A continuous-variable quantum-inspired algorithm for classical image segmentation

Abstract

The probabilistic nature of quantum particles, state space, and the superposition principle are among the important concepts in quantum mechanics. A framework was previously developed by the authors that allowed to take advantage of these quantum aspects in the field of image processing. This was done by modeling each image’s pixel by a two-state quantum system which allowed efficient single-object segmentation. However, the extension of the framework to multi-object segmentation would be highly complex and computationally expensive. In this paper, we propose a classical image segmentation algorithm inspired by the continuous-variable quantum theory that overcomes the challenges in extending the framework to multi-object segmentation. By associating each pixel with a quantum harmonic oscillator, the space of coherent states becomes continuous. Thus, each pixel can evolve from an initial state to any of the continuous coherent states under the influence of an external resonant force. The Hamiltonian operator is designed to account for this force and is derived from the features extracted at the pixel. Therefore, the system evolves from an initial ground state to a final coherent state depending on the image features. Finally by calculating the fidelity between the final state and a set of reference states representing the objects in the image, the state with the highest fidelity is selected. The collective states of all pixels produce the final segmentation. The proposed method is tested on a database of synthetic and natural images, and compared with other methods. Average sensitivity and specificity of 97.86% and 99.61% were obtained respectively indicating the high segmentation accuracy of the algorithm.

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Funding

AY is supported by an Australian Government Research Training Program Scholarship. This work is supported by the National Natural Science Foundation of China under Grant No. 61463016, 61763014, National Key R&D Plan under Grant No. SQ2018YFC120002 and “Science and technology innovation action plan” of Shanghai in 2017 under Grant No. 17510740300.

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Correspondence to Akram Youssry.

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Appendices

Appendix: 1. Proof of Eq. 16

It is required to prove that given the Hamiltonian

$$ H=\hbar\omega\left( \hat{a}^{\dagger}\hat{a}+\frac{1}{2}\right) + \hbar\left( f(t)\hat{a}+f^{*}(t)\hat{a}^{\dagger}\right), $$
(28)

such that

$$ f(t)=f_{0} e^{i\omega t}, $$
(29)

and the initial state

$$ |\psi(0)\rangle=|0\rangle, $$
(30)

then the final state obtained after evolution is a coherent state

$$ |\alpha\rangle=e^{-\frac{1}{2}|\alpha|^{2}} \sum\limits_{n=0}^{\infty} {\frac{\alpha^{n}}{\sqrt{n!}}}|n\rangle. $$
(31)

The Hamiltonian in Eq. 28 can be written in the form

$$ H=H_{0}+V, $$
(32)

where

$$ H_{0}=\hbar\omega\left( \hat{a}^{\dagger}\hat{a}+\frac{1}{2}\right) $$
(33)

is the free field Hamiltonian which is time independent, while

$$ V=\hbar\left( f(t)\hat{a}+f^{*}(t)\hat{a}^{\dagger}\right) $$
(34)

is the interaction potential which is time dependent. As it is known in the theory of quantum mechanics, there are three pictures (representations) of states and operators:

  • The Schrödinger picture in which the states are time dependent, while the operators are time independent.

  • The Heisenberg picture in which the states are time independent, while the operators are time dependent

  • The Dirac (interaction) picture, where both states and operators are time dependent.

The three representations are equivalent to each other and can be converted easily from one picture to another. The state represented in the interaction picture |ψ(t)〉I is related to Schrödinger’s picture |ψ(t)〉S via the transformation

$$ |\psi(t)\rangle_{I}=U_{0}^{\dagger} (t)|\psi(t)\rangle_{S}, $$
(35)

where

$$ U_{0}=e^{-\frac{i}{\hbar}H_{0} t}=e^{-i\omega t \left( \hat{a}^{\dagger}\hat{a}+\frac{1}{2}\right)} $$
(36)

is the free field evolution operator. The interaction potential in the interaction picture VI(t) is related to the Schrödinger’s picture V0 via the transformation

$$ V_{I}(t)=U_{0}^{\dagger} (t) V_{0} U_{0}(t). $$
(37)

By substituting Eqs. 36 in Eq. 37, we get

$$ V_{I}(t)=e^{i\omega t \left( \hat{a}^{\dagger}\hat{a}+\frac{1}{2}\right)} \hbar\left( f(t)\hat{a}+f^{*}(t)\hat{a}^{\dagger}\right) e^{-i\omega t \left( \hat{a}^{\dagger}\hat{a}+\frac{1}{2}\right)}. $$
(38)

This equation is expanded as

$$ V_{I}(t)=\hbar f(t) e^{i\omega t \hat{a}^{\dagger}\hat{a}}\hat{a}e^{-i\omega t \hat{a}^{\dagger}\hat{a}} +\hbar f^{*}(t) e^{i\omega t \hat{a}^{\dagger}\hat{a}}\hat{a}^{\dagger} e^{-i\omega t \hat{a}^{\dagger}\hat{a}}. $$
(39)

The operator part in each of the two terms is in the form esABesA, where s = iωt, \(A=\hat {a}^{\dagger }\hat {a}\), and \(B=\hat {a}\) for the first term and \(\hat {a}^{\dagger }\) for the second term. This allows us to use the Baker-Campbell-Hausdorff (BCH) formula

$$ \begin{array}{@{}rcl@{}} e^{sA} B e^{-sA}&=&B+[A,B]s+[A,[A,B]]\frac{s^{2}}{2!}\\ &&+[A,[A,[A,B]]]\frac{s^{3}}{3!}+\cdots. \end{array} $$
(40)

So, we need to calculate these nested commutators. Rewriting the commutator relation in Eq. 5, in the form \(\hat {a}\hat {a}^{\dagger }-\hat {a}^{\dagger }\hat {a}=1\), we obtain

$$ [\hat{a}^{\dagger} \hat{a},\hat{a}]=\hat{a}^{\dagger} \hat{a} \hat{a}- \hat{a} \hat{a}^{\dagger} \hat{a}=\hat{a}^{\dagger} \hat{a} \hat{a}-(1+\hat{a}^{\dagger} \hat{a})\hat{a}=-\hat{a}. $$
(41)
$$ [\hat{a}^{\dagger} \hat{a},[\hat{a}^{\dagger} \hat{a},\hat{a}]]=[\hat{a}^{\dagger} \hat{a},-\hat{a}]=-[\hat{a}^{\dagger} \hat{a},\hat{a}]=\hat{a}. $$
(42)

and so on. The nested commutators form an alternating series between \(\hat {a}\) and \(-\hat {a}\). By substituting the result of Eq. 41 in the BCH formula of Eq. 40, we get

$$ \begin{array}{@{}rcl@{}} e^{i\omega t \hat{a}^{\dagger}\hat{a}}\hat{a}e^{-i\omega t \hat{a}^{\dagger}\hat{a}}\!&=&\hat{a}-(i\omega t)\hat{a}+\frac{(i\omega t)^{2}}{2!}\hat{a}-\frac{(i\omega t)^{3}}{3!}\hat{a}+\cdots \end{array} $$
(43)
$$ \begin{array}{@{}rcl@{}} &=&\hat{a}\!\left( \!1 + (-i\omega t) + \frac{(-i\omega t)^{2}}{2!} + \frac{(-i\omega t)^{3}}{3!} + {\cdots} \!\right) \end{array} $$
(44)
$$ \begin{array}{@{}rcl@{}} &=&\hat{a}e^{-i\omega t}, \end{array} $$
(45)

where the last equation comes from the Taylor expansion of the exponential function. By taking the adjoint we obtain

$$ e^{i\omega t \hat{a}^{\dagger}\hat{a}}\hat{a}^{\dagger} e^{-i\omega t \hat{a}^{\dagger}\hat{a}}=\hat{a}^{\dagger} e^{i\omega t}. $$
(46)

Now we can insert the results into Eq. 39 to get

$$ V_{I}(t)=\hbar\left( f(t)\hat{a}e^{-i\omega t}+f^{*}(t)\hat{a}^{\dagger} e^{i\omega t}\right). $$
(47)

This interaction part does not generally commute in time [VI(t1),VI(t2)]≠ 0, which makes the exact solution difficult to obtain; however, if the external force is resonant with the oscillator so that it takes the form in Eq. 29, and f0 chosen to be real, we obtain the interaction Hamiltonian in the form

$$ V_{I}(t)=\hbar f_{0} \left( \hat{a}+\hat{a}^{\dagger}\right), $$
(48)

which clearly commutes in time, so we can obtain a closed-form solution in the interaction picture

$$ \begin{array}{@{}rcl@{}} |\psi(t)\rangle_{I}=U_{I}|\psi(0)\rangle_{I}&=&e^{-\frac{i}{\hbar}V_{I} t}|\psi(0)\rangle_{I} \end{array} $$
(49)
$$ \begin{array}{@{}rcl@{}} &=&e^{-i f_{0} t (\hat{a} +\hat{a}^{\dagger})}|\psi(0)\rangle_{I}. \end{array} $$
(50)

We can now return back to Schrödinger’s picture for the initial and final states by using Eq. 35

$$ U_{0}^{\dagger}(t)|\psi(t)\rangle_{S}=e^{-i f_{0} t (\hat{a} +\hat{a}^{\dagger})}U_{0}^{\dagger}(0)|\psi(0)\rangle_{S}, $$
(51)

which is equivalent to

$$ |\psi(t)\rangle_{S}=U_{0}(t) e^{-i f_{0} t (\hat{a} +\hat{a}^{\dagger})}U_{0}^{\dagger}(0)|\psi(0)\rangle_{S}. $$
(52)

Since |ψ(0)〉S = |0〉, and \(U_{0}^{\dagger }(0)=1\), therefore

$$ |\psi(t)\rangle_{S}=e^{-i \omega t \left( \hat{a}^{\dagger} \hat{a} +\frac{1}{2}\right)} e^{-i f_{0} t (\hat{a} +\hat{a}^{\dagger})}|0\rangle. $$
(53)

For two operators A and B, if [A,B]≠ 0 and [A, [A,B]] = [B, [A,B]] = 0, then the following identity holds

$$ e^{A+B}=e^{A} e^{B} e^{-\frac{1}{2}[A,B]}=e^{B} e^{A} e^{\frac{1}{2}[A,B]}. $$
(54)

The second exponential clearly satisfies these two conditions with \(A=(-i f_{0} t)\hat {a}\) and \(B=(-i f_{0} t)\hat {a}^{\dagger }\); consequently the final state becomes

$$ |\psi(t)\rangle_{S}=e^{-i \omega t \left( \hat{a}^{\dagger} \hat{a} +\frac{1}{2}\right)} e^{-i f_{0} t \hat{a}^{\dagger}} e^{-i f_{0} t \hat{a}} e^{-\frac{(f_{0} t)^{2}}{2}}|0\rangle. $$
(55)

Rearranging the non-operator terms,

$$ |\psi(t)\rangle_{S}=e^{-\frac{i \omega t}{2}} e^{-\frac{(f_{0} t)^{2}}{2}} e^{-i \omega t \hat{a}^{\dagger} \hat{a}} e^{-i f_{0} t \hat{a}^{\dagger}} e^{-i f_{0} t \hat{a}} |0\rangle. $$
(56)

The first term is an overall phase shift that does not change the state, so it can be neglected to get

$$ |\psi(t)\rangle_{S}=e^{-\frac{(f_{0} t)^{2}}{2}} e^{-i \omega t \hat{a}^{\dagger} \hat{a}} e^{-i f_{0} t \hat{a}^{\dagger}} e^{-i f_{0} t \hat{a}} |0\rangle. $$
(57)

The last exponential can be expanded as a Taylor series:

$$ |\psi(t)\rangle_{S}=e^{-\frac{(f_{0} t)^{2}}{2}} e^{-i \omega t \hat{a}^{\dagger} \hat{a}} e^{-i f_{0} t \hat{a}^{\dagger}} \sum\limits_{n=0}^{\infty} {\frac{(-i f_{0} t)^{n}}{n!}\hat{a}^{n}} |0\rangle. $$
(58)

All the terms in the series vanish due to definition of annihilation operator except for the first term (n = 0) which is equal to 1 so

$$ |\psi(t)\rangle_{S}=e^{-\frac{(f_{0} t)^{2}}{2}} e^{-i \omega t \hat{a}^{\dagger} \hat{a}} e^{-i f_{0} t \hat{a}^{\dagger}} |0\rangle. $$
(59)

Again expanding the last exponential as a Taylor series we get

$$ |\psi(t)\rangle_{S}=e^{-\frac{(f_{0} t)^{2}}{2}} e^{-i \omega t \hat{a}^{\dagger} \hat{a}} \sum\limits_{n=0}^{\infty} {\frac{(-i f_{0} t)^{n}}{n!}(\hat{a}^{\dagger})^{n}} |0\rangle. $$
(60)

By the repeated application of the creation operator on the vacuum state and using Eq. 2,

$$ |\psi(t)\rangle_{S}=e^{-\frac{(f_{0} t)^{2}}{2}} e^{-i \omega t \hat{a}^{\dagger} \hat{a}} \sum\limits_{n=0}^{\infty} {\frac{(-i f_{0} t)^{n}}{n!}\sqrt{n!}}|n\rangle. $$
(61)

Due to linearity, we can operate on the series term by term to get

$$ |\psi(t)\rangle_{S}=e^{-\frac{(f_{0} t)^{2}}{2}} \sum\limits_{n=0}^{\infty} {\frac{(-i f_{0} t)^{n}}{n!}\sqrt{n!}}e^{-i \omega t \hat{n}}|n\rangle. $$
(62)

Given that |n〉 are eigenstates of the number operator \(\hat {n}=\hat {a}^{\dagger } \hat {a}\), with eigenvalues n, the expression becomes

$$ |\psi(t)\rangle_{S}=e^{-\frac{(f_{0} t)^{2}}{2}} \sum\limits_{n=0}^{\infty} {\frac{(-i f_{0} t)^{n}}{n!}\sqrt{n!}}e^{-i \omega t n}|n\rangle. $$
(63)

Simplifying this expression, we get

$$ |\psi(t)\rangle_{S}=e^{-\frac{(f_{0} t)^{2}}{2}} \sum\limits_{n=0}^{\infty} {\frac{(-i f_{0} t e^{-i \omega t})^{n}}{\sqrt{n!}}}|n\rangle. $$
(64)

Now let

$$ \alpha = -i e^{-i \omega t} f_{0} t\to|\alpha|^{2}=(f_{0} t)^{2}, $$
(65)

then the final state becomes

$$ |\psi(t)\rangle_{S}=e^{-\frac{1}{2}|\alpha|^{2}} \sum\limits_{n=0}^{\infty} {\frac{\alpha^{n}}{\sqrt{n!}}}|n\rangle. $$
(66)

This is exactly the definition of the coherent state |α〉, which is required to prove.

Appendix: 2. Proof of invariance of scaling factor in the reference states

Given that the fidelity between a final state |α〉 and two reference states |β1〉 and |β2〉 with equal magnitudes and different phases (i.e lying on the same circle in the complex plane) satisfy that

$$ F_{1}>F_{2}. $$
(67)

It is required to prove that scaling the reference states do not change that relation. Evaluating the fidelity yields

$$ \langle{\alpha|\beta_{1}}\rangle>\langle{\alpha|\beta_{2}}\rangle, $$
(68)

or,

$$ e^{-\frac{1}{2}|\alpha-\beta_{1}|^{2}}>e^{-\frac{1}{2}|\alpha-\beta_{2}|^{2}}. $$
(69)

This is equivalent to

$$ \begin{array}{@{}rcl@{}} |\alpha-\beta_{1}|^{2} &<& |\alpha-\beta_{2}|^{2} \end{array} $$
(70)
$$ \begin{array}{@{}rcl@{}} (\alpha-\beta_{1})(\alpha-\beta_{1})^{*} &<& (\alpha-\beta_{2})(\alpha-\beta_{2})^{*} \end{array} $$
(71)
$$ \begin{array}{@{}rcl@{}} -\alpha\beta_{1}^{*}-\beta_{1}\alpha^{*}+|\beta_{1}|^{2} &<&-\alpha\beta_{2}^{*}-\beta_{2}\alpha^{*}+|\beta_{2}|^{2}. \end{array} $$
(72)

Since |β1|2 = |β2|2, then

$$ \begin{array}{@{}rcl@{}} -\alpha\beta_{1}^{*}-\beta_{1}\alpha^{*}&<&-\alpha\beta_{2}^{*}-\beta_{2}\alpha^{*} \end{array} $$
(73)
$$ \begin{array}{@{}rcl@{}} \frac{\alpha}{\alpha^{*}}&<&-\frac{\beta_{1}-\beta_{2}}{\beta_{1}^{*}-\beta_{2}^{*}}. \end{array} $$
(74)

If scaling is done, such that β1γβ1 and β2γβ2 for positive γ, then the above equation does not change. Consequently, decisions based on fidelity do not change. This can be generalized to any number of reference states.

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Youssry, A., El-Rafei, A. & Zhou, RG. A continuous-variable quantum-inspired algorithm for classical image segmentation. Quantum Mach. Intell. 1, 97–111 (2019). https://doi.org/10.1007/s42484-019-00009-2

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Keywords

  • Quantum-inspired algorithms
  • Coherent states
  • Quantum harmonic oscillator
  • Signal processing