CN109636869B - Dynamic PET image reconstruction method based on non-local total variation and low-rank constraint - Google Patents
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Abstract
The invention discloses a dynamic PET image reconstruction method based on non-local total variation and low-rank constraint, which utilizes the segmentation smooth characteristic and the space-time correlation of a PET image and introduces the low-rank constraint and the non-local total variation constraint at the same time, realizes the reconstruction of the dynamic PET image, can remove the step effect and keep fine details, and is beneficial to improving early focus detection. The method optimizes the space-time correlation of the solution through the constraint of low rank and sparsity, namely, the background components and the detail components of the image are reconstructed through a combined model; meanwhile, in the reconstruction of the dynamic PET image, in order to fully consider the structural smoothness characteristic of image data, the invention introduces non-local total variation constraint, and recovers more image details and removes the step effect by utilizing the image redundancy characteristic. Compared with the prior art, the method can provide more accurate reconstructed images to improve lesion detection, and has better robustness to noise.
Description
Technical Field
The invention belongs to the technical field of PET imaging, and particularly relates to a dynamic PET image reconstruction method based on non-local total variation and low-rank constraint.
Background
Dynamic Positron Emission Tomography (DPET) enables monitoring of the spatiotemporal distribution of radiolabeled tracers in vivo, which has the potential to improve early detection, cancer characterization and treatment response assessment. The DPET uses a detector system placed around an object so as to obtain a plurality of different angle views in a series of time frames, and a radiotracer concentration map can be reconstructed by using the projection data, namely dynamic PET image reconstruction, and the dynamic processes of tracer uptake and metabolism can be better evaluated through time series images, so that the DPET has important application value in scientific research and clinical application.
PET image reconstruction is a pathologic inverse problem, and it is standard practice to use a regularization term to constrain the solution of the problem, so that the inverse problem is adaptive. In addition to traditional algorithms such as ML-EM (maximum likelihood-effective) and the like, the first type of algorithm is space smooth constraint, one of the algorithms is a Maximum A Posteriori (MAP) algorithm, and the design of penalty items for keeping smoothness of a region and sharp change of an edge is always the key point of PET (positron emission tomography) reconstruction research; another approach is to use the full variation constraint (TV) to boost the structural smoothness property of PET images, however TV-based models assume that each image pixel always has a diffusion direction of edges and gradients, which may lead to a step effect. The second category of algorithms is to exploit both temporal and spatial information to improve the quality of dynamic PET image reconstruction, temporal constraints are typically used to improve the additional robustness of the spatial solution, including spatio-temporal spline models, tracing dynamics, wavelet transforms, etc., whereas tracing dynamics methods assume that all voxels are well modeled by the same set of model dynamics, which may not be the case in practice. In addition, the wavelet transform method still leaves a selectable space for selecting the E-spline wavelet parameter vector and the appropriate wavelet coefficients, and in terms of early lesion detection problems using PET images, although the supplemental information derived from CT and MRI images is incorporated into the regularization constraints during reconstruction, it does not provide information about organ and lesion metabolism, and therefore lacks sufficient information to guide the reconstruction of dysfunctional regions. Some studies have shown that using anatomical boundaries without accurate lesion contours does not improve the lesion detection or quantification task.
Disclosure of Invention
In view of the above, the invention provides a dynamic PET image reconstruction method based on non-local total variation and low-rank constraint, which utilizes the segmentation smoothing property and the space-time correlation of a PET image and introduces low-rank constraint and non-local total variation constraint at the same time, thereby realizing dynamic PET image reconstruction, removing the step effect, keeping fine details and being beneficial to improving early lesion detection.
A dynamic PET image reconstruction method based on non-local total variation and low-rank constraint comprises the following steps:
(1) detecting biological tissues injected with the radioactive medicament by using a detector, dynamically acquiring coincidence counting vectors corresponding to each moment, and combining the coincidence counting vectors into a coincidence counting matrix Y;
(2) combining dynamic PET image sequences into a PET concentration distribution matrix X, and establishing a PET measurement equation according to a PET imaging principle;
(3) introducing low-rank constraint to the PET measurement equation to obtain a dynamic PET image reconstruction model M1 based on the low-rank constraint;
(4) non-local total variation constraint is carried out on the low-rank part and the sparse part of each frame of image, and a dynamic PET image reconstruction model M2 based on the non-local total variation constraint is further obtained;
(5) the objective function of the dynamic PET reconstruction obtained by combining the dynamic PET image reconstruction models M1 and M2 is as follows:
s.t.L+S＝X
wherein: | | | represents the nuclear norm, | | | | the non-woven vision1Representing a 1-norm, L being a low rank portion containing periodically varying radioactivity concentrations in the image background, S being a sparse portion containing radioactivity concentrations of non-homogeneous tissue with different metabolic rates, JNLTV(L) and JNLTV(S) is the result of non-local total variation of the low rank part L and the sparse part S, lambda, mu and alphaLAnd alphaSAll are weight coefficients, Ψ (Y | X) is a likelihood function for X and Y;
(6) and (4) carrying out optimization solution on the objective function to obtain a PET concentration distribution matrix X, so as to restore a dynamic PET image sequence.
Further, the expression of the PET measurement equation is as follows:
Y＝DX+R+S
wherein: d is a system matrix, and R and S are measurement noise matrices reflecting random events and scattering events, respectively.
Further, the expression of the dynamic PET image reconstruction model M1 is as follows:
M1＝||L||*+λ||S||1+μΨ(Y|X)
s.t. L+S＝X
further, the expression of the dynamic PET image reconstruction model M2 is as follows:
M2＝αLJNLTV(L)+αSJNLTV(S)+μΨ(Y|X)
s.t. L+S＝X
further, the likelihood function Ψ (Y | X) is expressed as follows:
wherein: dijIs the value of the ith row and jth column element in the system matrix D, yimFor matching the value of the element in the ith row and the mth column in the count matrix Y, xjm is the element value of the jth row and mth column in the PET concentration distribution matrix X, rim is the element value of the ith row and mth column in the measurement noise matrix R, simFor measuring the element value of the ith row and the mth column in the noise matrix S, I, J and M are natural numbers, I is more than or equal to 1 and less than or equal to I, J is more than or equal to 1 and less than or equal to J, M is more than or equal to 1 and less than or equal to M, I is the dimension which accords with the counting vector, J is the number of rows of the PET concentration distribution matrix X, namely the number of pixels of the PET image, and M is the number of columns of the PET concentration distribution matrix X, namely the sampling time length.
Further, in the step (6), an ADMM (Alternating Direction Method of Multipliers) algorithm is adopted to perform optimization solution on the objective function; the low-rank constraint problem is subjected to iterative optimization solution by using a singular value threshold algorithm and a soft shrinkage algorithm, the likelihood function problem is subjected to iterative optimization solution by using an EM (Expectation maximization) algorithm, and the non-local total variation constraint problem is subjected to iterative optimization solution by using a gradient descent method.
Different from the existing method taking time prior and space prior as two different constraints, the method optimizes the space-time correlation of the solution by using one constraint of low rank and sparsity, namely, reconstructs the background component and the detail component of the image by using a joint model; meanwhile, in the reconstruction of the dynamic PET image, in order to fully consider the structure smoothness characteristic of image data, the invention introduces non-local total variation (NLTV) constraint, and recovers more image details and removes the step effect by utilizing the image redundancy characteristic. Compared with TV, NLTV integrates local and non-local correlation of an image matrix, the invention can provide more accurate reconstructed images to improve lesion detection and has better robustness to noise.
According to the method, non-local total variation constraint is introduced into a low-rank matrix recovery analysis framework so as to ensure the structural smoothness and clear boundaries of regions of interest (ROIs) in the PET image; the low rank constraint of the image sequence eliminates noise through inherent averaging in tissues, and simultaneously introduces non-local total variation to improve the spatial resolution of the PET image is also the innovation point of the invention; the low rank and sparse matrix decomposition can provide random noise components for the NLTV constraint, which can provide enhanced clean PET images with confirmation of random noise information and in turn assist in the decomposition of the low rank matrix and sparse matrix to obtain a reconstructed result that better conforms to reality. By combining the performance of the invention in a simulation data experiment, the invention can obtain more accurate reconstruction results by comparing the results with results of an ML-EM algorithm and an LRTV algorithm (based on low rank and total variation constraint), and has important practical application value for improving early focus detection, evaluating dynamic processes of tracer uptake and metabolism and the like.
Drawings
Fig. 1 is a schematic flow chart of a dynamic PET image reconstruction method according to the present invention.
Fig. 2(a) is a template image of the monte carlo simulated Zubal chest data.
Fig. 2(b) is a template image of Hoffman brain data.
FIG. 3(a) is a real image of Zubal thorax data frame 8.
FIG. 3(b) shows a data count rate of 1X 107And simulating a PET image result of the Zubal chest data reconstruction frame 8 by using an ML-EM method for Monte Carlo.
FIG. 3(c) shows a data count rate of 1X 107Next, 8 th frame PET image results of Zubal thoracic data reconstruction were simulated for Monte Carlo using LRTV method.
FIG. 3(d) shows a data count rate of 1X 107The method is adopted to simulate the 8 th frame PET image result reconstructed by Zubal thorax data for Monte Carlo.
FIG. 4(a) is a result of an enlarged image of the portion outlined in FIG. 3 (a).
FIG. 4(b) is the image result of the enlarged portion outlined in FIG. 3 (b).
FIG. 4(c) is the image result of the enlarged portion outlined in FIG. 3 (c).
FIG. 4(d) is the image result of the enlarged portion outlined in FIG. 3 (d).
FIG. 5(a) shows a data count rate of 1X 107Deviation line plot of the lower ROI2 per frame Zubal thorax data image results.
FIG. 5(b) shows a data count rate of 1X 107Variance line plot of the lower ROI2 per frame Zubal thorax data image results.
FIG. 6(a) shows a data count rate of 1X 107The 14 th frame PET image result of Zubal thorax data reconstruction is simulated by adopting an ML-EM method for Monte Carlo.
FIG. 6(b) shows a data count rate of 1X 107The 14 th frame PET image result of Zubal thoracic data reconstruction was simulated for Monte Carlo using the LRTV method.
FIG. 6(c) shows the data count rate is 1X 107The method is adopted to simulate the result of the PET image of the 14 th frame reconstructed by Zubal thoracic cavity data for Monte Carlo.
FIG. 7(a) shows a data count rate of 1X 106The 14 th frame PET image result of Zubal thorax data reconstruction is simulated by adopting an ML-EM method for Monte Carlo.
FIG. 7(b) shows the data count rate of 1X 106The 14 th frame PET image result of Zubal thoracic data reconstruction was simulated for Monte Carlo using the LRTV method.
FIG. 7(c) shows a data count rate of 1X 106The method is adopted to simulate the result of the PET image of the 14 th frame reconstructed by Zubal thoracic cavity data for Monte Carlo.
Fig. 8(a) is a real image of frame 11 of Hoffman brain data.
FIG. 8(b) shows a data count rate of 1X 107And simulating the result of the PET image of the 11 th frame reconstructed by Hoffman brain data on Monte Carlo by adopting an ML-EM method.
FIG. 8(c) shows a data count rate of 1X 107And simulating the PET image result of the reconstructed Hoffman brain data of the 11 th frame by using a Monte Carlo method.
FIG. 8(d) shows a data count rate of 1X 107Lower application of the inventionThe method simulates the result of the 11 th frame PET image reconstructed by Hoffman brain data on Monte Carlo.
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.
As shown in FIG. 1, the dynamic PET image reconstruction method based on non-local total variation and low rank constraint of the invention comprises the following steps:
s1, establishing a measurement data matrix Y and a system matrix D according to a dynamic PET scanning mode.
Dividing a scanning process of the dynamic PET into a certain number of time frames according to needs, and constructing a measurement data matrix Y of the dynamic PET according to a time sequence of a coincidence counting vector acquired by a detector in each time frame; and counting the probability that the emitted photons at each pixel point are received by each detector, thereby obtaining a system matrix D.
And S2, establishing a dynamic PET imaging model.
For dynamic PET imaging, it requires a series of successive time frame samples of the time information of the tracer. For each individual time frame, the projection data y represents the sum of coincidence events captured by each detector during that time period, i.e., y ═ y { (y)iI ═ 1,2, …, I }, where I is the total number of detectors; the corresponding radiodensity image is recorded as x ═ { x ═ xjJ ═ 1,2, …, J }, where J is the total number of pixels; the relationship between the measurement data y and the unknown concentration profile x is:
due to the independence of the poisson hypothesis, we have a likelihood function of y:
to facilitate the solution, we log the likelihood function and minimize the negative log of the likelihood function:
for dynamic PET, we can combine all the time frame information to get a data matrix. For the m-th frame of projection data, vector ymRepresents the m-th column of the data matrix Y,
yimrepresenting projection data captured by the ith detector for the mth frame; in the same way as above, the first and second,
xjmrepresenting the radioactivity concentration of the jth pixel point of the mth frame; we sum the negative log-likelihood functions for each frame to yield:
wherein: y isimRepresenting the m-th frame of projection data ymThe value of the i-th detector of (c),
is the m-th frame
The ith entry of (2).
And S3, low-rank and sparse constraint.
Performing low rank and sparse decomposition on PET images, wherein L component comprises periodic variation of radioactivity intensity in background, and S component refers to those tissues with different metabolic rates, and establishing the following decomposition model based on the L component and the S component:
scaling the objective function (5) to a convex optimization problem:
wherein: | L | Lily calculation*Represents the kernel norm of the matrix L, i.e. the sum of the singular values of L. | S | non-woven phosphor1L represents S1And (4) norm.
And S4, non-local total variation constraint.
Let P be [ P ]1,p2,…,pm,…,pM]Representing a sequence of M frames, in which
Is the m-th frame image vector. Each column is divided into two rows
Conversion to its matrix form
In which UxV ═ J, is converted
Now in space
In (1). We sequentially apply to each frame of the image matrix
Carrying out non-local total variation constraint:
wherein: u, v are pixels in omega spacePoints, which are natural numbers, w (u, v) is a non-negative symmetric non-local weight function, GδIs a gaussian kernel with a standard deviation of δ and h is a filter parameter.
And S5, reconstructing the LRNLTV of the dynamic PET image.
Applying non-local total variation constraints to decomposed L and S respectively, and combining the low rank and sparse constraints before, we have the following objective functions:
s.t. L+S＝X
wherein: λ, μ, αLAnd alphaSAre all weight coefficients.
And S6, optimizing algorithm based on the augmented Lagrange multiplier method.
Introducing auxiliary variables P and Q, and writing an augmented Lagrangian function of an objective function (8):
wherein: z, ZL，ZSBeing Lagrangian multipliers, betaL、βSIs a penalty parameter; the problem is decomposed into five sub-problems to be solved, and the sub-problems are divided into three categories according to the property of each sub-problem.
6.1 solving the L, S subproblem:
for the L subproblem, other variables are fixed, only items related to L are extracted, and a formula is arranged, wherein the formula comprises the following components:
solving equation (10) using singular value thresholding, then the iterative update with L is:
wherein: a. theε(Γ)＝UBε(s)VT，UsVTIs the singular value decomposition of Γ, in which the soft-reduction algorithm Bε(s) is:
also for the S subproblem, the simplified formula is followed by:
solving equation (12) using a soft-shrink algorithm, then the iterative update with S is:
6.2 solving the X sub-problem:
first a negative likelihood function Ψ (w | X) for the hidden variable w is written, where
Representing the emitted photon from pixel j detected at coincidence line i for the mth frame:
next X is optimized using the EM algorithm.
E, step E: taking the condition expectation of w as a matter of course,
and insert it into Ψ (w | X); then we have an alternative function phi (X; X)k)：
And M: calculating phi (X; X)k) For xjmAnd is made equal to 0; after simplification, x is foundjmThe solution of (a) is the root of the following quadratic equation:
equation (15) is a convex function, and we update x by taking the larger heeljm：
s.t. ajm＝β，
Wherein: []jmRepresenting the jth entry of the matrix.
6.3 solving the P, Q sub-problem:
for the P sub-problem, other variables are also fixed, only the item related to L is extracted, and the formulation is finished, including:
we solve equation (17) using the gradient descent method, first we write Euler-Lagrange of equation (7):
for fixed u, there are:
we will refer to each column of the L matrix
Conversion to its matrix form
ZLIs also converted to its matrix form, where U × V ═ J, and L and ZLEach frame of (a) is processed in turn; for each frame of the image, there is
The iteration of (2) is more recent:
wherein,
is a step of gradient descent. After all frames have been processed, the matrix is processed
Conversion back to vector form
L can be recovered and P and Z can be recovered in the same mannerLThen the P sub-problem is solved.
For the Q sub-problem, since it has the same form as the P sub-problem, it is solved in the same way; wherein the lagrange multiplier is updated as usual.
The following is that we performed experiments on Zubal thoracic and Hoffman brain template data simulated by Monte Carlo to verify the accuracy of the system reconstruction results of the present invention. Fig. 2(a) and 2(b) are schematic templates of Zubal thoracic and Hoffman brain data, respectively, used for experiments, dividing the different regions into three regions of interest (where ROI4 is the background region). The experimental operating environment is as follows: 8G memory, 3.40GHz, 64-bit operating system, CPU is intel i 7-3770; the model of the simulated PET scanner is Hamamatsu SHR-22000, and the designed radionuclide and medicine are18F-FDG, set sinogram for 64 projection angles, data results collected by 64 beams at each angle, and the size of system matrix D is 4096 × 4096. In this test, the test is carried out on a 1X 10 substrate6、1×107The projection data at two different count rates were tested.
The PET image results of the inventive reconstruction framework are compared with the image results of the ML-EM and LRTV reconstruction methods, respectively, using the same measured data matrix Y and system matrix D to control the comparability of the results. As can be seen from fig. 3(a) to fig. 3(d), the noise of the image result of the frame reconstructed by the present invention in the region is significantly smaller than that of the images reconstructed by the other two methods, and the functional region is smoother while ensuring the edge contrast. Fig. 4(a) to fig. 4(d) are enlarged image results of the framed parts of fig. 3(a) to fig. 3(d), respectively, and it is obvious that the image results reconstructed by the present invention have clearer ROI boundary information, and have obvious improvement effect on early lesion detection. Fig. 5(a) -5 (b) show the quantization error of each frame of ROI2 for Zubal thoracic data, further illustrating the accuracy of the reconstruction results of the present invention. FIG. 6(a) to FIG. 6(c) and FIG. 7(a) to FIG. 7(c) show the respective count rates of 1X 107And 1X 106The reconstruction results of the following three reconstruction methods on the 14 th frame of Zubal thoracic cavity data prove that the reconstruction results have robustness to noise, and the further analysis of the quantization results is shown in Table 1. FIGS. 8(a) -8 (d) are reconstruction experiments performed on Hoffman brain data demonstrating the reconstruction of the present inventionThe result is robust to different template data.
TABLE 1
The embodiments described above are presented to facilitate one of ordinary skill in the art to understand and practice the present 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 dynamic PET image reconstruction method based on non-local total variation and low-rank constraint comprises the following steps:
(1) detecting biological tissues injected with the radioactive medicament by using a detector, dynamically acquiring coincidence counting vectors corresponding to each moment, and combining the coincidence counting vectors into a coincidence counting matrix Y;
(2) combining dynamic PET image sequences into a PET concentration distribution matrix X, and establishing a PET measurement equation according to a PET imaging principle;
(3) introducing low-rank constraint to the PET measurement equation to obtain a dynamic PET image reconstruction model M1 based on the low-rank constraint;
(4) non-local total variation constraint is carried out on the low-rank part and the sparse part of each frame of image, and a dynamic PET image reconstruction model M2 based on the non-local total variation constraint is further obtained;
(5) the objective function of dynamic PET reconstruction obtained by combining the dynamic PET image reconstruction models M1 and M2 is as follows:
s.t.L+S＝X
wherein: | | non-woven hair*Represents the kernel norm, | | | | luminance1Representing a 1-norm, L being a low rank portion containing periodically varying radioactivity concentrations in the image background, S being a sparse portion containing radioactivity concentrations of non-homogeneous tissue with different metabolic rates, JNLTV(L) and JNLTV(S) is the result of non-local total variation of the low rank part L and the sparse part S, lambda, mu and alphaLAnd alphaSAll are weight coefficients, Ψ (Y | X) is a likelihood function for X and Y;
(6) and (4) carrying out optimization solution on the objective function to obtain a PET concentration distribution matrix X, so as to restore a dynamic PET image sequence.
2. The dynamic PET image reconstruction method according to claim 1, characterized in that: the expression of the PET measurement equation is as follows:
Y＝DX+R+S
wherein: d is a system matrix, and R and S are measurement noise matrices reflecting random events and scattering events, respectively.
3. The dynamic PET image reconstruction method according to claim 1, characterized in that: the expression of the dynamic PET image reconstruction model M1 is as follows:
M1＝||L||*+λ||S||1+μΨ(Y|X)
s.t.L+S＝X。
4. the dynamic PET image reconstruction method according to claim 1, characterized in that: the expression of the dynamic PET image reconstruction model M2 is as follows:
M2＝αLJNLTV(L)+αSJNLTV(S)+μΨ(Y|X)
s.t.L+S＝X。
5. the dynamic PET image reconstruction method according to claim 2, characterized in that: the likelihood function Ψ (Y | X) is expressed as follows:
wherein: dijIs the value of the ith row and jth column element in the system matrix D, yimFor the value of the element in the ith row and mth column in the coincidence count matrix Y, xjmIs the value of the jth row and mth column element r in the PET concentration distribution matrix XimFor measuring the value of the element, s, in the ith row and mth column of the noise matrix RimFor measuring the element value of the ith row and the mth column in the noise matrix S, I, J and M are natural numbers, I is more than or equal to 1 and less than or equal to I, J is more than or equal to 1 and less than or equal to J, M is more than or equal to 1 and less than or equal to M, I is the dimension which accords with the counting vector, J is the number of rows of the PET concentration distribution matrix X, namely the number of pixels of the PET image, and M is the number of columns of the PET concentration distribution matrix X, namely the sampling time length.
6. The dynamic PET image reconstruction method according to claim 1, characterized in that: in the step (6), an ADMM algorithm is adopted to carry out optimization solution on the objective function; the low-rank constraint problem is subjected to iterative optimization solution by using a singular value threshold algorithm and a soft shrinkage algorithm, the likelihood function problem is subjected to iterative optimization solution by using an EM (effective electromagnetic) algorithm, and the non-local total variation constraint problem is subjected to iterative optimization solution by using a gradient descent method.
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