Phase diagram of a higher-order active contour energy Aymen El - - PowerPoint PPT Presentation

Outline

Phase diagram of a higher-order active contour energy

Aymen El Ghoul : Ariana (INRIA/I3S), URISA (Sup’Com Tunis) Josiane Zerubia and Ian Jermyn : ARIANA (INRIA, I3S) Ziad Belhadj : URISA (Sup’Com Tunis)

July 26, 2007

Motivations Stability analysis of a circle Stability analysis of a long bar Summary and future work

Outline

1

Motivations Region modelling Problem statement

2

Stability analysis of a circle Variational study Stability conditions

3

Stability analysis of a long bar Variational study Stability conditions Experimental results

4

Summary and future work

Motivations Stability analysis of a circle Stability analysis of a long bar Summary and future work

Outline

1

Motivations Region modelling Problem statement

2

Stability analysis of a circle Variational study Stability conditions

3

Stability analysis of a long bar Variational study Stability conditions Experimental results

4

Summary and future work

Motivations Stability analysis of a circle Stability analysis of a long bar Summary and future work Region modelling

Probabilistic framework - variational approach

Problem Find the region R in the image domain containing a particular entity (road network, tree crowns. . . ) Probabilistic framework MAP estimate: ˆ R = arg maxR∈R P(R|I, K) Using Bayes’s theorem: ˆ R = arg maxR∈R P(I|R, K)P(R|K) Variational approach based on energy minimization The probabilities can be written in terms of ‘energies’: exp (−E) Energy minimization: ˆ R = arg minR∈R

• Eimage(I|R, K) + Eprior(R|K)

Motivations Stability analysis of a circle Stability analysis of a long bar Summary and future work Region modelling

HOAC model, [Rochery et al. 06]

Eg(γ) = λCL(γ) + αCA(γ)

• Snakes [Kass et al. 1988]

− βC 2

• dt dt′ ˙

γ(t′) · ˙ γ(t) Φ(|γ(t) − γ(t′)|)

1 2 3 4 5 6 7 0.2 0.4 0.6 0.8 1 Distance entre deux points R = |γ(t)−γ(t’)| Φ(R)

d ε

Motivations Stability analysis of a circle Stability analysis of a long bar Summary and future work Problem statement

Problem statement

Motivations Stability analysis of a circle Stability analysis of a long bar Summary and future work Problem statement

Scaling

The model has 5 parameters (λC, αC, βC, d, ǫ), so it is difficult to analyse. Scaling:

We take ǫ = d. The interaction function can be written as a function of ˆ γ = γ

d ,

Let ˆ α = αCd

λC , ˆ

β = βCd

λC and ˆ

Φ(z) = Φ( z

d ). The HOAC model

becomes: ˆ Eg(ˆ γ) = L(ˆ γ) + ˆ αA(ˆ γ) − ˆ β 2

• dt dt′ ˙

ˆ γ(t′) · ˙ ˆ γ(t) ˆ Φ(|ˆ γ(t) − ˆ γ(t′)|) , ˆ Eg depends only on the two scaled parameters ˆ α and ˆ β.

Stability analysis (Taylor series expansion): Eg(γ) = Eg(γ0 + δγ) = Eg(γ0) + δγ|δEg δγ γ0 + 1 2δγ|δ2Eg δγ2 |δγγ0

Motivations Stability analysis of a circle Stability analysis of a long bar Summary and future work

Outline

1

Motivations Region modelling Problem statement

2

Stability analysis of a circle Variational study Stability conditions

3

Stability analysis of a long bar Variational study Stability conditions Experimental results

4

Summary and future work

Motivations Stability analysis of a circle Stability analysis of a long bar Summary and future work Variational study

Parametrization

Circle parametrization γ0(t) = (r0(t), θ0(t)) = (r0, t) where t ∈ [−π, π[ γ(t) = γ0(t) + δγ(t) = (r(t), θ(t)) = (r0 + δr(t), θ0(t) + δθ(t)) The energy Eg is invariant to tangential changes so δθ(t) = 0. Perturbation definition The operator F = δ2Eg

δγ2 is invariant to ‘translation’ on the

circle, So the Fourier basis diagonalizes the operator F, So, perturbations are given in terms of Fourier coefficients by : δr(t) =

k akeir0kt where k = m r0 and m ∈ Z.

Motivations Stability analysis of a circle Stability analysis of a long bar Summary and future work Variational study

Energy of a perturbed circle [Horvath et al. 06]

Eg(r) = Eg(r0 + δr) = e0(r0) + a0e1(r0) + 1 2

• k

|ak|2e2(k, r0) .

Motivations Stability analysis of a circle Stability analysis of a long bar Summary and future work Variational study

Circle energy e0

0.5 1 1.5 2 2.5 3 2 4 6 8 10 12 14 16 18 20 r0 e0 0.5 1 1.5 2 2.5 3 2 4 6 8 10 12 14 r0 e0 0.5 1 1.5 2 2.5 3 −2 2 4 6 8 10 r0 e0

αC = 1, βC = 1.54 αC = 1, βC = 1.84 αC = 1, βC = 2.03

0.5 1 1.5 2 2.5 3 5 10 15 20 25 30 r0 e0 0.5 1 1.5 2 2.5 3 5 10 15 20 25 30 35 r0 e0 0.5 1 1.5 2 2.5 3 −30 −25 −20 −15 −10 −5 5 r0 e0

αC = 1, βC = 1.10 αC = 1, βC = 0.88 αC = −1, βC = 1

Motivations Stability analysis of a circle Stability analysis of a long bar Summary and future work Stability conditions

First stability condition: e1 = 0

βC(λC, αC, r0) = λC + αCr0 G10(r0) ⇐ ⇒ ˆ β(ˆ α, ˆ r0) = 1 + ˆ α ˆ r0 ˆ G10(ˆ r0)

1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 r0 βC (βC

i ,r0 i )

Motivations Stability analysis of a circle Stability analysis of a long bar Summary and future work Stability conditions

Second stability condition: e2 positive definite

0.5 1 1.5 2 2.5 3 2 4 6 8 10 12 14 16 r0 e0 1 2 3 4 5 6 7 8 −50 50 100 150 200 250 r0 k e2

αC = 1, r∗

0 = 1

e2(αC, k, r∗

0 ) > 0

e2(αC, k, r0) > 0 , ∀k = 1 r0 ⇐ ⇒ ˆ αa(ˆ r0, k) > f(ˆ r0, k) , ∀k = 1 r0 .

Motivations Stability analysis of a circle Stability analysis of a long bar Summary and future work Stability conditions

Bounds on the parameter ˆ α

ˆ r a 0.5 1 1.5 2 2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 ˆ r f 0.5 1 1.5 2 2.5 −3 −2 −1 1 2 3

Graph of a Graph of f

ˆ r fa 0.5 1 1.5 2 2.5 −8 −6 −4 −2 2 4 6 8 ˆ r ˆ α 0.5 1 1.5 2 2.5 −10 −8 −6 −4 −2 2 4 6

Graph of f/a Bounds of ˆ α

Motivations Stability analysis of a circle Stability analysis of a long bar Summary and future work Stability conditions

Circle phase diagram

1 2 3 4 5 6 7 1 2 3 4 5 6 7

asinh(αC) asinh(βC)

0.69 < ˆ r0 < ∞

Motivations Stability analysis of a circle Stability analysis of a long bar Summary and future work

Outline

1

Motivations Region modelling Problem statement

2

Stability analysis of a circle Variational study Stability conditions

3

Stability analysis of a long bar Variational study Stability conditions Experimental results

4

Summary and future work

Motivations Stability analysis of a circle Stability analysis of a long bar Summary and future work Variational study

Bar parametrization

µ = 1 µ = 2 1\2 −1\2

γ0,µ(tµ) =

• x0,µ(tµ)

= ±µ l tµ , tµ ∈ [−0.5, 0.5] y0,µ(tµ) = ±µ

w0 2

where ±µ =

• +1

si µ = 1 −1 si µ = 2

Motivations Stability analysis of a circle Stability analysis of a long bar Summary and future work Variational study

Perturbation definition

δγµ(tµ) = (δxµ(tµ), δyµ(tµ)). Tangential perturbations make no difference: δxµ(tµ) = 0, Perturbations are given in terms of Fourier coefficients (bar ‘translation’ invariance) by: δyµ(tµ) =

kµ aµ,kµeikµltµ

where kµ = 2πmµ

l

, mµ ∈ Z, Contour expression: γµ(tµ) = γ0,µ(tµ) + δγµ(tµ) =

• xµ(tµ)

= ±µ l tµ yµ(tµ) = ±µ

w0 2 + kµ aµ,kµeikµltµ .

Motivations Stability analysis of a circle Stability analysis of a long bar Summary and future work Variational study

Energy of a perturbed long bar

Eg(γ) l = e0(w0) +

• a1,0 − a2,0
• e1(w0)

+ 1 2

• k
• |a1,k|2 + |a2,k|2

e20 + (a1,ka2,k + a1,−ka2,−k)e21 = e0 + e1

• a1,0 − a2,0
• + 1

2

• k

a∗

k e2 at k .

ak =

• a∗

1,k

a2,k

• , e2 =

e20 e21 e21 e20

• .

Motivations Stability analysis of a circle Stability analysis of a long bar Summary and future work Variational study

Bar energy e0

0.5 1 1.5 2 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 w0 e0 0.5 1 1.5 2 0.5 1 1.5 2 2.5 w0 e0 0.5 1 1.5 2 −1 −0.5 0.5 1 1.5 2 2.5 w0 e0

αC = 1, βC = 0.66 αC = 1, βC = 1.84 αC = 1, βC = 2.47

0.5 1 1.5 2 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 w0 e0 0.5 1 1.5 2 2 2.5 3 3.5 w0 e0 0.5 1 1.5 2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 w0 e0

αC = 1, βC = 0.52 αC = 1, βC = 0.35 αC = −1, βC = −0.66

Motivations Stability analysis of a circle Stability analysis of a long bar Summary and future work Stability conditions

First stability condition: e1 = 0

βC(λC, αC, w0) = αC G10(w0) ⇐ ⇒ ˆ β(ˆ α, ˆ w0) = ˆ α ˆ G10( ˆ w0)

0.5 0.52 0.54 0.56 0.58 0.6 0.62 0.64 0.66 0.68 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 βc w0 (βC

i ,w0 i )

Motivations Stability analysis of a circle Stability analysis of a long bar Summary and future work Stability conditions

Second stability condition: e2 positive definite

e2 is positive definite iff its eigenvalues, λ+(k) and λ−(k), are strictly positive for all k. λ+ = e20 + e21 and λ− = e20 − e21,

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 w0 e0 5 10 15 10 20 30 40 50 60 70 80 90 100 m Energie

αC = 1, βC = 0.66, λ+(αC, k, w∗

0) > 0,

w∗

0 = 1.2

λ−(αC, k, w∗

0) > 0

Motivations Stability analysis of a circle Stability analysis of a long bar Summary and future work Stability conditions

Second stability condition: e2 positive definite

λ± = λCk2 + βC(αC, w0)G±(d, ǫ, w0, k), (λC > 0) Scaling property: ⇒ ˆ λ±(ˆ α, ˆ w0, k) = k2 +

ˆ α ˆ G10( ˆ w0)

ˆ G±( ˆ w0, k) Second order stability condition: ˆ λ±(ˆ α, ˆ w0, k) > 0, ∀k ⇔ ˆ α ˆ G±( ˆ w0, k) > −k2 ˆ G10( ˆ w0), ∀k .

Motivations Stability analysis of a circle Stability analysis of a long bar Summary and future work Stability conditions

Phase diagram of a long bar

ˆ w0 ˆ α 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 1 2 3 4 5 6 7

asinh(αC) asinh(βC)

1 2 3 4 5 6 7 −1 1 2 3 4 5 6 7

Bounds of the parameter ˆ α 0.88 < ˆ w0 < 1.3

Motivations Stability analysis of a circle Stability analysis of a long bar Summary and future work Experimental results

Phase diagram of HOAC

Circle model Bar model

1 2 3 4 5 6 7 1 2 3 4 5 6 7

asinh(αC) asinh(βC) asinh(αC) asinh(βC)

1 2 3 4 5 6 7 −1 1 2 3 4 5 6 7

0.69 < ˆ r0 < ∞ 0.88 < ˆ w0 < 1.3

Motivations Stability analysis of a circle Stability analysis of a long bar Summary and future work Experimental results

Gradient descent evolution: circles

Motivations Stability analysis of a circle Stability analysis of a long bar Summary and future work Experimental results

Gradient descent evolution: bars

Motivations Stability analysis of a circle Stability analysis of a long bar Summary and future work

Outline

1

Motivations Region modelling Problem statement

2

Stability analysis of a circle Variational study Stability conditions

3

Stability analysis of a long bar Variational study Stability conditions Experimental results

4

Summary and future work

Motivations Stability analysis of a circle Stability analysis of a long bar Summary and future work

Summary and future work

Summary Circle phase diagram for tree detection, Bar phase diagram for road extraction. Future work Analyse the region of the phase diagram which gives both stable circles and stable bars. Fourth order expansion for more complex shapes.