Ultrasound image segmentation using an active contour model and learning-structured inference

Abstract

Automated segmentation of medical ultrasound (US) images is a challenging problem due to the complicated features of lesions, inconsistent lesions across individuals, and the high segmentation accuracy requirement. From recently published papers in this area, the active contour model (ACM) and machine learning method produce more accurate lesion segmentation results than previous methods. This paper proposes a novel image segmentation approach that integrates an ACM with a generalized linear model (GLM) and forms learning-structured inference. Compared with the GLM, the proposed method can solve the problems of initialization and the local minimum of the ACM. Furthermore, rather than using the ACM as a postprocessing tool, we integrate it into the training phase to fine-tune the GLM. This step allows the use of unlabeled data during training in a semisupervised setting. The integrated model requires only one image as the training set and is not as sensitive to labeled data as other methods. The proposed method is verified using US images, and the results show that the proposed method can produce accurate segmentation results.

Appendices

Appendix 1

1.1 Symbol Definitions

The parameters used in the proposed method are as follows:

$$ \mathrm{area}\ \mathrm{in}\kern0.5em {\Omega}_1:{A}_o={\int}_{\Omega}H\left(\phi (x)\right) dx $$

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$$ \mathrm{area}\ \mathrm{in}\kern0.5em {\Omega}_2:{A}_b={\int}_{\Omega}\left[1-H\left(\phi (x)\right)\right] dx $$

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$$ \mathrm{area}\ \mathrm{in}\kern0.75em \Omega :\mathrm{A}={\int}_{\Omega} dx $$

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$$ \mathrm{sum}\ \mathrm{of}\ \mathrm{the}\ \mathrm{in}\mathrm{tensity}\ \mathrm{in}\kern0.5em {\Omega}_1:{S}_o={\int}_{\Omega}I(x)H\left(\phi (x)\right) dx $$

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$$ \mathrm{sum}\ \mathrm{of}\ \mathrm{the}\ \mathrm{in}\mathrm{tensity}\ \mathrm{in}\kern0.5em {\Omega}_2:{S}_b={\int}_{\Omega}I(x)\left[1-H\left(\phi (x)\right)\right] dx $$

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$$ {\Omega}_1:{c}_o={S}_o/{A}_o={\int}_{\Omega}I(x)H\left(\phi (x)\right) dx/{\int}_{\Omega}H\left(\phi (x)\right) dx $$

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$$ \mathrm{average}\ \mathrm{in}\mathrm{tensity}\ \mathrm{in}\kern0.5em {\Omega}_2:{c}_b={S}_b/{A}_b={\int}_{\Omega}I(x)\left[1-H\left(\phi (x)\right)\right] dx/{\int}_{\Omega}\left[1-H\left(\phi (x)\right)\right] dx $$

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Appendix 2

1.1 Minimum of the proposed energy functional (11)

According to the gradient descent method, we can obtain the following:

$$ \partial \phi /\partial t=-\partial E\left(\phi, \wp \right)/\partial \phi =2{\wp}_o{\wp}_b\left({c}_o-{c}_b\right)\left({\partial c}_o/\partial \phi -{\partial c}_b/\partial \phi \right) $$

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where

$$ {\partial c}_o/\partial \phi =\left({S}_o^{\prime }{A}_o-{S}_o{A}_o^{\prime}\right)/{A}_o^2=\left({S}_o^{\prime }-{c}_o{A}_o^{\prime}\right)/{A}_o=\delta \cdotp \left(\left(I-{c}_o\right)/{A}_o\right) $$

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$$ {\partial c}_b/\partial \phi =\left({S}_b^{\prime }{A}_b-{S}_b{A}_b^{\prime}\right)/{A}_b^2=-\left({S}_b^{\prime }-{c}_b{A}_b^{\prime}\right)/{A}_b=-\delta \cdotp \left(\left(I-{c}_b\right)/{A}_b\right) $$

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The corresponding derivative is:

$$ {\displaystyle \begin{array}{l}\partial \phi /\partial t=-\partial E\left(\phi, \wp \right)/\partial \phi =2{\wp}_o{\wp}_b\left({c}_o-{c}_b\right)\left({\partial c}_o/\partial \phi -{\partial c}_b/\partial \phi \right)\\ {}=2{\wp}_o{\wp}_b\left({c}_o-{c}_b\right)\left(\delta \left(\left(I-{c}_o\right)/{A}_o\right)+\delta \left(\left(I-{c}_b\right)/{A}_b\right)\right)=2\delta {\wp}_o{\wp}_b\left({c}_o-{c}_b\right)\left(\left(I-{c}_o\right)/{A}_o+\left(I-{c}_b\right)/{A}_b\right)\end{array}} $$

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