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7/31/2019 Expos Anti Ago

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Mathematical Image Segmentation:

Variational and Level Set Techniques.

Adrin Galdrn1,2, David Pardo2,3, Artzai Picn1

1Tecnalia Research & Innovation, Parque Tecnolgico de Bizkaia, Zamudio, Spain.

2Basque Country University (UPV/EHU), Bilbao, Spain. 3IkerBasque Foundation, Bilbao, Spain.

16 de septiembre de 2011
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Tecnalia + UPV + Image Processing
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Image Processing

Great Variety of Tasks

Image Filtering and

Enhancement

Image Features

Extraction

Segmentation

Registration

ClassificationRecognition
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Mathematical Image Segmentation

Many types of images

Light Intensity Images (LI), Color Images (RGB, RGBA, CIELAB), Magnetic

Resonance Images (MRI), X-Ray Images, Ultra-Sound Images, Diffusion

Tensor Images (DTI)

Biomedical Images extracted form the Bioimage Suite of image processing developed at Yale Un iversity, www.bioimagesuite.org
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Different Approaches

An important decision:

Continuous Vs Discrete

Classical algorithms are simpler, but have some heavy disadvantages.

Algorithm complexity is greatlyincreased with the complexity

of the application.

Not able to evolve to adapt

new image data or conditions.

Difficult to include prior oracquired knowledge not related

with image data.

Must be adapted for use in

other applications.
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PDE-based Approach

Variational Techniques

Consider the image as the graph of a function I : R2/R3 X.Define an Energy modeling the problem, and find the

minimum of that functional. For that, differentiate the functional,

find Euler-Lagrange equations, and solve numerically.

PDEs are not only used for

Segmentation

Noise Removal, Image

Restoration, Image

Classification,. . . .

C. Ballester, M. Bertalmio, V. Caselles, G. Sapiro, and J. Verdera,

IMA Report 2000, IEEE Trans. IP 200
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Most Classical Model: Snakes

The idea

Put a closed parametric curve c on the image, and make it

evolve in time, to get close to the contour of the object we want

to segment.

Minimize

J(c) =

ba

|c(q)|2 dq+

ba

|c(q)|2 dq+

ba

1

1 + |I(c(q))|2dq


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Snakes: Details

J(c; c) = . . . = b

a

c + civ + EI

c

Euler-Lagrange for Snakes

c(q) + civ(q) + EI(c(q)) = 0c(a) = c(b), c(a) = c(b)

Gradient Flow for Snakes

c

t= c(t, q) + civ(t, q) + EI(c(t, q))

c(0, q) = c(q), c(t, a) = c(t, b)c(t, 0) = c0(t) c

(t, a) = c(t, b)

(where c0 is the initial curve, supposed to be enclosing the object to be detected.)


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Pros, Cons, Limitations

Snakes Model

Pros

Low complexity.Easy to introduce prior

knowledge.

Well established

technique, lots ofpublications.

User interactivity

Cons

It depends on the choice of

the parametrization.

The sampling affects the

final result

Very sensitive to initial

conditions.

Unable to perform

topological changes.


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Level Set Method

What is the Level Set Method?

An implicit data representation of a hypersurface, a

set of PDEs that govern how the surface moves, and

the corresponding numerical methods for

implementing this on computers. Stanley Osher

Many Different Applications

Geometry, Fluid Dynamics,

Optimal Design, Seismic

Analysis, Robotics,

Semiconductors, Optimal

Control, Image

Processing. . .Level set fluid simulation, R. Fedkiw et al.

(http://physbam.stanford.edu/ fedkiw/)


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Level Set Method

In the most simple case, represent a plane curve as the set of

zeros of : R2 R.

An initial curve c(p, t) = (x(p, t), y(p, t)), parameterized by p, ismoving across the image following a velocity fieldV : R2 R2. Parameter t controls the temporal evolution.


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Level Set Method

tracks c(t) through its 0-level set (c(t), t) = 0, so:

t(x(t), y(t), t) +

x

x

t+

y

y

t= t +

x

t,

y

t

c= 0

The time derivative c is the velocity at which it is evolving, so:

t + V = 0.

Decomposing the velocity field in its normal and tangential

components, V = Vn n + Vt t, and using n = /||,

t + Vn|| = 0

Level Set Method: How does it Apply to Image


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Segmentation?

We build an Integral Cost Function J : C R that depends on the

image, and pushes the curve towards high gradients, or region

dissimilarities.

Dictionary

Heaviside Function, H() =

0, if 01, if < 0

Normal, n = ||

Curvature, = div

||

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Segmentation?

Geodesic Active Regions

Extending Snakes, we can arrive to

minC(q) g(|I(C(q))|) |C(q)|dqwhere g(t) = 1/(1 + t2)

Level Set Version

t= g(I) || +g(I)



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Segmentation?

Active Contours without edges

It is easy to introduce simple global statistics,

E(c1, c2) = (c1)2H()+(c2)

2(1H())+|H()|dxdy

where c1is the mean inside , c2 is the mean outside , and |H()| is the length of the zero level set of .

We can make the energy function dependant on more complex

probabilistic and statistical tools (prior knowledge, histogram,

texture, 3D shape, location of landmarks).



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Segmentation?

Level Set Segmentation


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Level Set Method: Advantages

Advantages of the Implicit Representation

The Level Set Method does not

suffer of

Parametrization-Dependency

problems.It is able to account for

Changes of the Topology

automatically: the embedding

function can vary smoothly,

while its zero level set inherits

non-continuous changes.

The resulting law of evolution

for a surface becomes a

Hamilton-Jacobi PDE, a

classical equation that has

been intensively studied. The

Level Set Method allows us to

introduce the machinery of theNumerical Analysis, both for its

theoretical study and for the

computational solution.

Straightforward Generalization

to higher dimensions, since the

formulas for curvature and

normal vector are not

dimension-dependent.


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Conclusions

Mathematical and statistical modeling techniques are essential

to deal with complex applications in Image Processing and

Computer Vision.

The Level Set Method is a powerful tool that has been applied in

different scientific fields.

Due to its numerous advantages over traditional methods, it is

also being applied by the Image Processing community.

The goal is to be able to combine eperience and expert

knowledge in statistical tools with advanced numerical analysis

techniques to develop Image Segmentation Algorithms that rely

on the Level Set Framework.


19/19

Mathematical Image Segmentation:

Variational and Level Set Techniques.

Adrin Galdrn1,2, David Pardo2,3, Artzai Picn1

1Tecnalia Research & Innovation, Parque Tecnolgico de Bizkaia, Zamudio, Spain.

2Basque Country University (UPV/EHU), Bilbao, Spain. 3IkerBasque Foundation, Bilbao, Spain.

16 de septiembre de 2011
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