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Submitted on 14 Nov 2016

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Towards a unified bayesian geometric framework fortemplate estimation in Computational Anatomy

Nina Miolane, Xavier Pennec, Susan Holmes

To cite this version:Nina Miolane, Xavier Pennec, Susan Holmes. Towards a unified bayesian geometric framework fortemplate estimation in Computational Anatomy. International Society of Bayesian Analysis: WorldMeeting, Jun 2016, Cagliari, Italy. �hal-01396716�

• Modal approximation in (1): exp −𝑑𝐺2 𝑔,𝑔0

2𝜂2≃ 𝛿𝑔0 i.e. 𝜂 ≃ 0

• Adding regularization in (1): +𝜎2

𝜂2𝑑𝐺2 𝑔𝑖 , 𝑔0

References: [1] Miolane, Holmes, Pennec. Biased estimators on quotient spaces (2015). [2] Allassonniere, S., Amit, Y., Trouve, A.: Towards a coherent statistical framework for dense

deformable template estimation (2007). [3] Devilliers, Allassonniere, Pennec, Trouve. Frechet means top and quotient space might not be consistent: a case study (2015).

Different estimators of the template’s shape

Computational Anatomy aims to model and analyze the variability of the human anatomy. Given a set of medical images of the same organ, the first step is the estimation of the mean

organ’s shape. This mean anatomical shape is called the template in Computer vision or Medical imaging. The estimation of a template/atlas is central because it represents the starting

point for all further processing or analyses. In view of the medical applications, evaluating the quality of this statistical estimate is crucial. How does the estimated template behave for

varying amount of data, for small and large level of noise? We present a geometric Bayesian framework which unifies two estimation problems that are usually considered distinct: the

template estimation problem and manifold learning problem - here associated to estimating the template’s orbit. We leverage this to evaluate the quality of the template estimator.

Comparison of the estimators

Template estimation in Computational Anatomy

Towards a unified bayesian geometric framework

for template estimation in Computational Anatomy

Nina Miolane, Xavier Pennec, Susan Holmes nina.miolane@inria.fr

?

M: space of the images 𝑋𝑖’sG: Lie group of transformations

Action of 𝐺 on 𝑀: 𝜌:𝑀 × 𝐺 → 𝑀 denoted: 𝑋, 𝑔 → 𝜌 𝑋, 𝑔𝑄: shape space, quotient of 𝑀 by 𝐺

𝑋𝑖 = 𝝆(𝑻, 𝒈𝒊) + 𝜖𝑖where 𝑔𝑖~𝒩(𝑔0, 𝜂) i.i.d. and 𝜖𝑖 ~𝒩(0, 𝜎) i.i.d.

Generative model of organs’ shapes

Template estimation as a non-linear model of Errors-in-Variables

MLE-F: Fast but inconsistent

MLE-S: Consistent but slow

Intra-subject Inter-subjects

Images from: [Talbot and al 2013][Lorenzi and al, 2011][Gerber and al, 2010][Margeta and al, 2011]

Electromechanical model of the heart

Aging model of the brain Brain manifold learning

Computational Physiology Computational Anatomy

Organ shape analysis

𝑡

Computational Medicine relying on medical images

First step: template shape computation Second step: analysis

Template shape

Non-linear model of Errors-in-Variables

𝑔𝑖

𝑔𝑖

𝜖𝑖

𝜖𝑖

𝑋𝑖 = 𝝆(𝑻, 𝒈𝒊) + 𝜖𝑖where 𝑔𝑖~𝒩(𝑔0, 𝜂) i.i.d. and 𝜖𝑖 ~𝒩(0, 𝜎) i.i.d.

𝑻

Regression curve

parameterized by 𝑻

𝜌(𝑇, 𝑔𝑖)

𝜌(𝑇, 𝑔𝑖)

𝑋𝑖

𝑋𝑖

Goal:

Estimate the template 𝑻

Goal:

Estimate the curve

parameterized by 𝑻

Unification through Geometric Statistics

Space of images 𝑀

Orbit of template

under the Lie group action

Template shape

Maximum-Likelihood (MLE-F)

Functional model: 𝑔𝑖’s are parameters Structural model: 𝑔𝑖’s are random variables

Maximum-Likelihood: Expectation-Maximization algorithm (MLE-S) 𝑇 = argminT

𝑖=1

𝑛

min𝑔𝑖∈𝐺

𝑑𝑀2 𝜌 𝑇, 𝑔𝑖 , 𝑋𝑖

Frechet mean in the shape spaceNo closed form solution

Likelihood: 𝐿 = Π𝑖=1𝑛 exp −

𝑑𝑀2 𝜌 𝑋𝑖,𝑔𝑖 ,𝑇

2𝜎2Likelihood: 𝐿 = Π𝑖=1

𝑛 𝑔∈𝐺 exp −𝑑𝑀2 𝜌 𝑋𝑖,𝑔 ,𝑇

2𝜎2exp −

𝑑𝐺2 𝑔,𝑔0

2𝜂2𝑑𝑔

(1) ∀𝑖, 𝑔𝑖 = argmin𝑔∈𝐺 𝑑𝑀2 𝜌 𝑇, 𝑔𝑖 , 𝑋𝑖

(2) 𝑇 = argminT 𝑖=1𝑛 𝑑𝑀

2 𝜌 𝑇, 𝑔𝑖 , 𝑋𝑖

(1) Expectation

(2) Maximization

Adding priors: 𝑝 𝑇 = cte. exp −𝑑𝑄 𝑇,𝑇0

2

2𝜎𝑇2 reweights metric in shape space; 𝑝 𝑔0 = cte. exp −

𝑑𝑂 𝑔0,𝑔02

2𝜎𝑔02 reweights metric in the orbit; 𝑝 𝜎 = cte.

exp − 𝜎2

2𝜎𝜎2

2𝜎𝜎2

𝑎𝜎

; 𝑝 𝜂 = cte.exp −

𝜂2

2𝜎𝜂2

2𝜎𝜂2

𝑎𝜂

Maximum-a-Posteriori (MAP-F)

Acknowledgment: Participation in this conference was supported by the "NSF @ISBA junior travel support”

𝑟

P.d.f. of the radii

For 𝑇 = 1 and 𝑚 = 3

𝜎 = 3

0 𝜎

Bias 𝑇, 𝑇

For 𝑇 = 1 and 𝑚 = 3

1 2 3

1

2

3

4

0.2

0.6

1.0

1.4

2 4 86 10

Improvement using the Bayesian framework: fast and inconsistency substantially reduced

𝑛10 20 30 40 50

0%

4%

8%

𝑛10 20 30 40 50

0%

4%

8%

0

𝑻 𝑻Bias 𝑇, 𝑇 for MLE-F

For 𝑇 = 1 and 𝑚 =2

Bias 𝑇, 𝑇 for MAP-F

For 𝑇 = 1 and 𝑚 =2