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 twostate quantum system which allowed efficient singleobject segmentation. However, the extension of the framework to multiobject segmentation would be highly complex and computationally expensive. In this paper, we propose a classical image segmentation algorithm inspired by the continuousvariable quantum theory that overcomes the challenges in extending the framework to multiobject 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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Appendices
Appendix: 1. Proof of Eq. 16
It is required to prove that given the Hamiltonian
such that
and the initial state
then the final state obtained after evolution is a coherent state
The Hamiltonian in Eq. 28 can be written in the form
where
is the free field Hamiltonian which is time independent, while
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
where
is the free field evolution operator. The interaction potential in the interaction picture V_{I}(t) is related to the Schrödinger’s picture V_{0} via the transformation
By substituting Eqs. 36 in Eq. 37, we get
This equation is expanded as
The operator part in each of the two terms is in the form e^{sA}Be^{−sA}, 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 BakerCampbellHausdorff (BCH) formula
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
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
where the last equation comes from the Taylor expansion of the exponential function. By taking the adjoint we obtain
Now we can insert the results into Eq. 39 to get
This interaction part does not generally commute in time [V_{I}(t_{1}),V_{I}(t_{2})]≠ 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 f_{0} chosen to be real, we obtain the interaction Hamiltonian in the form
which clearly commutes in time, so we can obtain a closedform solution in the interaction picture
We can now return back to Schrödinger’s picture for the initial and final states by using Eq. 35
which is equivalent to
Since ψ(0)〉_{S} = 0〉, and \(U_{0}^{\dagger }(0)=1\), therefore
For two operators A and B, if [A,B]≠ 0 and [A, [A,B]] = [B, [A,B]] = 0, then the following identity holds
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
Rearranging the nonoperator terms,
The first term is an overall phase shift that does not change the state, so it can be neglected to get
The last exponential can be expanded as a Taylor series:
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
Again expanding the last exponential as a Taylor series we get
By the repeated application of the creation operator on the vacuum state and using Eq. 2,
Due to linearity, we can operate on the series term by term to get
Given that n〉 are eigenstates of the number operator \(\hat {n}=\hat {a}^{\dagger } \hat {a}\), with eigenvalues n, the expression becomes
Simplifying this expression, we get
Now let
then the final state becomes
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 β_{1}〉 and β_{2}〉 with equal magnitudes and different phases (i.e lying on the same circle in the complex plane) satisfy that
It is required to prove that scaling the reference states do not change that relation. Evaluating the fidelity yields
or,
This is equivalent to
Since β_{1}^{2} = β_{2}^{2}, then
If scaling is done, such that β_{1} → γβ_{1} and β_{2} → γβ_{2} 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., ElRafei, A. & Zhou, RG. A continuousvariable quantuminspired algorithm for classical image segmentation. Quantum Mach. Intell. 1, 97–111 (2019). https://doi.org/10.1007/s42484019000092
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Keywords
 Quantuminspired algorithms
 Coherent states
 Quantum harmonic oscillator
 Signal processing