level set methods in medical image analysis · ized models for applications in visualization and...
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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|
ITCS 6010:Biomedical Imaging and Visualization 3 Evolving Interfaces:Level Set Methods
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
ITCS 6010:Biomedical Imaging and Visualization 7 Evolving Interfaces:Level Set Methods
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,
ITCS 6010:Biomedical Imaging and Visualization 8 Evolving Interfaces:Level Set Methods
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
ITCS 6010:Biomedical Imaging and Visualization 9 Evolving Interfaces:Level Set Methods
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.
ITCS 6010:Biomedical Imaging and Visualization 10 Evolving Interfaces:Level Set Methods
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
ITCS 6010:Biomedical Imaging and Visualization 11 Evolving Interfaces:Level Set Methods
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)|
ITCS 6010:Biomedical Imaging and Visualization 12 Evolving Interfaces:Level Set Methods
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
ITCS 6010:Biomedical Imaging and Visualization 14 Evolving Interfaces:Level Set Methods
Example: Medical Image Segmentation
Topological Split
ITCS 6010:Biomedical Imaging and Visualization 15 Evolving Interfaces:Level Set Methods