Statistical image segmentation with Bayesian approach Tapio Helin - - PowerPoint PPT Presentation

Introduction Hierarchical stochastic model Conclusions

Statistical image segmentation with Bayesian approach

Tapio Helin

Institute of Mathematics, Helsinki University of Technology

Workshop on Inverse and Partial Information Problems: Methodology and Applications Linz, October 28, 2008

in collaboration with

Professor Matti Lassas, Helsinki University of Technology

Introduction Hierarchical stochastic model Conclusions

Outline

1

Introduction

2

Hierarchical stochastic model

3

Conclusions

Introduction Hierarchical stochastic model Conclusions

Outline

1

Introduction

2

Hierarchical stochastic model

3

Conclusions

Introduction Hierarchical stochastic model Conclusions

Hierarchical priors

Suppose some parameter of the prior distribution is unknown or it is preferable not to approximate it exactly. The prior is called hierarchical if these parameters are modelled also with random variables. One way to think about it: suppose the distribution of U depends

• n V

M = AU + E ⇒ M = A U V

• + E

Introduction Hierarchical stochastic model Conclusions

Data segmentation

In 1984 S. Geman and D. Geman proposed a statistical approach to image denoising. Let U,M and E be Rn-valued random variables such that M = AU + E, where E ∼ N(0, σ2I). They introduced a {0, 1}n -valued random variable L that described the edge set. The a priori distribution for the pair (U, L) was πpr(u, ℓ) ∝ exp

• −
• i
• α(1 − ℓi+ 1

2 )(ui+1 − ui)2 + βℓi+ 1 2

• .

Introduction Hierarchical stochastic model Conclusions

Data segmentation

In this setting the posteriori distribution is π(u, ℓ|m) ∝ exp (−E(u, ℓ, m)) where the free energy E(u, ℓ, m) is given by E(u, ℓ, m) =

• i
• α
• 1 − ℓi+ 1

2

• (ui+1 − ui)2 +

+βℓi+ 1

2 +

1 2σ2 ((Au)i − mi)2

• .

G-G proposed to maximize the πpost, i.e., to solve the MAP estimate.

Introduction Hierarchical stochastic model Conclusions

Mumford-Shah -functional

In 1989 Mumford and Shah presented the following continuous version of G-G’s method arg min

u,K

• T\K

|Du|2dx + ♯(K) +

• T

|u − m|2dx (1) where K is the ”discontinuity” or the ”jump-set” of a piecewise regular function u(x).

Introduction Hierarchical stochastic model Conclusions

Ambrosio-Tortorelli

Ambrosio and Tortorelli proposed approximating Mumford-Shah functional by elliptic functionals Fǫ(u, v) =

• T

(v2+ǫ2)|Du|2+ǫ|Dv|2+ (1 − v)2 4ǫ +|Au−m|2dx. Now Fǫ Γ-converges in L1(T) × L1(T) -topology to the weak formulation of Mumford-Shah -functional: F(u, v) =

• T\Su

|Du|2dx + ♯(Su) +

• T

|Au − m|2dx where u ∈ SBV(T) and v(x) = 1 almost everywhere. Otherwise F(u, v) = ∞.

Introduction Hierarchical stochastic model Conclusions

Ambrosio-Tortorelli

Minimizing

• T

(v2 + ǫ2)|Du|2 + ǫ|Dv|2 + (1 − v)2 4ǫ + |u − m|2dx results to m u v

Introduction Hierarchical stochastic model Conclusions

Ambrosio-Tortorelli

If this arg min

u,v∈H1(T)

• T

(v2 + ǫ2)|Du|2 + ǫ|Dv|2 + (1 − v)2 4ǫ + |Au − m|2dx is what you want to happen qualitatively - how to do it?

Introduction Hierarchical stochastic model Conclusions

Outline

1

Introduction

2

Hierarchical stochastic model

3

Conclusions

Introduction Hierarchical stochastic model Conclusions

Motivation

Suppose you are given a linear inverse problem M = AU + E in Rn with a hierarchical prior such that U ∝ N(u0, CU(v)) and V ∝ N(v0, CV ) and E white noise.

Introduction Hierarchical stochastic model Conclusions

Motivation

Then πpost(u, v | m) ∝ exp(−1 2E(u, v)) with a free energy of the form E(u, v) = log (det CU(v)) + u − u0, CU(v)−1(u − u0)+ + v − v0, C −1

V (v − v0) + Au − m 2.

Introduction Hierarchical stochastic model Conclusions

More motivation

Put u0 = 0, v0 = 1, CV = 1 4ǫId − ǫ∆n −1 and CU(v) = (−Dn(ǫ2+v2)Dn)−1 Then the free energy is close to A-T functional, in fact, E(u, v) = log (det CU(v)) + u, −Dn(ǫ2 + v2)Dnu +v − 1, 1 4ǫId − ǫ∆n

• (v − 1) + Au − m 2

= −

• T

n log(ǫ2 + v2)dx + Fǫ(u, v).

Introduction Hierarchical stochastic model Conclusions

Things to consider

One could ask e.g. (i) is the corresponding L2-valued model well-defined? (ii) how to discretize invariantly? (iii) how does the posterior distribution behave asymptotically? (iv) are we still doing what we were supposed to?

Introduction Hierarchical stochastic model Conclusions

About MAP estimate

Consider the minimization problem min

u,v∈H1(T)∩Xn

• T

−n log(ǫ2 + v2) + (ǫ2 + v2)|Dnu|2+ + 1 4ǫ(1 − v)2 + ǫ|Dnv|2 + |Au − m|2dx By letting u = 0 and v = 1 we notice that the minimum value decreases as n gets larger. Also we see that minimizers diverge.

Introduction Hierarchical stochastic model Conclusions

Definition of V

To simplify our notations we assume that the probability space has the structure Ω = Ω1 × Ω2, Σ = Σ1 × Σ2 and P = P1 × P2. Define V as Gaussian random variable on L2(T) with mean v0 = 1 and covariance operator CV = 1

4ǫId − ǫ∆

−1. Assume that V = V (ω2). Lemma For 0 < α < 1

2 we have V ∈ C 0,α(T) almost surely.

Introduction Hierarchical stochastic model Conclusions

Definition of U

Define a mapping U for every test function φ ∈ C ∞(T) φ → {ω → W (ω1), AV (ω2)φ | ω = (ω1, ω2) ∈ Ω}, where Av = (˜ D∗(ǫ2 + v2)˜ D)−1/2. Lemma The mapping U is a generalized random variable. Corollary We have U ∈ L2(T) almost surely.

Introduction Hierarchical stochastic model Conclusions

Exponential moment

When dealing with additive Gaussian noise the following property is needed. Theorem For every b > 0 there exists a constant Cb > 0 such that the exponential moments satisfy Eeb(U,V )L2×L2 < Cb and Eeb(Un,Vn)L2×L2 < Cb. for all n ∈ N.

Introduction Hierarchical stochastic model Conclusions

CM estimates converge

Corollary Given the introduced prior, Gaussian noise and weakly converging discretization scheme, the CM estimates converge. A numerical example: deconvolution m u v

Introduction Hierarchical stochastic model Conclusions

Outline

1

Introduction

2

Hierarchical stochastic model

3

Conclusions

Introduction Hierarchical stochastic model Conclusions

Conclusions

Summary: (1) We introduced a new prior model for segmenting signals. (2) CM estimates converge for a linear problem. (3) Connection to Mumford-Shah functional. Future work includes (1) understanding the limiting estimate better, (2) analysing error and (3) understanding MAP estimates better.

Introduction Hierarchical stochastic model Conclusions

If you want to know more...

• M. Lassas, E. Saksman, S.Siltanen: Discretization invariant

Bayesian inversion and Besov space priors, submitted.

• S. Geman, D. Geman: Stochastic Relaxation, Gibbs Distributions,

and the Bayesian Restoration of Images. IEEE Trans. PAMI, PAMI-6(6), 1984.

• D. Mumford, J. Shah: Optimal approximation by piecewise smooth

functions and associated variational problems. Comm. Pure Appl. Math., 1989.

• L. Ambrosio, V.M.Tortorelli: Approximation of functionals

depending on jumps by elliptic functionals via Γ-convergence.

• Comm. Pure Appl. Math., 1990.