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Region-Based Active Contours with Exponential Family Observations

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Abstract

In this paper, we focus on statistical region-based active contour models where image features (e.g. intensity) are random variables whose distribution belongs to some parametric family (e.g. exponential) rather than confining ourselves to the special Gaussian case. In the framework developed in this paper, we consider the general case of region-based terms involving functions of parametric probability densities, for which the anti-log-likelihood function is a special case. Using shape derivative tools, our effort focuses on constructing a general expression for the derivative of the energy (with respect to a domain), and on deriving the corresponding evolution speed. More precisely, we first show by an example that the estimator of the distribution parameters is crucial for the derived speed expression. On the one hand, when using the maximum likelihood (ML) estimator for these parameters, the evolution speed has a closed-form expression that depends simply on the probability density function. On the other hand, complicating additive terms appear when using other estimators, e.g. method of moments. We then proceed by stating a general result within the framework of multi-parameter exponential family. This result is specialized to the case of the anti-log-likelihood function with the ML estimator and to the case of the relative entropy. Experimental results on simulated data confirm our expectations that using the appropriate noise model leads to the best segmentation performance. We also report preliminary experiments on real life Synthetic Aperture Radar (SAR) images to demonstrate the potential applicability of our approach.

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Abbreviations

Ω I :

The image domain

Ω:

A region of the image

∂Ω:

The boundary of the region Ω

|Ω|:

Region size ∫ Ω d x

I(x):

The intensity of the pixel at the location x

k(x,Ω):

The region descriptor of Ω

k b (x):

The boundary descriptor of ∂Ω

Ω in :

Inside region

Ω out :

Outside region

〈J′(Ω),V〉:

The Eulerian derivative of domain energy criterion J(Ω)

p(y,η):

Probability density function (pdf) of the random variable Y

η :

Hyper-parameter vector of the pdf

\(\mathbb{E}{[Y]}\) :

Expectation of the random variable Y

k′(x,Ω,V):

The domain derivative of the function k

speed(x,Ω):

Evolution speed of the active contour

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Authors and Affiliations

  1. GREYC UMR CNRS 6072, 6 bd Marechal Juin, 14050, Caen Cedex, France

    François Lecellier, Jalal Fadili & Marinette Revenu

  2. LIMOS UMR CNRS 6158, Université Blaise Pascal, Complexe des Cézeaux, 63177, Aubiere, France

    Stéphanie Jehan-Besson

  3. Laboratoire J.A. Dieudonné UMR CNRS 6621, Parc Valrose, 06108, Nice Cedex, France

    Gilles Aubert

  4. CHU de Caen, Avenue Georges Clemenceau, 14000, Caen, France

    Eric Saloux

Authors
  1. François Lecellier
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  2. Jalal Fadili
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  3. Stéphanie Jehan-Besson
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  4. Gilles Aubert
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  5. Marinette Revenu
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Correspondence to François Lecellier.

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Lecellier, F., Fadili, J., Jehan-Besson, S. et al. Region-Based Active Contours with Exponential Family Observations. J Math Imaging Vis 36, 28–45 (2010). https://doi.org/10.1007/s10851-009-0168-8

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