Coarse-to-fine tuning knowledgeable system for boundary delineation in medical images

Abstract

Medical ultrasound image segmentation is crucial to the clinical diagnosis of planning for medical diseases. However, this task is challenging because of the missing/ambiguous edges and inhomogeneous intensity distribution of organs usually noted in ultrasound images. In this study, we devised a new coarse-to-refined architecture for different organ segmentation tasks. Our presented model has four merits: first, our work exploits the benefit of artificial intelligence algorithm to intelligently locate the target area and the feature of the principal curve to intelligently approach the center of data points in the refinement step. Second, we designed an enhanced polygon tracking model to increase our algorithm’s accuracy and efficiency. Third, to ensure population diversity and achieve optimal model initialization, we improved the traditional quantum evolution network both the numerous operator and global optimum search algorithm. Fourth, we devised an interpretable mathematical mapping function to smoothen the contour of the region of interest, which is expressed through the neural network model parameters. Outcomes on different types of datasets indicate that our developed model achieves excellent segmentation capability, yielding an average Dice similarity coefficient, Jaccard similarity coefficient, and accuracy of 94.1% ± 3.9%, 92.4% ± 4.7%, and 93.6% ± 4.1%, respectively.

Graphical abstract

Appendix

1.1 Quantum computing

The most important theory of quantum computing is to transform the relative phase between quanta using the ground states to compensate for the interference of the superposition state through the quantum gate. Here are some important concepts of quantum computing.

Chromosome coding based on real numbers

A quantum bit is the smallest unit of information saved in a quantum computer with two states (“0” and “1” states). Let α and β be complex numbers that denote the probability amplitudes of the “0” and “1” states, respectively. Meanwhile, the likelihood of measuring |0〉 and |1〉 are |α|² and |β|², respectively. The quantum bit is denoted as follows:

$$\left| \psi \right\rangle =\alpha \left| 0 \right\rangle +\beta \left| 1 \right\rangle$$

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where it satisfies the rule that: |α|² + |β|² = 1.

Q-bit representation

The main purpose of Q-bit representation is to indicate a linear superposition of states. Composed of a string of m Q-bits, the Q-bit individual set is denoted as follows.

$$q=\left[ {\left. {\begin{array}{*{20}{c}} {{\alpha _1}} \\ {{\beta _1}} \end{array}} \right|\left. {\begin{array}{*{20}{c}} {{\alpha _2}} \\ {{\beta _2}} \end{array}} \right|\left. {\begin{array}{*{20}{c}} \cdots \\ \cdots \end{array}} \right|\begin{array}{*{20}{c}} {{\alpha _m}} \\ {{\beta _m}} \end{array}} \right]$$

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To better understand this, suppose that in the two-Q-bit system, there exists an individual $\left[ {\left. {\begin{array}{*{20}{c}} {1/\sqrt 2 } \\ {1/\sqrt 2 } \end{array}} \right|\begin{array}{*{20}{c}} {1/2} \\ {\sqrt 3 /2} \end{array}} \right]$ whose states are shown as $\frac{{\sqrt 2 }}{4}\left| {00} \right\rangle +\frac{{\sqrt 6 }}{4}\left| {01} \right\rangle +\frac{{\sqrt 2 }}{4}\left| {10} \right\rangle +\frac{{\sqrt 6 }}{4}\left| {11} \right\rangle$. Therefore, we can calculate the corresponding probabilities of each state $\left| {00} \right\rangle ,\left| {01} \right\rangle ,\left| {10} \right\rangle ,\left| {11} \right\rangle$ with 1/8, 3/8, 1/8, and 3/8, respectively. Overall, this individual includes information on four states.

Quantum rotation gate

The quantum rotation gate is very important to update the quantum gate operation, as it has a great impact on improving the performance of QEN. Using the rotation angle$\Delta {\theta _i},i=1,\dots,m$, the quantum rotation gate $R({\theta _i})$ in QEN is defined as $R({\theta _i})=\left[ {\begin{array}{cc} {\cos (\Delta {\theta _i})}& -{\sin (\Delta {\theta _i})} \\ {\sin (\Delta {\theta _i})}&{\cos (\Delta {\theta _i})} \end{array}} \right]$.

Quantum rotation space transformation

Using the quantum rotation gate $R({\theta _i})$, the Q-bit of an individual is calculated as $\left[ {\begin{array}{*{20}{c}} {\alpha _{i}^{\prime}} \\ {\beta _{i}^{\prime}} \end{array}} \right]=U\left( {\Delta {\theta _i}} \right)\left[ {\begin{array}{*{20}{c}} {{\alpha _i}} \\ {{\beta _i}} \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} {\cos (\Delta {\theta _i})}&{ - \sin (\Delta {\theta _i})} \\ {\sin (\Delta {\theta _i})}&{\cos (\Delta {\theta _i})} \end{array}} \right]\bullet \left[ {\begin{array}{*{20}{c}} {{\alpha _i}} \\ {{\beta _i}} \end{array}} \right]$.

1.2 QEN

We suppose that QEN has a population of Q-bit individuals, $Q(g)=(q_{1}^{g},q_{2}^{g},\dots,q_{n}^{g})$ at generation g, n is the population size, and $q=\{ q_{{i1}}^{g},q_{{i2}}^{g},..,q_{{im}}^{g}\} =\left[ {\left. {\begin{array}{c} {\alpha _{{i1}}^{g}} \\ {\beta _{{i1}}^{g}} \end{array}} \right|\left. {\begin{array}{c} {\alpha _{{i2}}^{g}} \\ {\beta _{{i2}}^{g}} \end{array}} \right|\left. {\begin{array}{*{20}{c}} \cdots \\ \cdots \end{array}} \right|\begin{array}{*{20}{c}} {\alpha _{{im}}^{g}} \\ {\beta _{{im}}^{g}} \end{array}} \right]$ is a Q-bit individual set.

(Step 1) Quantum mutation operation

The mutation operator is expressed as follows.

$$v_{{i,j}}^{g}=\alpha _{{z1,j}}^{g}+F(\alpha _{{z2,j}}^{g} - \alpha _{{z3,j}}^{g})$$

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where both i and j are integers and are in the range of [1, n] and [1, m], respectively. The integers z1, z2, and z3 are selected randomly and are in the range of [1, n]. The control variable F, which controls the amplification of the differential variation, is determined within the range of [0, 1].

(Step 2) Quantum crossover operation

The crossover operator is denoted as follows:

$$u_{{i,j}}^{{g+1}}=\left\{ {\begin{array}{*{20}{c}} {v_{{i,j}}^{g}{\text{ }}if(rand[0,1] \leq CR)} \\ {\alpha _{{i,j}}^{g}{\text{ }}if(rand[0,1]>CR)} \end{array}} \right.$$

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where CR is the crossover probability and lies within the range of [0, 1]. If CR is too large, the QEN turns to fast convergence. If CR is too small, the model needs to spend more time looking for the minimum problem in the QEN.

(Step 3) Quantum selection operation

This step is adopted to decide whether or not the outcome in this step should become the best Q-bit individual of generation g + 1. A selection scheme is implemented between a Q-bit (or binary) individual and its crossover vector. In this stage, a greedy selection operator is adopted, which is denoted as follows.

$$\alpha _{i}^{{g+1}}=\left\{ {\begin{array}{*{20}{c}} {u_{i}^{g}{\text{ }}if{\text{ }}f(u_{i}^{g})<f(\alpha _{i}^{g})} \\ {\alpha _{i}^{g}{\text{ }}otherwise} \end{array}} \right.$$

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and then,

$$q_{i}^{{g+1}}=\left\{ {\begin{array}{*{20}{c}} {u_{i}^{{{\text{g+}}1}}{\text{ }}if{\text{ }}f(u_{i}^{{g+1}})<f(\alpha _{i}^{{g+1}})} \\ {\alpha _{i}^{{g+1}}{\text{ }}otherwise} \end{array}} \right.$$

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