CN110659698A - Dictionary learning method for PET image reconstruction - Google Patents

Dictionary learning method for PET image reconstruction Download PDF

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CN110659698A
CN110659698A CN201910940260.XA CN201910940260A CN110659698A CN 110659698 A CN110659698 A CN 110659698A CN 201910940260 A CN201910940260 A CN 201910940260A CN 110659698 A CN110659698 A CN 110659698A
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刘华锋
叶慧慧
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Shenzhen Research Institute Of Zhejiang University
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Abstract

The invention discloses a dictionary learning method facing PET image reconstruction, which comprises the steps of establishing a mathematical model of a reconstruction problem, adding structural dictionary constraint, and reconstructing a PET image based on the constraint of the structural dictionary; in the process of reconstructing the PET image by combining Poisson and dictionary constraint, an EM algorithm is adopted for optimization solution. Therefore, the invention effectively utilizes the structural dictionary constraint, and improves the problems of low resolution and noise interference of the result generated by the computer in the process of reconstructing the PET image; compared with the prior reconstruction method, the invention has the advantage that the better reconstruction effect can be obtained.

Description

Dictionary learning method for PET image reconstruction
Technical Field
The invention belongs to the technical field of PET imaging, and particularly relates to a dictionary learning method for PET image reconstruction.
Background
Positron Emission Tomography (PET) is a medical imaging technology based on nuclear physics and molecular biology, and can observe the metabolic activity of cells from a molecular level, thereby providing an effective basis for detection and prevention of early diseases. Before the PET measurement is performed, the subject first inhales or injects a radiopharmaceutical containing a positive electron species corresponding to the location to be measured, which is usually generated by a cyclotron. After a short period of time, the radionuclide reaches the corresponding region and is absorbed by the target tissue, at which time scanning can take place. As the radionuclide decays, positrons are emitted outward, and after traveling a short distance, they encounter electrons in the tissue and annihilate. Upon annihilation, a pair of photons moving in opposite directions are emitted, each with an energy of 511 keV. Therefore, the detector detects the emitted photon pair in the measured object to determine the position of the annihilation event. Typically, if two oppositely located photons are detected within a set time window (e.g., 10ns), we consider the two photons to be generated in the same annihilation event, and belong to a true coincidence, the positron emission event is recorded. The set of all positron emission events can be approximately equal to the line integral of the radionuclide concentration distribution, with the greater the number of events recorded, the greater the degree of approximation.
The PET image has the advantages of high sensitivity and high specificity, but because the radionuclide is affected by human tissues to generate serious attenuation, and the attenuation correction method is complex and high in cost, the image reconstructed from the measured data has low resolution and is slightly blurred. Conventionally, a statistical iteration method is often adopted for radioactive concentration distribution reconstruction, and the iteration method is based on a statistical model, has good adaptability to incomplete data, and gradually becomes a research focus of a PET reconstruction algorithm, wherein the research focus comprises famous MLEM (maximum likelihood expectation maximization), MAP (maximum posterior), and SAGE (penalty likelihood) algorithms. How to obtain more accurate and clear reconstructed images is a hot point of research.
Disclosure of Invention
Aiming at the technical problems in the prior art, the invention provides a dictionary learning method for PET image reconstruction, which can obtain a high-quality PET reconstructed image.
A dictionary learning method for PET image reconstruction comprises the following steps:
(1) detecting biological tissues injected with the radioactive medicament by using a detector, acquiring coincidence counting vectors corresponding to each crystal block of the detector, and further constructing a coincidence counting matrix y of the PET;
(2) according to the PET imaging principle, a PET measurement equation is established as follows; introducing Poisson noise constraint to the measurement equation to obtain a Poisson model L (x) of the PET;
y＝Gx+r+v
wherein: g is a system matrix, x is a PET concentration distribution matrix, and r and v are measurement noise matrices related to reflection coincidence events and scattering coincidence events respectively;
(3) introducing structural dictionary constraint into the Poisson model L (x), and obtaining a PET image reconstruction model based on the structural dictionary constraint as follows:
Figure BDA0002220561370000021
wherein: λ is a weight coefficient, Rsparse (x, α) is a penalty term for the PET concentration distribution matrix x and the sparse coefficient matrix α;
(4) and (4) carrying out optimization solution on the PET image reconstruction model to obtain a PET concentration distribution matrix x, and further reconstructing to obtain a PET image.
The expression of the Poisson model l (x) is as follows:
Figure BDA0002220561370000031
wherein: y isiTo fit the coincidence count vector corresponding to the ith crystal block in the count matrix y,to count the vector yiAverage of all elements in, niThe total number of crystal blocks of the detector.
The penalty term RsparseThe expression of (x, α) is as follows:
Figure BDA0002220561370000032
wherein: d is a structural dictionary, EsTo segment the operator, Esx is the s-th n-dimensional submatrix in the PET concentration distribution matrix x, mu is a weight coefficient, ns＝(m-n+1)2M is the dimension of the PET concentration distribution matrix x, n is the dimension of the predetermined submatrix (generally 7 or 8), alphasFor the s-th column in the sparse coefficient matrix alphaSparse coefficient vector, | αs||0Representing a sparse coefficient vector alphasNumber of middle non-zero elements, | | | | | non-phosphor2Representing the L2 norm.
In order to ensure that the positions of non-zero elements in the sparse matrix a are kept unchanged in the process of updating the dictionary, a mask matrix M is added in the dictionary updating stage, and the formula becomes:
Figure BDA0002220561370000033
the elements in the mask matrix M are not 0, i.e. 1, therefore, the constraint α M is 0 such that all 0's in α remain intact. Although the problem in the above equation is simpler than the overcomplete dictionary learning problem, it is still non-convex and difficult to solve. We can propose a different way of iteratively solving this constraint problem, by fixing α and minimizing with respect to D, and then fixing D to update α.
The specific method for carrying out optimization solution on the PET image reconstruction model in the step (4) is as follows: and (3) solving the PET concentration distribution matrix x by the following iterative equation based on an EM (Expectation Maximization) algorithm:
Figure BDA0002220561370000041
Figure BDA0002220561370000042
Figure BDA0002220561370000043
wherein:
Figure BDA0002220561370000044
and
Figure BDA0002220561370000045
are all the intermediate variables of the series of the Chinese characters,
Figure BDA0002220561370000046
for the (k + 1) th iteration PET concentration distribution matrix xk+1The value of the j-th element in (b),
Figure BDA0002220561370000047
for the kth iteration PET concentration distribution matrix xkValue of the jth element of (1), esljTo segment operator EsRow i and column j element values in (1),
Figure BDA0002220561370000048
Figure BDA0002220561370000049
nj＝m2，Esxkfor the kth iteration PET concentration distribution matrix xkOf [ E ] an s-th n-dimensional sub-matrix, [ E ]sxk]lIs a sub-matrix EsxkThe value of the ith element in (1), nl＝n2，
Figure BDA00022205613700000410
For the structural dictionary D and the k-th iteration sparse coefficient vectorMultiplying the l-th element value, g, in the matrixijFor the ith row and jth column element values in the system matrix G,
Figure BDA00022205613700000412
representing the PET concentration distribution matrix x from the k-th iterationkThe number of photons detected by the ith crystal block of the detector is emitted from the jth element in (a), and k is the number of iterations.
The number of photons
Figure BDA00022205613700000413
The expression of (a) is as follows:
Figure BDA0002220561370000051
wherein: r isiAnd viThe ith element value in the measurement noise matrices r and v, respectively.
The dictionary and a sparse representation matrix obtained after the last iteration are used, and before OMP algorithm sparse coding is used, the method selects the maximum t/3(t is a target base number) coefficients from sparse matrices obtained after the previous D iteration (in the previous tracking stage) to initialize the current sparse matrix. The process then performs coefficient enhancement and optimization similar to the CoSaMP and SP algorithms. The k-th iteration sparse coefficient vector
Figure BDA0002220561370000052
Using an OMP (Orthogonal Matching Pursuit) algorithm to solve the following objective equations:
Figure BDA0002220561370000053
wherein: ρ is a preset minimum value.
The method comprises the steps of establishing a mathematical model of a reconstruction problem, adding structural dictionary constraint, and reconstructing a PET image based on the structural dictionary constraint; in the process of reconstructing the PET image by combining Poisson and dictionary constraint, an EM algorithm is adopted for optimization solution. Therefore, the invention effectively utilizes the structural dictionary constraint, and improves the problems of low resolution and noise interference of the result generated by the computer in the process of reconstructing the PET image; compared with the prior reconstruction method, the invention has the advantage that the better reconstruction effect can be obtained.
Drawings
Fig. 1 is a CT chest slice image.
FIG. 2 is a matrix diagram of a structural dictionary trained from the CT chest slice image of FIG. 1.
Figure 3 is a truth image for a pulmonary phantom.
Figure 4 is a PET image of a lung phantom reconstructed using the ML-EM algorithm.
Figure 5 is a PET image of a lung phantom reconstructed using the method of the invention.
Detailed Description
In order to more specifically describe the present invention, the following detailed description is provided for the technical solution of the present invention with reference to the accompanying drawings and the specific embodiments.
A positron emission tomography scanner detects radioactive signals emitted by a human body, forms original projection lines after being processed by a coincidence and acquisition system, and stores the original projection lines in a computer hard disk in a sinogram mode; and calling a correlation module by taking a dictionary matrix D learned by the sinogram from CT and a known system matrix G as input items for the originally acquired sinogram.
The invention relates to a PET image reconstruction method based on a structure dictionary, which comprises the following steps:
s1, establishing a basic model of a reconstruction problem according to a PET detection principle;
s2, introducing Poisson and a structure dictionary to optimize a problem model;
s3, initializing, setting a weight coefficient lambda, and setting an initial value x, alpha, wherein x is FBP (x); α is 0;
and S4, starting the set initial point, and calculating according to an EM algorithm. In step E, we use the estimated k-th iteration time xkAnd the sinogram y to estimate the hidden variable wij. Will be estimated
Figure BDA0002220561370000061
Substituted into Ωx(wijX, α), we get the equation of immediacy
Figure BDA0002220561370000062
In the M step, by making the equation in time
Figure BDA0002220561370000063
Is equal to zero, we find a new xk+1；
S5, judging whether an iteration stop condition x is less than 10-3If the condition is not satisfied, step S4 is executed, and if the condition is satisfied, the iteration is stopped, and step S6 is executed.
S6, updating alpha; when the step S7 judges that the parameters are unqualified, returning the updated parameters to the step S4 for circulation;
s7, judging whether the iteration stop condition x is less than 10-4If the condition is not satisfied, executing step S4, and if the condition is satisfied, stopping the iteration; and further obtaining a PET concentration distribution vector X to realize PET imaging.
To complete the reconstruction of the PET image, the basic model of the PET detection process is based on the following equation:
y＝Gx+r+v
wherein: g is a system matrix, y is a corrected coincidence count vector, x is a PET concentration distribution vector, and r and v represent measurement noise matrices for reflection coincidence events and scattering coincidence events, respectively.
The expression of the Poisson model for the coincidence count vector is as follows:
Figure BDA0002220561370000071
wherein:
Figure BDA0002220561370000072
representing y obeys an average ofBased on the independent poisson assumption, the likelihood equation Pr (y | x) of y is expressed as follows:
Figure BDA0002220561370000074
wherein: l (x) is the objective function of the reconstruction problem, yiMeasured value of the i-th detector, niRepresenting the total number of detector crystal blocks.
The expression for adding the structural dictionary constraint is as follows:
Figure BDA0002220561370000075
wherein: l (x) is an objective function of the reconstruction problem, αsIs a sparse coefficient, D is a structural dictionary, EsIs a partition matrix, and λ is a weightCoefficient nsIs the number of blocks, λ and μ are both weight coefficients.
The structure dictionary D is a matrix obtained from CT images using the K-SVD Algorithm according to the document published by Aharon in 2006 (reference: K-SVD: An Algorithm for Designing over dictionary for sparse Representation). Fig. 1 is a CT chest slice image, and fig. 2 is a structural dictionary matrix graph trained from the CT chest slice image.
We first solve the x sub-problem, rewriting the objective function to the form:
Figure BDA0002220561370000081
wherein: y isiMeasured by the i-th detector, asIs a sparse coefficient, D is a structural dictionary, EsIs a partition matrix, λ is a weight coefficient, nsIs the number of blocks, λ and μ are both weight coefficients.
Next, a hidden variable w is introducedij：
Figure BDA0002220561370000082
Wherein: hidden variable wijRepresenting the number of photons emitted from voxel j that are detected by the detector crystal block i, gijIs the ijth entry in the system matrix G.
If wijAnd is known, then the equation has a solution at this time. Afterwards, the EM algorithm is performed in two steps:
step E: using the estimated time x of the k-th iterationkAnd the sinogram y to estimate the hidden variable wij. Will be estimatedSubstituted into Ωx(wijX, α), the equation of real time is obtained
Figure BDA0002220561370000092
Step M: by making equations of immediacy
Figure BDA0002220561370000093
Is equal to zero, a new x is found.
In step E, the current estimate x is knownjAnd sinogram y, estimate wijThe desired formula of (a) is:
Figure BDA0002220561370000094
in step M, it is not possible to directly pair
Figure BDA0002220561370000095
This term is derived, so it replaces itself with its convexly separable proxy equation. Firstly, the first step is to
Figure BDA0002220561370000096
And [ Esx]lThe following is rewritten:
Figure BDA0002220561370000097
wherein: [ E ]sx]lRepresents EsItem l, n of xlRepresents EsThe total number of elements in the matrix of x,
Figure BDA0002220561370000098
is the value of the currently estimated image x at the jth pixel point, esljIs a matrix EsFor all j, epsilonsljAre all greater than 0.
Due to ([ E)sx]l-[Dαs]l)2Is a convex function, so:
Figure BDA0002220561370000099
substitution equation
Figure BDA00022205613700000910
In the method, a convex separable proxy equation omega (x, x) of the method is obtainedk)：
Figure BDA0002220561370000101
Calculating the partial derivative of x:
Figure BDA0002220561370000102
Figure BDA0002220561370000103
is a solution of a second order polynomial:
Figure BDA0002220561370000104
Figure BDA0002220561370000106
from the above process, the sub-problem of x is not directly solved, and we need to converge to an approximate solution through continuous iteration of the steps E and M.
Good initial parameters enable the relaxation algorithm to speed up the convergence speed of the algorithm without increasing RMSE. Initializing the OMP with the largest t/3 coefficients in the last iteration and running the standard OMP to find the remaining 2t/3 coefficients can gracefully improve RMSE and run time. With the CoefROMP algorithm, the desired coding for coefficient reuse is formed. Then we choose the best Matching Pursuit (OMP) algorithm to solve the alpha sub-problem.
Figure BDA0002220561370000111
Wherein:αsis a sparse coefficient, D is a structural dictionary, EsIs a partition matrix, | | αs||0Representing the sparse coefficient αSThe number of the non-zero elements in (E) is a minimum value.
We used a digital phantom model experiment of the lungs, which contains some regions of high concentration, to demonstrate the effectiveness of this embodiment. The experimental operating environment is as follows: 4G memory, 2.30GHz, 64-bit operating system, CPU is intel i 5.
The PET image reconstruction method based on the structure dictionary in the embodiment is compared with the reconstruction result of the traditional ML-EM (maximum expectation-maximization algorithm), the same observation value Y and the same system matrix G are used for ensuring the comparability of the result, and the specific parameters are set as follows: y is a sinogram acquired by n x n dimensions, wherein m is n x n, and G is a system matrix calculated in advance by m x m dimensions; where n is 128, i.e., m is 16384.
For the verification of the quality of the reconstructed image, 128 projection angles of high-dimensional raw acquisition data are adopted, each angle is 128 beams, namely m is 16384, and the size of the reconstructed image is 128 × 128, namely the dimension is 16384; the initial value setting is the same as above. Fig. 3 to 5 are schematic diagrams showing a comparison of a true value image, an image reconstructed based on a conventional ML-EM method, and an image reconstructed based on the present embodiment, respectively, and it can be seen visually that compared with the result of the ML-EM, the image reconstructed based on the present embodiment has an unsatisfactory ML-EM algorithm effect, and the reconstructed image has blurred images, many noise points, and very unclear edges. In contrast, the regions of the picture reconstructed by our algorithm have clear boundaries and the interior of the picture is smooth.
For the same data, the comparison was performed by the present embodiment and the conventional ML-EM method, respectively, as shown in table 1; the deviation, variance and root mean square error of the reconstruction result from the true value are smaller than those of the traditional ML-EM method by applying the embodiment, thereby demonstrating the feasibility of the technical scheme of the invention in the aspects of improving the accuracy and reducing the noise.
TABLE 1
Method of producing a composite material Deviation of Variance (variance) Root mean square error
ML-EM 0.1919 0.0543 0.2331
The present embodiment 0.1153 0.0341 0.1845
The embodiments described above are intended to facilitate one of ordinary skill in the art in understanding and using the invention. It will be readily apparent to those skilled in the art that various modifications to the above-described embodiments may be made, and the generic principles defined herein may be applied to other embodiments without the use of inventive faculty. Therefore, the present invention is not limited to the above embodiments, and those skilled in the art should make improvements and modifications to the present invention based on the disclosure of the present invention within the protection scope of the present invention.

Claims (6)

1. A dictionary learning method for PET image reconstruction is characterized by comprising the following steps:
(1) detecting biological tissues injected with the radioactive medicament by using a detector, acquiring coincidence counting vectors corresponding to each crystal block of the detector, and further constructing a coincidence counting matrix y of the PET;
(2) according to the PET imaging principle, a PET measurement equation is established as follows; introducing Poisson noise constraint to the measurement equation to obtain a Poisson model L (x) of the PET;
y＝Gx+r+v
wherein: g is a system matrix, x is a PET concentration distribution matrix, and r and v are measurement noise matrices related to reflection coincidence events and scattering coincidence events respectively;
(3) introducing structural dictionary constraint into the Poisson model L (x), and obtaining a PET image reconstruction model based on the structural dictionary constraint as follows:
Figure FDA0002220561360000011
wherein: λ is a weight coefficient, Rsparse(x, α) are penalty terms for the PET concentration distribution matrix x and the sparse coefficient matrix α;
(4) and (4) carrying out optimization solution on the PET image reconstruction model to obtain a PET concentration distribution matrix x, and further reconstructing to obtain a PET image.
2. The method for learning a dictionary for PET image reconstruction according to claim 1, wherein: the expression of the Poisson model l (x) is as follows:
wherein: y isiTo fit the coincidence count vector corresponding to the ith crystal block in the count matrix y,to count the vector yiAverage of all elements in, niThe total number of crystal blocks of the detector.
3. The method for learning a dictionary for PET image reconstruction according to claim 2, wherein: the penalty term RsparseThe expression of (x, α) is as follows:
Figure FDA0002220561360000021
wherein: d is a structural dictionary, EsTo segment the operator, Esx is the s-th n-dimensional submatrix in the PET concentration distribution matrix x, mu is a weight coefficient, ns＝(m-n+1)2M is the dimension of the PET concentration distribution matrix x, n is the dimension of the preset submatrix, alphasIs the s-th column sparse coefficient vector in the sparse coefficient matrix alpha, | alphas||0Representing a sparse coefficient vector alphasNumber of middle non-zero elements, | | | | | non-phosphor2Representing the L2 norm. Introducing a mask matrix and iteratively solving alphasAnd D.
4. The PET image reconstruction-oriented dictionary learning method according to claim 3, characterized in that: the specific method for carrying out optimization solution on the PET image reconstruction model in the step (4) is as follows: and (3) solving the PET concentration distribution matrix x by the following iterative equation based on the EM algorithm:
Figure FDA0002220561360000023
Figure FDA0002220561360000024
wherein:and
Figure FDA0002220561360000026
are all the intermediate variables of the series of the Chinese characters,
Figure FDA0002220561360000027
for the (k + 1) th iteration PET concentration distribution matrix xk+1The value of the j-th element in (b),for the kth iteration PET concentration distribution matrix xkValue of the jth element of (1), esljTo segment operator EsRow i and column j element values in (1),
Figure FDA0002220561360000032
nj＝m2，Esxkfor the kth iteration PET concentration distribution matrix xkOf [ E ] an s-th n-dimensional sub-matrix, [ E ]sxk]lIs a sub-matrix EsxkThe value of the l-th element in (b),
Figure FDA0002220561360000033
for the structural dictionary D and the k-th iteration sparse coefficient vector
Figure FDA0002220561360000034
Multiplying the l-th element value, g, in the matrixijFor the ith row and jth column element values in the system matrix G,
Figure FDA0002220561360000035
representing the PET concentration distribution matrix x from the k-th iterationkThe number of photons detected by the ith crystal block of the detector is emitted from the jth element in (a), and k is the number of iterations.
5. The PET image reconstruction-oriented dictionary learning method according to claim 4, characterized in that: the number of photons
Figure FDA0002220561360000036
The expression of (a) is as follows:
Figure FDA0002220561360000037
wherein: r isiAnd viThe ith element value in the measurement noise matrices r and v, respectively.
6. The method for PET image reconstruction as recited in claim 4, wherein: the dictionary and a sparse representation matrix obtained after the last iteration are used, and before OMP algorithm sparse coding is used, the method selects the maximum t/3(t is a target base number) coefficients from sparse matrices obtained after the previous D iteration (in the previous tracking stage) to initialize the current sparse matrix. The process then performs coefficient enhancement and optimization similar to the CoSaMP and SP algorithms. Finally, the k-th iteration sparse coefficient vector
Figure FDA0002220561360000038
And solving the following target equation by adopting an OMP algorithm to obtain:
wherein: ρ is a preset minimum value.
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