss, ~Srr, ~Srs, ~Sss, ...)

� G can be a variety of functions, depending on the application.

ITCS 6010:Biomedical Imaging and Visualization 4 Evolving Interfaces:Level Set Methods

Formulation of Interface Propagation

� Consider a boundary, curve (2D) or surface(3D), separating two re-gions, evolving along its normal direction

� The front evolution is according to a speed function, F

F = F (L, G, I)

where

◦ L represents local properties, such as curvature, normal direc-tion

◦ G represents global properties, such as overall shape or posi-tion of moving front

◦ I represents independent properties

� In evolving interface problems, the challenge lies in designing thespeed function, F .

ITCS 6010:Biomedical Imaging and Visualization 5 Evolving Interfaces:Level Set Methods

Boundary Value Problem

� Assume F > 0, i.e. the front always moves outward.

� Compute arrival function T (x, y) of the front, given in 1D bydistance = rate ∗ time,

FdT

dx= 1

� Level set function φ is static, containing a family of level sets, foreach time t. In general,

φ(~x(t)) = k(t) ⇒ ∇φ(~x).~v =dk(t)

dtITCS 6010:Biomedical Imaging and Visualization 6 Evolving Interfaces:Level Set Methods

Boundary Value Problem

� Trade the moving boundary problem to something completely static(overlay a grid on the domain).

� Consider an evolving circle: results in a conical shaped level setfunction

� Motivation:

◦ Topological changes are handled by φ (or T (x, y), front obtainedas level set of φ

◦ Efficient techniques to compute φ

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Initial Value Problem

� Assume the front moves with speed F that is neither strictly positiveor negative

� Front can move forward or backward, can pass over a point multipletimes

� T (x, y) is no longer a single-valued function

� Handled by using a one-parameter family of embeddings, i.e., φ(x, t)changes over time, ~x remains on the zero (or some k) level set of φas it moves,

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Initial Value Problem

� The level set value of a particle on the front with path x(t) is alwayszero,

φ(x(t), t) = 0

By the chain rule,

φt +∇φ(x(t), t).~v = 0

and is the level-set equation of Osher-Sethian

� Initial value problem is computationally more expensive, however itis considerably more flexible

� Advantages:

◦ Topologically flexible, can represent complex shapes

◦ Shape complexity only restricted by the sampling resolution

◦ Wide range of applications, computational physics, image pro-cessing, medical image analysis, vision

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Application:Medical Image Analysis

� Similar to deformable models, level sets have been used in semi-automatic segmentaion.

� Powerful, as it can accommodate topological changes withoutchange in parameterization.

� Easily extensible to higher dimensional datasets

� Need to define appropriate speed functions.

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Application:Medical Image Analysis

� Front represents boundary of an evolving shape, must stop in thevicinity of the desired object boundaries.

� Final configuration is when all points of the front come to a stop.

� Speed function F is usually defined as the sum of an advection andgeometry terms

F = FA + FG

� FA is independent of the moving front’s geometry, is similar to aninflation (compression) force, while FG depends on the image ge-ometry, such as local curvature (regularizing term).

� The level set equation is given by

φt + FA|∇φ| + FG|∇φ| = 0

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Application:Medical Image Analysis

� Case 1: FG = 0:

◦ Can add a negative speed FI such as

FI(x, y) =−FA

(M1 −M2){|∇Gσ ∗ I(x, y)| −M2}

where M1, M2 are the max and min values of image gradient.

◦ FI is in the range [−FA, 0]

� Case 2: FG 6= 0:

◦ Can multiply F = FA + FG by

kl(x, y) =1

1 + |∇Gσ ∗ I(x, y)|

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Example: Medical Image Segmentation

Recovery of Stomach Shape from Abdominal Section (CT)ITCS 6010:Biomedical Imaging and Visualization 13 Evolving Interfaces:Level Set Methods

Example: Medical Image Segmentation

Smoothness Control in Shape Recovery

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Example: Medical Image Segmentation

Topological Split

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