Ω min E E ( ( = ∂Ω Ω Ω () ) E ) = = ∪Γ∪Ω ∫ ∫ k k + ( E ( x x , , y y , ) Ω ds ) ds + ...

(1)

In Details: Segmentation

Active Contours

28/11/14

!

Principle

min ( E

_image

+ E

regularisation

+ ... )

Gradient-based approach:

Region-based approach:

E ( ∂Ω ) = k

_b

( x, y) ds

∂Ω

∫

E ( Ω

_i

) = k

_b

( x, y , Ω

_i

) ds

Ω

∫

Ω=Ω_in ∪ Γ ∪Ω_out

Aymeric Histace -‐ HDR

Defense 24

(2)

In Details: Active Contour

Shape Prior

28/11/14 Main Idea

•In Medical Image, the shape of the structure to segment is often “known”

Cardiac MRI

Microconfocal images

SA LA

X-Ray Radiography

(hip bone)

Proposal

E ( ) λ

_r

⁼ ^E

^prior

( ) ^λ

^r

⁺ ^E

^image

( ) ^λ

^r

• Question: Knowing the

“mean” shape of an

object, can we integrate it in an AC segmentation process ?

with a reduced shape descriptor (statistical

learning)

λ

_r

Defense 25

(3)

In Details: Active Contour

Shape Prior

28/11/14

Shape Space Representation Shape descriptor

Shape Ω

λ

_i ^{such as}

0 5

N = N₀ =20 N₀ =40

λ

_i

i=1 N₀

∑

Legendre moments Zernike

moments

Statistical Learning (PCA)

The chicken image set used to build the statistical shape model

( )( )

1

1 ^N^S ^T

i i

S i

N ₌

=

å

- -

Q λ λ λ λ

( )

,

T

r i = i -

λ P λ λ

Test image

1

1 ^N^S

i S i

N ₌

=

å

λ λ

Moments

Defense 26

(4)

In Details: Active Contour

Shape Prior

28/11/14

Shape Space of Moments

,1 1 ,2 2

r r

l l

= × + × +

λ p p λ

Eigen Shapes

( ) ( ^, ²)

1

; ,

NS

r r r i

i

p s

=

=å

λ N λ λ

( ) (

^, ²

)

1

ln ^S ; ,

N

prior r r r i

i

E s

=

æ ö

= - ç ÷

è

å

ø

λ N λ λ

( ) ( ^, ²)

1

; ,

NS

r r r i

i

p s

=

=å

λ N λ λ

Iterations shown in the feature space spanned by the first two principal axes

How it works

Final Result

Different from Template Matching

E

( )

λ_r ⁼ ^E^prior

( )

^λ^r ⁺ ^E^image

( )

^λ^r

Defense 27

(5)

In Details: Active Contour

Shape Prior

28/11/14

Results 1 (Legendre Moments)

Chan-Vese method

Multi-reference shape prior

method (Foulonneau 09)

The proposed method

Original binary image

Image with Gaussian noise

Image with structural noise

Image with Gaussian and structural noise

Defense 28

(6)

In Details: Active Contour

Shape Prior

28/11/14 Results 2

Question 2: Answer 1 Space Shape of Legendre Moments makes possible Shape

Prior integration different from template matching

CAIP 2011, Journal of Mathematical Imaging and Vision 2013

Defense 29

(7)

And Then…

28/11/14

CAD: Main Contributions

Gaussian Noise

Stochastic Resonance Non-Linear PDE

∂I

∂t =div g

(

_η( ∇I )^∇I

)

)) , ( ( )

(u g u x y

g_h = +h

Active Contour With Shape Prior E=E_prior+E_image

Shape learning

Shape descriptor (Legendre)

Up :Foulonneau, Down :Our Approach

Alpha-Divergence Based Active Contour

ChanVese E =E_histogram+Eregularisation

Joint Optimization:

-Segmentation Process -Metric of the divergence

Coll. CREATIS David Rousseau

Coll. UCLan B. Matuszewski

Coll. I3S PhD Leila Meziou

Defense 30

(8)

In Details: Active Contour

Alpha-‐divergence

28/11/14

Histogram-Based Active Contour

E = E

_histogram

+ E

regularisation

D p (

₁

|| p

₂

, Ω ) ⁼ ^ϕ ( ^p

¹

^, ^p

²

^, ^λ )

ℜ^m

∫ ^d ^λ

ϕ a similarity function p_i the pdf of Ω_i λ the quantization level

"

#

$

%$

$ Divergence

with

Questions:

•How to model the pdf p_i(derivation constraints)?

•Which φ function?

Probability density function

Defense 31

(9)

In Details: Active Contour

Alpha-‐divergence

28/11/14

Histogram-Based Active Contour pdf modeling

(Parzen Window)

pˆ_i

(

λ^,Ω_i

)

⁼ ¹

Ω_i g_σ

(

I(x)− λ

)

^d^x

Ω

∫

_i

with g_σ a Gaussian kernel of variance σ

Divergence

Usually

• Kullback-Leibler

• Hellinger

• Ki²

Our proposal

• Alpha-divergence

α={0,1}

α=0.5 α=2

Defense 32

(10)

In Details: Active Contour

Alpha-‐divergence

28/11/14

Joint Optimization

Maximization of the divergence

Optimization of alpha-parameter

argmax_Γ

(

D_α

(

p_in || p_out,Ω

) )

^argmax^α

(

^D^α

⁽

^pⁱⁿ ^|| ^p^out^,^Ω

⁾ )

∂

α

∂t =−∂D_α

(

p_in || p_out,

α )

∂Γ

∂t =−∂_p_in_,_p_outD_α

(

p_in || p_out,

α )

$

%

&

'

&

Algorithm 1.α_t+1 (α_init = 1)

1.Γ_t+1 Aymeric Histace -‐ HDR

Defense 33

(11)

In Details: Active Contour

Alpha-‐divergence

28/11/14

Result 1 (Synthetic images)

Defense 34

(12)

In Details: Active Contour

Alpha-‐divergence

28/11/14

Result 2 (Synthetic images)

Defense 35

(13)

In Details: Active Contour

Alpha-‐divergence

28/11/14

Result 3 (Natural Images)

X-Ray images Videocapsule

(coloscopy)

ICIP 2011, ICASSP 2012, MIUA 2012 (Best Student Paper Award), Annals of BMVA 2013, ICIP 2014, IJBI 2014

Microconfocal images of cells

Question 2:

Answer 2 Alpha-divergence

are a flexible tool to cop with different noise

scenarios in medical image analysis (but not

only)

Defense 36

Ω min E E ( ( = ∂Ω Ω Ω () ) E ) = = ∪Γ∪Ω ∫ ∫ k k + ( E ( x x , , y y , ) Ω ds ) ds + ...

In Details: Segmentation

Active Contours

min ( E

+ E

+ ... )

E ( ∂Ω ) = k

( x, y) ds

∫

E ( Ω

) = k

( x, y , Ω

) ds

∫

In Details: Active Contour

Shape Prior

E ( ) λ

= E

( ) λ

+ E

( ) λ

λ

In Details: Active Contour

Shape Prior

λ

λ

∑

( )( )

å

( )

å

In Details: Active Contour

Shape Prior

( ) (

)

å

( )

( )

( )

In Details: Active Contour

Shape Prior

In Details: Active Contour

Shape Prior

And Then…

CAD: Main Contributions

(

)

In Details: Active Contour

Alpha-­‐divergence

E = E

+ E

D p (

|| p

, Ω ) = ϕ ( p

, p

, λ )

∫ d λ

In Details: Active Contour

Alpha-­‐divergence

(

)

(

)

∫

In Details: Active Contour

Alpha-­‐divergence

(

(

) )

(

(

) )

α

(

α )

(

α )

In Details: Active Contour

Alpha-­‐divergence

In Details: Active Contour

⁼ ^E

( ) ^λ

⁺ ^E

( ) ^λ

Alpha-‐divergence

, Ω ) ⁼ ^ϕ ( ^p

^, ^p

^, ^λ )

∫ ^d ^λ

Alpha-‐divergence

Alpha-‐divergence

⁽

⁾ )

Alpha-‐divergence

Alpha-‐divergence

Alpha-‐divergence