Method for generating tomographic image using quantum computer

Abstract

The present invention relates to a method for generating a tomographic image using a quantum computer and, more specifically, to a method for generating a tomographic image by quantum computing using a specific algorithm having limited conditions. The present invention further emphasizes the robustness and precision of the algorithm, thereby increasing the possible of effective application in the field of electron tomography (ET) in which accurate reconstruction is essential. In addition, by replacing an existing optimization algorithm with a quantum optimization algorithm, the present invention can be applied with great utility in fields related to medical image diagnosis, especially in fields such as breast cancer and cardiovascular diagnosis.

Description

Quantum computer tomographic image generation method

[0001] The present invention relates to a method for generating tomographic images using a quantum computer, and more specifically, to a method for generating tomographic images using quantum computing that utilizes a specific algorithm having restrictive conditions.

[0002] Unless otherwise indicated in this specification, the contents described in this section are not prior art for the claims of this application, and are not to be recognized as prior art simply because they are included in this section.

[0003] Computed tomography (CT) is used as an important tool for analyzing the internal structure of objects without destroying them. CT is widely used in biology, archaeology, earth science, and materials science to analyze the internal structure of objects.

[0004] Depending on the size of the object, three major CT imaging systems have been developed: helical CT, electron tomography (ET), and synchrotron X-ray tomography. Depending on the object size, the helical CT system is suitable for tens of centimeters, ET is suitable for sizes ranging from several nanometers to tens of nanometers, and synchrotron X-ray tomography is suitable for sizes ranging from several hundred nanometers to several centimeters.

[0005] Tomography systems using cone-beam sources are inexpensive to manufacture but struggle to reconstruct sharp CT images, whereas parallel-beam tomography systems are more expensive to produce but can reconstruct sharp CT images. Until recently, three main algorithms were used for CT image reconstruction: iterative algorithms, the Fast Fourier Transform (FFT), and artificial intelligence (AI). While the FFT can reconstruct CT images more sharply than other algorithms, it has several limitations. Therefore, the FFT is primarily used for data reconstruction in synchrotron systems.

[0006] Quantum optimization algorithms have evolved alongside advancements in quantum annealing. While D-Wave quantum processing units (QPUs) can perform computations fast enough, they do not provide a sufficient number of logical qubits. To address this issue, hybrid solvers have been developed for D-Wave systems, where a classical computer and a quantum annealer work together. (C. McGeoch, P. Farre, and W. Bernoudy, “D-Wave hybrid solver service+ Advantage: Technology update,” Tech. Rep., 2020) Up to 1,000,000 logical qubits can be used when computing optimization problems using hybrid solvers.

[0007] Furthermore, quantum optimization algorithms are continuously evolving. Of particular note is the discovery of a quantum optimization algorithm for linear systems by the inventors. (Korean Patent Publication No. 10-2024-0172854)

[0008] However, existing quantum optimization CT algorithms have a limitation in that they cannot integrate other optimization concepts because they optimize only one QUBO model for projected data.

[0009] The present invention aims to solve the problem of reconstructing a superior tomographic image by newly applying a quantum-optimized CT algorithm using quantum computing to improve upon the problems of the conventional technology described above.

[0010] Therefore, the objective of the present invention is to provide a method for generating tomographic images of excellent quality using a quantum optimized CT algorithm under specific conditions utilizing quantum computing.

[0011] Another objective of the present invention is to provide a method for generating a quantum computer tomography image that is reconstructed into a superior image using a method of applying constraints to an acquired tomography image or a new algorithm that synthesizes two or more QUBO models.

[0012] The purpose of the present invention is not limited to the purposes stated above, and should be understood to include all purposes that can be inferred from the configuration of the invention described in the detailed description or claims of the present invention, or all purposes that can be achieved by the description or technical concept of the present invention.

[0013] In order to solve the problem of the present invention as described above, the present invention

[0014] a) a step of acquiring a sinogram in a quantum superposition state for a target image acquisition object, and

[0015] b) a step of converting the entire or any part of the acquired sinogram into an energy optimization model using quantum compressed sensing, which is a method for minimizing the total change amount of a tomographic image in a quantum superposition state, and

[0016] c) A step of reconstructing a tomographic image using an energy minimization algorithm by excluding the remaining untransformed parts using constraints or masking.

[0017] A method for generating a quantum computer tomographic image including is provided.

[0018] According to a preferred embodiment of the present invention, in order to obtain a sinogram of a quantum superposition state in step a), a method of representing each pixel of a single-layer image in a quantum superposition state based on the following Equation 12 or Equation 12a may be used.

[0019] [Equation 12]

[0020]

[0021] Here, is for a tomographic image of a quantum superposition state ( Represents the pixel at the ) position, and the above mathematical formula 12 is satisfied is from 0 It can represent all integers up to and including the attenuation coefficient of a particular light source for the sample, including the X-ray mass attenuation coefficient of the sample,

[0022] [Mathematical Formula 12a]

[0023]

[0024] Here, is a mistake, Depending on the combination of the sample's The N damping coefficients that It represents all cases that can be expressed.

[0025] According to a preferred embodiment of the present invention, the step of converting to an energy optimization model in step b) may include a step of shaping a QUBO model, an Ising model, or a quadratic function model for a sinogram, and a method of performing quantum compression sensing in two or more identical or different energy optimization models selected from the QUBO model, the Ising model, or the quadratic function model.

[0026] According to a preferred embodiment of the present invention, the algorithm for energy optimization may use some or all of the sinogram values ​​with low error in Equation 15 below and utilize a quantum compression sensing method, which is a method for minimizing the total change amount of a tomographic image of a quantum superposition state in Equation 32 below, to construct two energy optimization models F.

[0027] [Equation 15]

[0028]

[0029] Here And am.

[0030] [Mathematical Formula 32]

[0031]

[0032] Here is a CT image in a quantum superposition state It is a value for the position.

[0033] According to a preferred embodiment of the present invention, the constraint is a projection angle and For each sinogram value of the location, the undefined sinogram A real sinogram We can set constraints to be equal to , and use the "epsilon test" to apply constraints such as the following mathematical expressions 16 and 17.

[0034] [Equation 16]

[0035]

[0036] [Equation 17]

[0037]

[0038] Here, is a quantum superimposed sinogram It means location is of the preprocessed sinogram Represents, , And represents the sum of the hardware errors of a quantum computer or annealer or any of those capable of quantum computing and the allowable errors of the solution.

[0039] According to a preferred embodiment of the present invention, the acquired sinogram may include a step of preprocessing by making the parts without samples zero and readjusting the columns of the experimental sinogram so that each density changes continuously, or by adjusting the acquired sinogram so that the sum of each column is constant.

[0040] According to a preferred embodiment of the present invention, the masking process may include a masking method that identifies and excludes a change region in which a change including a horizontal or vertical direction is not continuous in the calculation of an energy optimization model.

[0041] According to a preferred embodiment of the present invention, a method may be included for calculating an energy optimization model from the following mathematical expression 28 or 29 using an acquired sinogram and a preprocessed experimental sinogram for a tomographic image of the quantum superposition state.

[0042] [Equation 28]

[0043]

[0044] [Equation 29]

[0045]

[0046] Here is a quantum superimposed sinogram It means location is of the preprocessed sinogram Represents, If it is 0, all terms are included as constraints. In this case, all terms of the sinogram can be used.

[0047] According to a preferred embodiment of the present invention, the calculation of the energy optimization model may include a method of linearly combining two or more identical or different energy optimization models using the following mathematical expression 35.

[0048] [Mathematical Formula 35]

[0049]

[0050] Here is a constant and is an energy optimization model for a specific calculation, and can contain an energy optimization model of quantum compressive sensing that minimizes the total variation within a CT image.

[0051] The method for generating a tomographic image according to the present invention can very effectively generate and acquire an excellent image when the sample size is larger than the CCD size or the region of interest is narrow.

[0052] In addition, it not only reduces the number of projection images required for tomographic reconstruction by more than 80%, but also fundamentally prevents errors that generate motion artifacts during CT scans.

[0053] In addition, according to the present invention, by applying a quantum optimization CT algorithm to Shepp-Logan test samples and using constraints of a CQM solver, the CQM solver can not only reduce computational load but also obtain an optimized image.

[0054] In particular, the algorithm of the present invention can address the problem of circular artifacts in images when reconstructing sinograms containing horizontal error regions by focusing on each sinogram pixel that is less susceptible to error and generating an energy-optimized model, such as the QUBO model, the Ising model, or a quadratic function model. Therefore, even if an energy-optimized model is generated by excluding unreliable regions from the sinogram, qubit information can still be obtained from other reliable regions.

[0055] In addition, while conventional classical algorithms often struggle to reconstruct within an angle range of less than 180°, the algorithm of the present invention operates effectively even in such a reduced angle range, making it highly applicable to electronic tomography.

[0056] Therefore, the present invention can increase the applicability in the fields of synchrotron radiation accelerators or electron tomography (ET) where accurate reconstruction is essential by further emphasizing the robustness and precision of the algorithm, and furthermore, by replacing the existing optimization algorithm with a quantum optimization algorithm to solve the problem of low clarity of cone beam or fan beam CT images, it has the effect of generating and providing excellent images, especially in the field of medical imaging diagnosis such as breast cancer and cardiovascular diagnosis.

[0057] The effects of the present invention are not limited to the effects described above, and should be understood to include all effects that can be inferred from the configuration of the invention described in the detailed description or claims of the present invention.

[0058] Figure 1 is a 5 × 5 sized Shepp-Logan sample image with integer pixel values ​​from 0 to 3.

[0059] Figure 2 is a sinogram of the sample image of Figure 1, where the values ​​inside the red box of the sinogram are used to construct an energy optimization model (QUBO model) and the remaining pixels are used to generate constraints, and the energy optimization model and constraints can be freely selected from the sinogram.

[0060] Figure 3 is a sample image of size 100 × 100 with zero padding applied, obtained by converting a 100 × 100 Shepp-Logan image into a binary image.

[0061] Figure 4 shows two images reconstructed using two algorithms from a 30x30 Shepp-Logan image with pixel values ​​ranging from 0 to 7, where (a) image is reconstructed using a hybrid solver, and (b) image is reconstructed using integer variables and a CQM solver, and in fact, (b) is identical to the original Shepp-Logan image.

[0062] Figure 5 illustrates an artificial horizontal error region in a sinogram that can cause circular artifacts in a reconstructed image and shows the reconstructed result from this sinogram, where (a) is a sinogram with artificial errors and a graph of the average difference for each consecutive row, and (b) is an image reconstructed from the sinogram via FFT.

[0063] FIG. 6 is a result of reconstructing a 100×100 cross-sectional image of a tooth sample by applying an algorithm to a sinogram having five artificially inserted horizontal error regions as shown in (a) of FIG. 5, wherein (a) is a solution image obtained by applying FFT and thresholding techniques to the original sinogram before adding artificial circular artificial shading, (b) is a segmented image obtained using the algorithm and a hybrid solver from the sinogram of (a) in FIG. 5, (c) is a post-processed image of (b), and (d) is a result of superimposing the solution image of (c) and the result of the present invention.

[0064] FIG. 7 shows experimental results for a limited angle for a 50×50 cross-sectional image at axial level 15 of a tooth sample, which is an embodiment of the present invention, wherein (a) is a sinogram created by dividing 0° to 180° into 100 equal parts; (b) is the result of applying FFT using only the first 50 projection images in the sinogram (a); (c) is the result of applying FFT using 60 projection images; (d) is a solution image obtained by applying FFT to the entire sinogram and using a threshold setting method; (e) to (g) are the results obtained by applying the algorithm to 50 projection images using a hybrid solver, the post-processing results, and the results overlaid with the solution image, respectively; and (h) to (j) are the results obtained by applying the algorithm to 60 projection images using a hybrid solver, the post-processing results, and the results overlaid with the solution image, respectively.

[0065] Figure 8 shows the results of a restricted angle experiment on a 50×50 Shepp-Logan binary sample image using zero pads, where (a) is a sinogram generated using only the first 25 projection images (less than 90°) that are evenly divided from 0° to 180° out of 50 projection images, (b) is the result of applying FFT to the sinogram of (a), and (c) is the result of applying the present algorithm to the sinogram of (a). This figure is also the original sample image used in the experiment.

[0066] The present invention will be described in more detail below as one embodiment.

[0067] The descriptions and configurations illustrated in the drawings in this specification are merely the most preferred embodiments of the invention and do not represent all technical aspects of the invention; therefore, it should be understood that various equivalents and modifications that may replace them may exist at the time of filing this application. The embodiments described in this invention should be understood as illustrative in all respects and not restrictive, and the scope of the invention is defined by the claims set forth below rather than by the detailed description, and all modifications or variations derived from the meaning and scope of the claims and equivalent concepts should be interpreted as being included within the scope of the invention.

[0068] The present invention relates to a method for generating tomographic images with excellent results using a quantum optimized computed tomography algorithm with constraints.

[0069] The formulation of the second-order unconstrained binary optimization (QUBO) model for quantum linear systems applied in this invention suggests that quantum optimization algorithms can play an important role in the fields of science and engineering.

[0070]

[0071] In the computed tomography (CT) image reconstruction method developed by the inventors, a QUBO model was developed by applying a quantum linear system to clearly show the internal structure of the sample. To date, due to the limitations of logic qubits, there have been no cases of reconstructing ultra-high-definition CT images using quantum optimization algorithms.

[0072] In the present invention, a quantum CT image reconstruction algorithm using a quantum linear system provides reliability regarding the accuracy of CT images by utilizing the difference between the global minimum energy and the theoretical minimum energy.

[0073] According to a preferred embodiment of the present invention, an algorithm capable of using two or more energy optimization models for CT image reconstruction is used.

[0074] The algorithm proposed in this invention can formulate multiple energy optimization models in CT image reconstruction and verify them using a hybrid solver of the D-Wave system, thereby significantly improving the reliability of tomographic images prepared for various purposes.

[0075] A preferred embodiment of the present invention utilizes a method of quantizing pixels of a tomographic image in a quantum superposition state into mass attenuation coefficients including absorption coefficients or scattering coefficients during the process of refining a tomographic image using a CT image reconstruction algorithm.

[0076] The method for generating a quantum computed tomography image according to the present invention preferably comprises:

[0077] a) a step of acquiring a sinogram in a quantum superposition state for a target image acquisition object, and

[0078] b) a step of converting the entire or any part of the acquired sinogram into an energy optimization model using quantum compressed sensing, which is a method for minimizing the total change amount of a tomographic image in a quantum superposition state, and

[0079] c) A step of reconstructing a tomographic image using an energy minimization algorithm by excluding the remaining untransformed parts using constraints or masking.

[0080] It is characterized by including

[0081] In the present invention, the term "energy optimization model" is interpreted to include at least the QUBO model, the Ising model, and the quadratic function model, or all optimization models of a similar form. In addition, the 'energy optimization model' in the present invention may use at least two or more of the models exemplified above, and they may be the same or different. In the present invention, when describing such 'energy optimization models', in some cases, only the QUBO model is represented as a representative example, and this may be interpreted to include other forms of 'energy optimization models'.

[0082] In the present invention, the quantum computer or the annealer may be interpreted as including each or both of them.

[0083] According to another preferred embodiment of the present invention, specifically, the present invention

[0084] Step for obtaining a sinogram of a quantum superposition state,

[0085] Preprocessing step of the sinogram obtained from the experiment,

[0086] A step of shaping a QUBO model, an Ising model, or a quadratic function model using the obtained sinograms,

[0087] A step comprising a quantum compression sensing method in at least two identical or different energy optimization models among a QUBO model, an Ising model, or a quadratic function model, and

[0088] It may include a step of obtaining a qubit value by calculating it on hardware capable of quantum computing.

[0089] In order to explain the present invention, the image generation process is described in detail along with the following basic technical description.

[0090] QUBO and Ising models

[0091] In statistical mechanics, the Ising model serves as a mathematical framework for describing the dynamics of spin interactions within a system, particularly in the context of ferromagnetic materials. This model typically consists of a grid of discrete variables representing the orientation of magnetic spins, which can take on the values ​​+1 or -1. The physical problem described by the Ising model can be expressed mathematically using a Hamiltonian. The energy Hamiltonian can be formulated as follows:

[0092] Hereinafter, the numbers of mathematical expressions may be expressed as separate mathematical expressions or as numbers in parentheses next to the mathematical expressions.

[0093]

[0094] The QUBO model is a combinatorial optimization problem in computer science. Energy function Is dimensional binary vector space The set of real numbers in is defined as follows:

[0095]

[0096] Here represents an upper triangular matrix, and , is. The problem is the energy function The solution is to find the solution that minimizes . Therefore, the energy function can be re-formulated as follows:

[0097]

[0098] According to the present invention, the variables of the Ising model and variables of the QUBO model Is can be converted using .

[0099] Therefore, if a given problem can be expressed by one of two models, such as the QUBO model or the Ising model, it can also be expressed by the other model. For example, it may include a quadratic function model or a similar energy optimization model.

[0100]

[0101] Constrained Quadratic Model (CQM)

[0102] CQM has the following types of problems.

[0103] Minimize the goal

[0104]

[0105] <Subject to constraints>

[0106]

[0107] Here can be a binary, integer, or continuous variable, and and represents a real number, , M represents the total number of constraints.

[0108]

[0109] Quantum optimization model for linear systems

[0110] The QUBO model for solving linear systems is known in other literature. Matrix , a column vector of variables And the heat vector Given this, solving the linear system is Satisfying is used to obtain .

[0111] The least squares problem is to minimize As for finding , first, can be written as follows:

[0112] (6)

[0113] To generate a QUBO model for a linear system, one can consider the squared 2-norm.

[0114]

[0115] Here, Within the scope of If you appropriately utilize binary representation to express it, for example, by using and , can be substituted into the above equation. This mathematical equation provides a QUBO model for mathematical expression (9). Using the D-Wave QPU solver, the minimum value of QUBO consisting of first and second terms excluding the constant term You can obtain the solution qubit corresponding to.

[0116]

[0117] Radon transform and sinogram

[0118] The Radon transform is a mathematical representation of the method of obtaining projection data from a parallel beam X-ray source, which can be expressed as follows.

[0119]

[0120] Here is the slope of a straight line, is the intercept, represents the delta function.

[0121] Sinograms are generated by accumulating Radon transforms from various angles. Applying the same sample to different locations can result in different projection locations obtained through the Radon transform. In sinograms free of motion artifacts, the shape of the projected object follows the ideal sinogram pattern, which is essential for accurate CT image reconstruction.

[0122]

[0123] Quantum Optimization CT Algorithm

[0124] According to the present invention, a quantum optimization CT algorithm can reconstruct a CT image by minimizing the energy of a QUBO model, an Ising model, or a quadratic function model that can express the entire sinogram. This algorithm begins by expressing each pixel of the CT image to be reconstructed as a combination of qubits. A CT image expressed as a qubit ( Superimposed sinogram by applying the Radon transform to (indicated by ). Obtain. Then, the sinogram obtained from the experiment If expressed as such, the ultimate goal is to express the energy function as the square of the difference between the superposition sinogram represented by the qubit and the actual sinogram, and to determine the qubit that minimizes this energy function.

[0125] The maximum pixel value person The reconstructed image is represented as qubits as follows:

[0126]

[0127] Here, m represents the number of qubits used to represent a single pixel, where the qubit is According to a preferred embodiment of the present invention, the acquisition of the sinogram of the quantum superposition state can be achieved by a method of representing each pixel of a single-layer image in a quantum superposition state based on the above Equation 12 or the following Equation 12a.

[0128] [Mathematical Formula 12a]

[0129]

[0130] In the above formula is a mistake, N damping coefficients of the sample depending on the combination It represents all cases that can be expressed.

[0131] Projection angle About of The i-th position is calculated as follows:

[0132]

[0133] Here is of the CT image ( ) Represents the location pixel, and Is Represents the corresponding coefficient obtained through Radon conversion affecting. Energy minimization model is given by the following equation:

[0134]

[0135] Here represents the change in projection angle, and N represents the number of pixels included in one projection angle of IP. Therefore, the minimum value of F can be obtained using D-Wave's hybrid solver.

[0136] The present invention presents an energy minimization algorithm having a constraint that reconstructs a CT image using only a portion of the sinogram. Instead of using the entire sinogram for all projection angles, an energy minimization model F can be constructed using only sinogram values ​​where the projection angle is 0° and the sinogram value is not 0, as in Equation (14).

[0137]

[0138] Here And am.

[0139] Also, projection angle When is not 0 For each sinogram value of the location, the undefined sinogram A real sinogram Set constraints to make it equal to . For more accurate calculations, you can apply the following constraints using the "epsilon test".

[0140]

[0141] Here , And represents the sum of the hardware error of the quantum computer or quantum annealer and the allowable error of the solution.

[0142] According to a preferred embodiment of the present invention, pixels of a sinogram are used by utilizing the above Equations 16 and 17, which are used in the calculation of energy optimization models such as the QUBO model, the Ising model, or the quadratic function model. It may include methods for using constraints on fields.

[0143] Additionally, even when the sinogram value is 0 at a projection angle of 0°, the same constraints described above are applied. To find the minimum of the energy function with the constraints applied, a hybrid solver is used to set the variables in two ways: binary variables are denoted as "Binaries," and integer variables are denoted as "Integers." Now, an energy minimization model that serves as the objective function for each variable is established.

[0144] According to a preferred embodiment of the present invention, a method for obtaining a sinogram of a quantum superposition state through mathematical projection of a cone beam, a fan beam, and a parallel beam from a tomographic image of the quantum superposition state may be included.

[0145] According to a preferred embodiment of the present invention, a method for converting an experimental sinogram may be included so that the sinogram of the generated quantum superposition state can display the sinogram obtained from a tomographic system.

[0146] According to a preferred embodiment of the present invention, the constraint is a projection angle When is not 0 For each sinogram value of the location, the undefined sinogram A real sinogram Constraints can be set to be equal to, and constraints such as those in Equations 16 and 17 above can be included using epsilon tests.

[0147] According to a preferred embodiment of the present invention, a pretreatment method may be included to make the parts without samples zero and to readjust the columns of an experimental sinogram so that each density changes continuously, or to adjust the acquired sinogram so that the sum of each column is constant.

[0148] According to a preferred embodiment of the present invention, the step of using the acquired sinogram may include a method for finding a masking region to be excluded from the calculation of an energy optimization model, such as a QUBO model, an Ising model, or a quadratic function model.

[0149] According to a preferred embodiment of the present invention, a method for finding a masking region may include a method of masking a portion related to a density that is abnormal or does not change continuously within a sinogram by determining the average of the difference between two adjacent rows of the sinogram, and not using it in an energy optimization model such as a QUBO model, an Ising model, or a quadratic function model.

[0150] According to a preferred embodiment of the present invention, a method for finding a masking region may include a method of determining a part of the sinogram where the change in density is irregular or a part where the change with adjacent regions is not continuous, and masking it so that it is not used in the calculation of an energy optimization model.

[0151] That is, the masking process may include a masking method that identifies and excludes regions of change where the change, including the horizontal or vertical, is not continuous in the calculation of the energy optimization model.

[0152]

[0153] CQM goals using binary variables

[0154] In the above mathematical formula (12), if each pixel of the CT image is represented as a binary variable, the energy function can be constructed as follows.

[0155]

[0156] By rearranging the first term of the above mathematical formula (19), the following equation can be obtained.

[0157]

[0158]

[0159] Simplifying the first term of the above mathematical formula (21) yields the following equation.

[0160]

[0161] In the case of binary variables since The energy minimization model F of the above mathematical formula (18) is converted into a form including linear and quadratic terms of binary variables.

[0162]

[0163] If we set aside the constant term, the remaining part serves as the objective function of CQM. Therefore, the minimum value of this objective function obtained in the present invention is am.

[0164]

[0165] CQM goals using variables

[0166] The integer variable is the pixel expressed in the above mathematical formula (12). a single integer variable It is represented as such. Substituting this into the above mathematical formula (15), the energy minimization model can be constructed as follows.

[0167]

[0168]

[0169] Here Is It represents the corresponding coefficients obtained using the Radon transform that affect the . Even when using integer variables, If we define as the objective function, the minimum value to be determined is It is maintained as such. Once the objective function of the CQM is formulated, constraints are generated for each variable using the above mathematical formulas (16) and (17), and these are added to the CQM to complete the CQM. The constraints below It is indicated as.

[0170] Implementation and Results

[0171] According to a preferred embodiment of the present invention, two sample images are reconstructed using D-Wave Ocean Software's hybrid solver, "Hybrid Solver for General CQM Problems, Version 1.12" and the sampler API, dimod.

[0172] <Creating Binary Variables>

[0173] qi = dimod.Binaries('qi')

[0174] Generate integer variables

[0175] zj = dimod.Integer('zj',upper_bound=max(pixel))

[0176] <Create Integer Variable>

[0177] cqm = dimod.ConstrainedQuadraticModel()

[0178] cqm.set_objective(F)

[0179] cqm.add_constraint( ), (for each constraint)

[0180] <Solved using a hybrid solver>

[0181] sampler = dwave.system.LeapHybridCQMSampler()

[0182] sampleset = sampler.sample_cqm(cqm)

[0183]

[0184] In the final output, the "sample set" can provide information about each occurrence, including the minimum value, the value of the variable generating the minimum value, and the state (true / false) of each constraint.

[0185]

[0186] Hereinafter, the image generation of the present invention is described as one embodiment. This embodiment is not intended to limit the present invention.

[0187]

[0188] [Example 1] Small image using two qubits

[0189] Using the first test dataset (see Fig. 1), we generated 5 × 5 Shepp-Logan image samples with integer pixel values ​​ranging from 0 to 3. These images can be represented by two qubits. The sinogram obtained by applying the Radon transform to these samples after zero-padding them in all directions is shown in Fig. 1. Here, we obtained projection images for 10 projection angles evenly distributed from 0° to 180°, and conducted experiments based on two types of variables.

[0190] Binary variables were generated using two qubits representing each pixel, and a total of 50 qubits were generated. For integer variables, 25 variables were generated by setting the upper limit to 3 so that each pixel could be represented by a single integer variable.

[0191] When setting the objective of CQM using the red area as shown in Figure 1, the desired minimum value in both methods was -284.

[0192] As shown in Figure 2, the CQM setup was completed by adding constraints to all pixels except those inside the red box. Then, the hybrid CQM solver calculated the target value for pixels satisfying the constraints. The binary version required a runtime of 0.001 seconds, while the integer version required 0.002 seconds. Despite configuring and solving the CQM with different variables, both methods successfully determined the minimum energy -284 and satisfied all constraints.

[0193] The solution array for the binary variable is [0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0] and the solution array for the integer variable is [0, 1, 1, 0, 2, 2, 1, 2, 1, 3, 3, 1, 2, 2, 2, [2, 2, 2, 1, 0, 1, 1, 0]. By sequentially selecting variables in pairs from the output of the binary variable and substituting them into q1+2q2, a new array with 25 components was obtained. This new array was identical to the array obtained from the output using integer variables. Additionally, the CT image was reconstructed using the solution variables of both methods, resulting in an image identical to that in Fig. 1.

[0194]

[0195] [Example 2] A binary image of size 100 × 100

[0196] The second test sample was a binary image requiring 10,000 variables. A 100 × 100 Shepp-Logan image was created, and all non-zero parts were converted to 1 to produce a grayscale image (see Fig. 3).

[0197] As in the first test, 10 projection angles were used. Since this is a binary image, setting the upper bound of the integer variables to 1 results in the same number of binary and integer variables, which is 10,000.

[0198] For this sample, the desired minimum was -251,578 when the objective was configured using non-zero values ​​of the sinogram at a projection angle of 0°. Both variables successfully determined their minimum values, satisfying all constraints.

[0199] When using binary variables, the runtime was 1.009 seconds, while for integer variables it was 1.281 seconds. This indicates that integer variables require a longer runtime than binary variables.

[0200]

[0201] [Example 3] Comparison of two algorithms

[0202] We compared the performance of generating a single energy optimization model (QUBO model) using the entire sinogram with the performance of generating a QUBO model using a subset of sinograms with multiple constraints.

[0203] For the experiment, we used a 30×30 Shepp-Logan image (see Fig. 4(b)) with integer pixel values ​​ranging from 0 to 7. Each pixel in this sample image is represented by a combination of qubits, and each pixel requires three qubits, resulting in a total of 2,700 qubits. A sinogram was generated using 30 full-width projections.

[0204] A single QUBO model can be constructed using the entire sinogram, and the global minimum energy can be calculated using a hybrid solver. Alternatively, the minimum value of the QUBO model can be calculated using a hybrid CQM solver by setting each pixel of the image sample to a variable with an integer value from 0 to 7. In this approach, the QUBO was formulated using only the sinogram at 0° and constraints on the remaining sinogram values.

[0205] Consequently, the method using the entire sinogram failed to find the global minimum energy, and the reconstructed image was not optimally reconstructed, as shown in Figure 4(a). In contrast, the method using the QUBO model on a portion of the sinogram with constraints successfully identified the global minimum energy of -60,107.

[0206] As shown in Figure 4(b), the reconstructed result confirmed a perfect reconstruction in which all constraints were satisfied.

[0207] However, the execution times of the two experiments showed that the hybrid solver took 6.819 seconds and the hybrid CQM solver took 33.979 seconds.

[0208]

[0209] [Example 4] Quantum supremacy results using noisy projection data

[0210] Two types of experiments were performed to demonstrate the quantum superiority of the quantum optimized tomography reconstruction algorithm.

[0211] The first experiment involved removing circular artifacts caused by noise during CT scans in existing algorithms. To clearly demonstrate the differences between the quantum algorithm and existing algorithms, artificial errors were introduced into the sinogram, as shown in Figure 5 (a).

[0212] When such errors occur consistently from multiple angles, a circular pattern appears in the reconstructed image as shown in Fig. 5(b). However, it has been confirmed that this problem can be solved by excluding unreliable regions with errors in the sinogram during the QUBO construction process, for example, because the algorithm according to the present invention focuses on each pixel of the image to be reconstructed when constructing the QUBO.

[0213] Therefore, to apply the algorithm of the present invention to a sinogram containing such horizontal error regions, the locations where the horizontal error regions occur in the projection image must first be identified. This can be achieved by identifying locations where density changes abruptly in the projection image.

[0214] Specifically, a density difference graph is generated using the difference between adjacent locations in the density graph of a single projected image, and this is used for all By averaging across, we obtain the graph represented by the blue line in (a) in Fig. 5. Examining this graph, we can see that the density difference changes sharply at the beginning and end of the horizontal error region. The position union of the horizontal error region identified in this way is defined as S, and the pixel values ​​in this region are excluded from the sinogram when generating QUBO. Consequently, the cost function F is defined as follows.

[0215]

[0216] A second experiment to demonstrate the algorithm's quantum superiority uses projection images captured only over a narrower angular range, rather than the full 0° to 180° angle.

[0217] maximum projection angle ( If expressed as ), for the angle variable and , the cost function F in this case is as follows.

[0218]

[0219] The results of the two experiments conducted to prove quantum supremacy can be confirmed in the above results.

[0220] According to one embodiment of the present invention, a method may be provided for calculating an energy optimization model, such as a QUBO model, an Ising model, or a quadratic function model, from Equation 28 or Equation 29 using a sinogram obtained by obtaining a tomographic image of a quantum superposition state and a preprocessed experimental sinogram.

[0221] According to one embodiment of the present invention, a method of excluding a specific location of a sinogram in order to use noise or constraints in the calculation of an energy optimization model such as a QUBO model, an Ising model, or a quadratic function model from the above mathematical expression 28 may be included, or a method of using only angles of a specific area may be included by applying the above mathematical expression 29.

[0222] According to one embodiment of the present invention, a quantum compressive sensing QUBO model Q2 for the total variation in a CT image is calculated and linearly combined with the QUBO model Q1, which is an example of an energy optimization model created using the above mathematical expressions 28 and 29. Therefore (single class QUBO model for CT image reconstruction (is a constant) It may include a method to obtain.

[0223] To demonstrate the quantum superiority of the quantum optimization tomography algorithm according to the present invention, a circular artificial shadow was artificially introduced into the sinogram of the sample.

[0224] The experiment was conducted under controlled conditions to accurately evaluate the impact of these artifacts on algorithm performance.

[0225] Figure 5(a) shows the sinogram used in the experiment to artificially generate circular artificial shadows and the corresponding results. The image used in the experiment represents a 100×100 cross-section of a tooth sample at the 27th axial level, and the projection angle is divided into 100 intervals from 0° to 360°. To generate the artificial artificial shadow, a 0 row with a length of 10 and an interval of 10 was introduced into the sinogram, as shown in Figure 5(a). The result of applying the FFT to this sinogram is shown in Figure 5(b). To better visualize the circular artificial shadow, the resulting image was restricted to pixel values ​​between 0 and 1.

[0226] When the algorithm was applied to the same sinogram, a result similar to that in Fig. 6(b) was obtained. The main structure was well restored, but many noticeable black and white dots appeared. After post-processing, the resulting image, such as in Fig. 6(c), shows a well-segmented cross-section of the tooth.

[0227] In fact, as can be seen in Fig. 6 (d), when the post-processed image and the solution image are superimposed, the difference can be observed only at the boundaries of the materials.

[0228]

[0229] [Example 5] Quantum superiority result for limited projection angle

[0230] To demonstrate the superiority of the algorithm in a smaller angle range, the maximum angle in this experiment was set to less than 180°.

[0231] The cross-sectional image of the tooth sample was set to 50×50 pixels and the axis level was 15.

[0232] To address concerns about using too few projection angles due to the smaller angle range, the range from 0° to 180° was initially divided into 100 angles and 100 projection images were created.

[0233] To generate the solution image, the FFT and threshold setting method were applied to the entire sinogram (see Fig. 7(d)). First, the experiment was performed using the initial 50 projection images corresponding to angles smaller than 90° out of 100. However, Fig. 7(e) shows the result of applying the algorithm, and a significant error appears in the upper left part.

[0234] To investigate the impact of using more angles, a second experiment was conducted using 60 projection images corresponding to angles less than 108°.

[0235] The results of applying FFT to 50 and 60 projected images are shown in Fig. 7(b) and Fig. 7(c), respectively. In both cases, the FFT results show that the pixel values ​​are outside the range between 0 and 1, making it impossible to identify the boundary shape.

[0236] On the other hand, as can be seen in (e) and (h) of FIG. 7, the method proposed in the present invention maintains the overall structure.

[0237] As can be seen in Figures 7(g) and 7(j), errors are found in the upper left and lower right regions when compared with the solution image after post-processing, but it can be seen that using 60 projection images results in less discrepancy with the solution image than using 50 projection images.

[0238] To perform experiments on problem-free sinograms, tests were conducted using 50×50 Shepp-Logan binary sample images with zero pads of thickness 11 added to all sides (see Fig. 8(c)).

[0239] In this case, as can be seen in (a) of Fig. 8, the projection angle was limited to 90° and only 25 projection images, which is half of the total 50, were used.

[0240] The result of applying FFT to these 25 projected images shows that, as can be seen in Fig. 10 (b), it is difficult to distinguish the outer boundaries of the top-left and bottom-right corners and the right ellipse among the two inner ellipses. The result of applying the algorithm in this case is shown in Fig. 8 (c), which exactly matches the sample images used in the experiment.

[0241]

[0242] Minimizing total variation in CT images

[0243] a CT image of When referring to pixel values, we mean total variation It can be defined as the sum of the differences between all adjacent pixel values ​​as shown below. It can also be expressed as the sum within the ROI (region of interest).

[0244]

[0245] Here, the QUBO formulation is possible by squaring the L2 norm.

[0246]

[0247] It can be calculated and the QUBO formulation can be obtained by substituting into the above mathematical formula (32).

[0248] If each of these is a QUBO model, the final QUBO model It can be obtained through a linear combination as follows. Here, each is a mistake.

[0249]

[0250] Practically speaking, here, is a constant and is an energy optimization model for a specific calculation, and can contain an energy optimization model of quantum compressive sensing that minimizes the total variation within a CT image.

[0251] According to a preferred embodiment of the present invention, through a CQM solver from Each can be set as a constraint.

[0252] In this case, the QUBO model for total variation and sequentially generate the QUBO model for each pixel of the remaining sinogram or pixels of a certain area. From sequentially It can be placed as.

[0253] According to one embodiment of the present invention, the energy minimization model by quantum compressive sensing, which is the minimization of the total variation in the CT image, is an energy optimization model such as the QUBO model by substituting the tomographic image in the quantum superposition state into the above mathematical expression (32). It may include a method to calculate.

[0254] According to a preferred embodiment of the present invention, the algorithm for energy optimization may construct two energy optimization models F by using some or all of the sinogram values ​​with low error in Equation 15 below and a quantum compressed sensing method that minimizes the total change amount of the tomographic image of the quantum superposition state in Equation 32 below.

[0255] In the calculation of the energy optimization model, the method may include a linear combination of at least two energy optimization models that are the same or different among the QUBO model, Ising model, or quadratic function model using the above mathematical formula (35). Additionally, according to a preferred embodiment of the present invention, since CT image reconstruction is possible with about 5% to 10% of the data in this method, several X-ray light sources and detectors (including charged coupled devices) in a CT system are used simultaneously to simultaneously acquire data that is completely free of errors regarding motion, thereby including an energy optimization model that includes quantum compressed sensing. It may include a method of obtaining a single-layer image using .

[0256]

[0257] Explanation of experimental results according to the examples

[0258] According to a preferred embodiment of the present invention, excellent tomographic images can be generated by reconstructing CT images using a QUBO model with a quantum optimization CT algorithm according to the present invention.

[0259] In one embodiment of the present invention, the minimum value of the QUBO model and the corresponding qubit solution value were obtained by utilizing a D-Wave solver. This algorithm can generally use the entire sinogram for all projection angles. However, the present invention obtained superior results by proposing a method to formulate a CQM (qubit solution method) that can use only a portion of the sinogram.

[0260] In addition, the approach according to the present invention has demonstrated the possibility of solving the problem by introducing additional constraints to compensate for the lack of reduced information, even though it uses less sinogram data. In particular, according to a preferred embodiment of the present invention, satisfactory results could be obtained even when appropriate constraints were provided and the objective function was changed from a squared form to a linear form that does not use squares, as in Equation (14).

[0261] Furthermore, in the present invention, even though this linear term was significantly reduced, the experiment continued to yield a successful solution.

[0262] According to the present invention, various experimental results have shown that the problem can be solved with only a sufficient number of constraints, regardless of the form of the objective function.

[0263] In particular, according to the present invention, the advantage of the objective function formulated as QUBO is that the minimum energy is It is known that... In addition, experiments were also conducted using constraints on 30 × 30 Shepp-Logan images with integer pixel values ​​ranging from 0 to 1023. In experiments using 900 integer variables and 9000 binary variables, the experiment using 10 times more binary variables required 10 times longer execution time than the experiment using integer variables. In terms of execution time, using integer variables was found to be more efficient.

[0264] According to one embodiment of the present invention, the CQM solver can use up to 100,000 constraint conditions.

[0265] In the present invention, a squared form was used as shown in Equation (14) above to utilize the global minimum energy of the QUBO model. The constraints of the CQM solver allowed the CT image to be reconstructed using only linear terms as shown in Equations (16) and (17).

[0266] According to one embodiment of the present invention, the number of required calculations does not decrease even if a CQM solver is used. Each constraint can be applied to a different quantum optimization model. To obtain a clear CT image of an actual sample, additional algorithms are required for high-performance CT image reconstruction, ring artifact correction, metal artifact correction, sinogram realignment to correct motion artifacts, and X-ray coherence effects. Some of these can be corrected by modifying the sinogram itself. However, there are limitations in correcting error information in the sinogram. Since existing quantum optimization CT algorithms use a single QUBO model for a given data, sinogram preprocessing is essential.

[0267] However, the approach proposed in the present invention, utilizing the CQM solver and the algorithm of the present invention, offers the potential to constrain the correction algorithm. In particular, tilt of the sample or rotation axis poses significant challenges in reconstructing clean CT images, but the present invention demonstrates that this problem can be overcome using a quantum optimization algorithm utilizing the CQM solver.

[0268] According to the present invention, it is preferable that the quantum optimization CT algorithm uses an energy optimization model (e.g., QUBO model) for the entire sinogram when reconstructing a CT image. When formulating the energy optimization model, only linear terms from the second column were used as constraints to perform conventional calculations. Assuming that the size of a CT image is nx×nx, the algorithm of the present invention calculates the energy optimization model for one projection image, so the overall calculation time is approximately nx times faster than the previous algorithm.

[0269] Another advantage of the present algorithm is that it can achieve more efficient results when the performance of a quantum computer or quantum annealer is the same. As the number of qubits used in the binary representation of each pixel increases, the maximum and minimum terms of the QUBO matrix coefficients increase and decrease accordingly. This makes it difficult to find the global minimum energy when calculating the QUBO model.

[0270] However, it can be confirmed that the algorithm according to the present invention can produce better results when calculating an energy optimization model (QUBO model) (see FIG. 4).

[0271] The final advantage of the algorithm according to the present invention is that it can efficiently use two or more energy optimization models, including the QUBO model.

[0272] Furthermore, X-ray projection images often contain various artifacts and noise. To address these challenges, separate energy optimization models are required to remove each artifact or error. However, the algorithm of the present invention formulates an energy optimization model for one projection image and uses linear terms for the remaining projection images. This allows for more efficient code management when multiple energy optimization models are required to solve a single problem involving multiple subproblems, by adding each energy optimization model as a constraint.

[0273] In particular, if there is an error in a specific energy optimization model, each energy optimization model with an error can be calculated independently without affecting other energy optimization models.

[0274] The algorithm of the present invention provides a method for efficiently removing artifacts or errors that may occur in X-ray projections containing constraints.

[0275] Since there are limitations to the global minimum energy calculation capability of hybrid solvers, there are still difficulties in using real data; however, the present invention suggests that this problem can be overcome by improving the performance of a quantum computer or quantum annealer.

[0276] When a sinogram containing horizontal error regions is reconstructed using existing algorithms, circular artificial shadows are generated in the reconstructed image. However, the algorithm of the present invention is effective in solving this problem by focusing on each sinogram pixel that is less affected by errors and generating an energy optimization model.

[0277] In particular, through an approach using the algorithm of the present invention, it is possible to exclude untrusted regions from the sinogram while simultaneously integrating qubit information from other trusted regions.

[0278] Therefore, even if an energy optimization model is generated by excluding unreliable regions from the sinogram, qubit information can be obtained from other reliable regions.

[0279] However, when the algorithm of the present invention is applied after removing unreliable regions as in Fig. 8(b), more black and white dots appear compared to the result without removal as in Fig. 6(b). This increase is likely due to the fact that the quantum algorithm failed to find a global minimum because the original data obtained by scanning actual tooth samples was not ideal. Nevertheless, the algorithm of the present invention achieved better results when there were fewer error regions.

[0280] Considering these advantages of the present invention, the algorithm of the present invention can be applied to a wider range of technologies beyond CT scanning. For example, since electron tomography is typically performed within an angular range of -70° to 70°, the ability of the algorithm of the present invention to operate effectively within an angular range of 140° can present a significant advantage. Figure 7 demonstrates that even in a 50×50 cross-sectional image of an actual tooth sample, meaningful segmentation results can be successfully obtained, allowing shape prediction using 60 angles less than 108°.

[0281] Moreover, experimental results on a 50×50 Shepp-Logan binary sample image yielded perfect segmentation results, with only 25 angles less than 90°. While existing classical algorithms often struggle to reconstruct angles less than 180°, the present algorithm works effectively even in this reduced angle range, demonstrating its high applicability to electron tomography.

[0282] These advantages of the present invention further emphasize the robustness and precision of the algorithm, enhancing its applicability in the field of electron tomography (ET), where accurate reconstruction is essential. Furthermore, by replacing existing optimization algorithms with quantum optimization algorithms to address the low contrast issues of CT images, such as cone-beam and fan-beam CT, it is expected to be highly useful in medical imaging diagnostics, particularly in areas such as breast cancer and cardiovascular diagnosis.

[0283] While the present invention has been described in detail using preferred embodiments, the scope of the present invention is not limited to the specific embodiments and should be construed in accordance with the appended claims. Furthermore, those skilled in the art will appreciate that numerous modifications and variations are possible without departing from the scope of the present invention.

Claims

1. a) A step of acquiring a sinogram of a quantum superposition state for a video image acquisition target, b) a step of converting the entire or a part of the acquired sinogram into an energy optimization model using quantum compressive sensing, which is a method of minimizing the total change in the tomographic image of the quantum superposition state, and c) A step of reconstructing a single-layer image using an energy minimization algorithm by using the remaining unconverted portion as a constraint or excluding it by masking. A method for generating a quantum computed tomography image including:

2. In claim 1, the method for generating a quantum computed tomography image, which comprises obtaining a sinogram of a quantum superposition state in step a) by representing each pixel of a tomographic image as a quantum superposition state based on the following mathematical expression 12 or mathematical expression 12a: [Equation 12] Here, is for the tomographic image of the quantum superposition state ( ) represents the pixel at the location, and the above mathematical expression 12 is Satisfied is from 0 It can represent all integers up to and including the attenuation coefficient of a particular light source for the sample, including the X-ray mass attenuation coefficient of the sample, [Equation 12a] Here, is a mistake, Depending on the combination of the sample It has N damping coefficients It represents all cases that can be expressed.

3. In claim 1, the step of converting into an energy optimization model in step b) includes the step of shaping a QUBO model, an Ising model, or a quadratic function model for a sinogram, and the step of performing quantum compressive sensing on two or more energy optimization models selected from among the QUBO model, the Ising model, or the quadratic function model. A method for generating a quantum computed tomography image.

4. In claim 1, the algorithm for energy optimization in step b) uses some or all of the sinogram values ​​with less error in the following mathematical expression 15, and uses a quantum compressive sensing method that minimizes the total change in the quantum superposition state tomographic image of the following mathematical expression 32, thereby constructing two energy optimization models F: [Equation 15] Here And am. [Equation 32] Here is a CT image of a quantum superposition state. It is a value for location.

5. In claim 1, the constraint is a projection angle and For each sinogram value of the location, the undefined sinogram is the actual sinogram A method for generating a quantum computed tomography image by setting constraints to be equal to and applying constraints such as the following mathematical expressions 16 and 17 using an epsilon test: [Equation 16] [Equation 17] Here, is a quantum superposed sinogram It means location is the preprocessed sinogram , represents, , And represents the sum of the hardware errors of a quantum computer or annealer or any of those capable of quantum computing and the allowable errors of the solution.

6. A method for generating a quantum computed tomography image, comprising a step of preprocessing the obtained sinogram by readjusting the columns of the experimental sinogram so that the portions without samples are made 0 and each density changes continuously, or by adjusting the obtained sinogram so that the sum of each column is constant.

7. A method for generating a quantum computed tomography image according to claim 1, wherein the masking process includes a masking method for determining and excluding a region of change in which changes including horizontal or vertical directions are not continuous in the calculation of an energy optimization model.

8. A method for generating a quantum computed tomography image, comprising: calculating an energy optimization model from the following mathematical expression 28 or 29 using an acquired sinogram and a preprocessed experimental sinogram for a quantum superposition state tomography image in claim 1: [Equation 28] [Equation 29] Here is a quantum superposed sinogram It means location is the preprocessed sinogram , represents, If it is 0, all terms are included as constraints. In this case, all terms of the sinogram can be used.

9. A method for generating a quantum computed tomography image, comprising a method of linearly combining at least two or more identical or different energy optimization models using the following mathematical expression 35 in calculating an energy optimization model according to claim 1: [Equation 35] Here is a constant is an energy optimization model for a specific calculation, and may include an energy optimization model of quantum compressive sensing that minimizes the total variation within a CT image.

Images