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Integrating a learned probabilistic model with energy functional for ultrasound image segmentation

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Abstract

The segmentation of ultrasound (US) images is steadily growing in popularity, owing to the necessity of computer-aided diagnosis (CAD) systems and the advantages that this technique shows, such as safety and efficiency. The objective of this work is to separate the lesion from its background in US images. However, most US images contain poor quality, which is affected by the noise, ambiguous boundary, and heterogeneity. Moreover, the lesion region may be not salient amid the other normal tissues, which makes its segmentation a challenging problem. In this paper, an US image segmentation algorithm that combines the learned probabilistic model with energy functionals is proposed. Firstly, a learned probabilistic model based on the generalized linear model (GLM) reduces the false positives and increases the likelihood energy term of the lesion region. It yields a new probability projection that attracts the energy functional toward the desired region of interest. Then, boundary indicator and probability statistical–based energy functional are used to provide a reliable boundary for the lesion. Integrating probabilistic information into the energy functional framework can effectively overcome the impact of poor quality and further improve the accuracy of segmentation. To verify the performance of the proposed algorithm, 40 images are randomly selected in three databases for evaluation. The values of DICE coefficient, the Jaccard distance, root-mean-square error, and mean absolute error are 0.96, 0.91, 0.059, and 0.042, respectively. Besides, the initialization of the segmentation algorithm and the influence of noise are also analyzed. The experiment shows a significant improvement in performance.

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Funding

This work was supported by the Natural Science Foundations of China (grant number 61801202) and Dalian Youth Science and Technology Star (grant number 2019RQ021).

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Authors and Affiliations

  1. Department of Computing and Information Technology, Liaoning Normal University, Dalian City, Liaoning Province, China

    Lingling Fang, Lirong Zhang & Yibo Yao

  2. Nanchang Institute of Technology, City, Nanchang, Jiangxi Province, China

    Lingling Fang

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  1. Lingling Fang
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  2. Lirong Zhang
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  3. Yibo Yao
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Correspondence to Lingling Fang.

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Appendices

Appendix 1

In this paper, the proposed optimal energy functional is defined as

$$ {\phi}^{\ast }=\arg \underset{\phi }{\operatorname{inf}}\left\{E\left(\phi \right)\right\} $$
(12)

where

$$ {\displaystyle \begin{array}{l}E\left(\phi \right)={E}_b\left(\phi \right)+{E}_p\left(\phi \right)\\ {}={\int}_{\Omega}\left[\cos \left(\pi \cdot \boldsymbol{\upzeta} +1/2\pi \right)+1\right]d\Omega +{\int}_{\Omega}\sqrt{{\boldsymbol{\upzeta}}_{\mathbf{1}}\cdot {\boldsymbol{\upzeta}}_{\mathbf{2}}}d\Omega \end{array}} $$
(13)

Differentiating (3) and (4) with respect to ϕ, we can obtain

$$ {\displaystyle \begin{array}{l}\partial {\boldsymbol{\upzeta}}_1/\partial \phi =\delta \left(\phi \right)\cdot \int \wp \cdot H\left(\phi \right)d\Omega -\wp \cdot \delta \left(\phi \right)\cdot \int H\left(\phi \right)d\Omega /{\left(\int H\left(\phi \right)d\Omega \right)}^2\\ {}=\delta \left(\phi \right)\cdot \left({\boldsymbol{\upzeta}}_1-\wp \right)/\int H\left(\phi \right)d\Omega =\delta \left(\phi \right)\cdot \left({\boldsymbol{\upzeta}}_1-\wp \right)/{A}_{+}\end{array}} $$
(14)

1.1 and

$$ {\displaystyle \begin{array}{l}\partial {\boldsymbol{\upzeta}}_2/\partial \phi =-\delta \left(\phi \right)\cdot \int \wp \cdot \left(1-H\left(\phi \right)\right)d\Omega -\wp \cdot \delta \left(\phi \right)\cdot \int \left(1-H\left(\phi \right)\right)d\Omega /{\left(\int \left(1-H\left(\phi \right)\right)d\Omega \right)}^2\\ {}=-\delta \left(\phi \right)\cdot \left(\wp +{\boldsymbol{\upzeta}}_2\right)/\int \left(1-H\left(\phi \right)\right)d\Omega =-\delta \left(\phi \right)\cdot \left(\wp +{\boldsymbol{\upzeta}}_2\right)/{A}_{-}\end{array}} $$
(15)

Next, the concrete derivation process is as follows:

$$ \partial {E}_b\left(\phi \right)/\partial \phi =\pi \cdot \sin \left(\pi \cdot \boldsymbol{\upzeta} +1/2\pi \right)\cdot \partial \boldsymbol{\upzeta} /\partial \phi $$
(16)

1.2 and

$$ {\displaystyle \begin{array}{l}\partial {E}_p\left(\phi \right)/\partial \phi \\ {}=1/2{\left({\boldsymbol{\upzeta}}_1{\boldsymbol{\upzeta}}_2\right)}^{\hbox{-} 1/2}\cdot \left(\partial {\boldsymbol{\upzeta}}_1/\partial \phi \cdot {\boldsymbol{\upzeta}}_2+\partial {\boldsymbol{\upzeta}}_2/\partial \phi \cdot {\boldsymbol{\upzeta}}_1\right)\\ {}=1/2\left(\sqrt{{\boldsymbol{\upzeta}}_2/{\boldsymbol{\upzeta}}_1}\cdot \partial {\boldsymbol{\upzeta}}_1/\partial \phi +\sqrt{{\boldsymbol{\upzeta}}_1/{\mathrm{P}}_2}\cdot \partial {\boldsymbol{\upzeta}}_2/\partial \phi \right)\end{array}} $$
(17)

By substituting (8) and (9) into (7) and combining the corresponding terms, we can obtain

$$ {\displaystyle \begin{array}{l}\partial E\left(\phi \right)/\partial \phi =\partial {E}_b\left(\phi \right)/\partial \phi +\partial {E}_p\left(\phi \right)/\partial \phi \\ {}=-\pi \cdot \sin \left(\pi \cdot \boldsymbol{\upzeta} +1/2\pi \right)\cdot \partial \boldsymbol{\upzeta} /\partial \phi +\delta \left(\phi \right)\cdot V\end{array}} $$
(18)

where

$$ {\displaystyle \begin{array}{l}V=1/2\left(\sqrt{{\boldsymbol{\upzeta}}_2/{\boldsymbol{\upzeta}}_1}\cdot \left({\boldsymbol{\upzeta}}_1-\wp \right)/A+-\sqrt{{\boldsymbol{\upzeta}}_1/{\boldsymbol{\upzeta}}_2}\cdot \left(\wp +{\boldsymbol{\upzeta}}_2\right)/A-\right)\\ {}=1/2\sqrt{{\boldsymbol{\upzeta}}_2/{\boldsymbol{\upzeta}}_1}\cdot {\boldsymbol{\upzeta}}_1/A+-1/2\sqrt{{\boldsymbol{\upzeta}}_2/{\boldsymbol{\upzeta}}_1}\cdot \wp /A+-1/2\sqrt{{\boldsymbol{\upzeta}}_1/{\boldsymbol{\upzeta}}_2}\cdot \wp /A--1/2\sqrt{{\boldsymbol{\upzeta}}_1/{\boldsymbol{\upzeta}}_2}\cdot {\boldsymbol{\upzeta}}_2/A-\\ {}=1/2\sqrt{{\boldsymbol{\upzeta}}_1{\boldsymbol{\upzeta}}_2}\cdot 1/A+-1/2\sqrt{{\boldsymbol{\upzeta}}_1{\boldsymbol{\upzeta}}_2}\cdot 1/A--1/2\cdot \wp \left(1/A+\cdot \sqrt{{\boldsymbol{\upzeta}}_2/{\boldsymbol{\upzeta}}_1}+1/A-\cdot \sqrt{{\boldsymbol{\upzeta}}_1/{\boldsymbol{\upzeta}}_2}\right)\\ {}=1/2{E}_p\left(\phi \right)\cdot \left(1/A+-1/A-\right)-1/2\cdot \wp \left(1/A+\cdot \sqrt{{\boldsymbol{\upzeta}}_2/{\boldsymbol{\upzeta}}_1}+1/A-\cdot \sqrt{{\boldsymbol{\upzeta}}_1/{\boldsymbol{\upzeta}}_2}\right)\end{array}} $$
(19)

Appendix 2

Theorem 1.

The energy function (9) is uniformly bounded in the Sobolev space Wlp(Ω).

Let Ω denote the bounded open subset in space R, and Γ be the locally integrable function in Sobolev space Wlp(Ω) [35] (Sobolev space is such a vector space of functions that equipped with a series of lp-norms). Then, the energy function (9) is set as

$$ \Gamma =\sup \left\{\left.\int \left({E}_b\left(\phi \right)+{E}_p\left(\phi \right)\right)\cdot \nabla \phi d\Omega \right|\phi =\left({\phi}_1,{\phi}_2,\cdots, {\phi}_N\right)\in {W}^{01}{\left(\phi \right)}^N,{\left|\phi \right|}_{W^{\infty}\left(\Omega \right)}<1\right\} $$
(20)

where dΩ is the Lebesgue measure Ω = sup {Ω+, Ω−}, and

$$ \nabla \phi =\sum \limits_{i=1}^N\partial {\phi}_i/\partial {x}_i. $$
(21)

Then, we can get that Γ ∈ W(Ω), ∇Γ ∈ W1(Ω), i.e.,

$$ -\int \left({E}_b\left(\phi \right)+{E}_p\left(\phi \right)\right)\cdot \nabla \phi d\Omega =\int \left(\nabla \left({E}_b\left(\phi \right)+{E}_p\left(\phi \right)\right)\right)\cdot \phi d\Omega $$
(22)

and the bounded variation space BV(Ω) is defined as

$$ BV\left(\Omega \right)=\left\{\left.\phi \right|\phi \in {W}^1\left(\Omega \right)\kern0.4em or\kern0.4em \Gamma \in {L}^1\left(\Omega \right)\right\}. $$
(23)

By the characteristics of the BV, one can get that if ϕ ∈ BV(Ω), then

$$ \Gamma \left({E}_b,{E}_p\right)={\int}_{-\infty}^{+\infty }{W}^1\left(\partial {\Omega}_{\sigma}\right) d\sigma . $$
(24)

Here, ∂Ωσ is the boundary, and W1(∂Ωσ) denotes the length of ∂Ωσ.Therefore, one can get that the energy function (9) is uniformly bounded in the Sobolev space Wlp(Ω).

1.1 Appendix 3

Theorem 2.

The convergence value is the minimum in the energy function (9).

Proof. From Theorem 1, one can get that there exists a minimal sequence {ϕn} ∈ Wlp(Ω), n ∈ Z. By the property of Sobolev Space Wlp(Ω), we can get

$$ \underset{n\to \infty }{\lim }E\left({\phi}_n\right)=\underset{w\to {H}^2(G)}{\operatorname{inf}}E\left(\phi \right) $$
(25)

Here, there exists a convergent subsequence ϕn that converges to ϕ, that is ϕn → ϕ. By the mandatory of function δ(ϕ) ⋅ V and the characteristics of a trigonometric function cos, we can get that ϕn is convergent to ϕ. By Fatou’s lemma, we can get \( \phi \le \underset{n\to \infty }{\lim}\operatorname{inf}{\phi}_n \), and ϕ convergences, and there exists a minimum. In conclusion, the energy function (9) is uniformly bounded, convergences, and there exists a minimum.

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Fang, L., Zhang, L. & Yao, Y. Integrating a learned probabilistic model with energy functional for ultrasound image segmentation. Med Biol Eng Comput 59, 1917–1931 (2021). https://doi.org/10.1007/s11517-021-02411-0

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