CN109636869A - The dynamic PET images method for reconstructing constrained based on non local full variation and low-rank - Google Patents
The dynamic PET images method for reconstructing constrained based on non local full variation and low-rank Download PDFInfo
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Abstract
The invention discloses a kind of dynamic PET images method for reconstructing constrained based on non local full variation and low-rank, this method utilizes the sectionally smooth characteristic and temporal correlation of PET image, low-rank constraint and non local full variational methods are introduced simultaneously, realize dynamic PET images reconstruction, alias can be removed and keep fine detail, be conducive to improve the detection of early stage lesion.The present invention optimizes the temporal correlation of solution by low-rank and this sparse constraint, that is to say through a conjunctive model and reconstructs the background component and details ingredient of image；Simultaneously in dynamic PET images reconstruction, for the structure smooth property for fully considering image data, invention introduces non local full variational methods, restore more image details using image redundancy characteristic and remove alias.Compared with the prior art, the present invention can provide more accurate reconstruction image to improve lesion detection, and have better robustness to noise.
Description
Technical field
The invention belongs to PET technical field of imaging, and in particular to a kind of to be moved based on what non local full variation and low-rank constrained
State PET image reconstruction method.
Background technique
Dynamic positron emission tomography scanning (DPET) can monitor the internal spatial and temporal distributions of radio-labeled tracers,
With early detection is improved, cancer characterizes and the potentiality of therapy response assessment.DPET uses the detector being placed on around object
System can be reconstructed using these data for projection and be put to obtain many different angular views in a series of time frames
Penetrating property tracer concentration figure is that dynamic PET images are rebuild, can preferably assess tracer uptake by time-series image
With the dynamic process of metabolism, there is significant application value in scientific research and clinical application.
PET image reconstruction is an ill inverse problem, and the way of standard is to carry out the solution of restricted problem using regular terms, from
And make inverse problem that there is well-posedness.In addition to traditional ML-EM scheduling algorithm, first kind algorithm is spatial smoothness constraints, therein
One is maximum a posteriori probability (MAP) algorithms, and design retains region smoothness and edge penalty term jumpy is always PET
Rebuild the emphasis of research；Another method is the structure smoothness properties using full variational methods (TV) Lai Tisheng PET image, however
The each image pixel of model hypothesis based on TV always has the dispersal direction of edge and gradient, this may result in ladder effect
It answers.Second class algorithm is to promote dynamic PET images reconstruction quality using time and spatial information simultaneously, time-constrain usually quilt
For additional robustness of room for promotion solution, including space-time Spline Model, tracer kinetics, wavelet transformation etc., however tracer is dynamic
Mechanics method assumes that all voxels are modeled well by identical model dynamics collection, this in practice may be really not so.
In addition, the method for wavelet transformation for selection E spline wavelets parameter vector and suitable wavelet coefficient still there are optional space,
In terms of carrying out early stage lesion test problems using PET image, although the supplement that will be obtained from CT and MRI image during reconstruction
Information has been merged into canonical constraint, but it is not provided with the information about organ and lesion metabolism, therefore lacks enough letters
Breath carrys out the reconstruction of guiding function abnormal area.It is some studies have shown that will not using the anatomical border of no accurate lesion outline
Improve lesion detection or quantification tasks.
Summary of the invention
In view of above-mentioned, the dynamic PET images reconstruction based on non local full variation and low-rank constraint that the present invention provides a kind of
Method, this method utilize the sectionally smooth characteristic and temporal correlation of PET image, while introducing low-rank constraint and non local full change
Divide constraint, realize dynamic PET images reconstruction, alias can be removed and keep fine detail, is conducive to improve sick in early days
Stove detection.
A kind of dynamic PET images method for reconstructing constrained based on non local full variation and low-rank, is included the following steps:
(1) biological tissue for being injected with radiopharmaceutical agent is detected using detector, dynamic acquisition obtains corresponding each
The coincidence counting vector at a moment, and these coincidence counting vectors are combined into coincidence counting matrix Y；
(2) make dynamic PET image combined sequence at PET concentration distribution matrix X, PET is established according to PET image-forming principle and is surveyed
Measure equation；
(3) low-rank constraint is introduced by measuring equation to the PET, obtains the dynamic PET images weight constrained based on low-rank
Established model M1；
(4) non local full variational methods are carried out by low-rank part to every frame image and sparse part, further obtained
Dynamic PET images reconstruction model M2 based on non local full variational methods；
(5) objective function for combining to obtain dynamic PET reconstruction for dynamic PET images reconstruction model M1 and M2 is as follows:
S.t.L+S=X
Wherein: | | | | * indicates nuclear norm, | | | |1Indicate 1- norm, L is that low-rank part contains week in image background
The radioactive concentration of phase variation, S are that sparse part contains the radioactive concentration of the heterogene structure with different metabolic rate,
JNLTV(L) and JNLTV(S) be respectively low-rank part L and sparse part S after non local full variation as a result, λ, μ, αLAnd αS?
For weight coefficient, Ψ (Y | X) is the likelihood function about X and Y；
(6) PET concentration distribution matrix X is obtained after carrying out optimization to above-mentioned objective function, so that reduction is set out
The PET image sequence of state.
Further, the expression formula of the PET measurement equation is as follows:
Y=DX+R+S
Wherein: D is sytem matrix, and R and S are respectively the measurement noise matrix for reflecting chance event and scattering events.
Further, the expression formula of the dynamic PET images reconstruction model M1 is as follows:
M1=| | L | |*+λ||S||1+μΨ(Y|X)
S.t. L+S=X
Further, the expression formula of the dynamic PET images reconstruction model M2 is as follows:
M2=αLJNLTV(L)+αSJNLTV(S)+μΨ(Y|X)
S.t. L+S=X
Further, the expression formula of the likelihood function Ψ (Y | X) is as follows:
Wherein: dijFor the i-th row jth column element value, y in sytem matrix DimIt is arranged for the i-th row m in coincidence counting matrix Y
Element value, xjM is jth row m column element value in PET concentration distribution matrix X, and rim is the i-th row m column in measurement noise matrix R
Element value, simFor the i-th row m column element value in measurement noise matrix S, i, j and m be natural number and 1≤i≤I, 1≤j≤
J, 1≤m≤M, I are the dimension of coincidence counting vector, line number, that is, PET image pixel that J is PET concentration distribution matrix X
Number, M are columns, that is, sampling time length of PET concentration distribution matrix X.
Further, ADMM (Alternating Direction Method of is used in the step (6)
Multipliers, alternating direction multiplier) algorithm to objective function carry out optimization；Wherein, low-rank restricted problem utilizes surprise
Different value thresholding algorithm and soft contraction algorithm are iterated Optimization Solution, and likelihood function problem utilizes EM (Expectation
MaximizationAlgorithm, expectation maximization) algorithm is iterated Optimization Solution, non local full variational methods problem benefit
Optimization Solution is iterated with gradient descent method.
Different from existing using time priori and spatial prior as the methods of two different constraints, the present invention by low-rank and
This sparse constraint to optimize the temporal correlation of solution, that is to say through a conjunctive model reconstruct the background of image at
Divide and details ingredient；Simultaneously in dynamic PET images reconstruction, for the structure smooth property for fully considering image data, the present invention
Non local full variation (NLTV) constraint is introduced, restore more image details using image redundancy characteristic and removes ladder effect
It answers.More accurate reconstruction can be provided by incorporating the part of image array and non local correlation, the present invention compared to TV, NLTV
Image has better robustness to noise to improve lesion detection.
Non local full variational methods are introduced into low-rank matrix and restored in analytical framework by the present invention, to ensure to feel in PET image
The structure flatness in interest region (ROIs) and clearly boundary；The low-rank constraint of image sequence passes through intrinsic average in tissue
Noise is eliminated, while to introduce non local full variation to improve PET image spatial resolution be also innovative point of the invention；Low-rank
It can be constrained with sparse matrix decomposition for NLTV and random noise component is provided, with the confirmation of random noise information, NLTV constraint
The clean PET image of enhancing can be provided, and help the decomposition of low-rank matrix and sparse matrix in turn, to obtain one
A reconstructed results for being more in line with truth.In conjunction with performance of the present invention in analogue data experiment, by being calculated with ML-EM
The Comparative result of method, LRTV algorithm (based on low-rank and full variational methods), the present invention can obtain accurate reconstructed results,
This has important practical application valence for the dynamic process etc. for improving the detection of early stage lesion, assessment tracer uptake and metabolism
Value.
Detailed description of the invention
Fig. 1 is the flow diagram of dynamic PET images method for reconstructing of the present invention.
Fig. 2 (a) is the template image of the thoracic cavity Monte Carlo simulation Zubal data.
Fig. 2 (b) is the template image of Hoffman brain data.
Fig. 3 (a) is the true picture of the 8th frame of the thoracic cavity Zubal data.
Fig. 3 (b) is that data counts rate is 1 × 107It is lower to use ML-EM method to the thoracic cavity Monte Carlo simulation Zubal data
The 8th frame PET image result rebuild.
Fig. 3 (c) is that data counts rate is 1 × 107It is lower to use LRTV method to the thoracic cavity Monte Carlo simulation Zubal data weight
The 8th frame PET image result built.
Fig. 3 (d) is that data counts rate is 1 × 107It is lower to use the method for the present invention to the thoracic cavity Monte Carlo simulation Zubal data
The 8th frame PET image result rebuild.
Fig. 4 (a) is by outlining the amplified image result in part in Fig. 3 (a).
Fig. 4 (b) is by outlining the amplified image result in part in Fig. 3 (b).
Fig. 4 (c) is by outlining the amplified image result in part in Fig. 3 (c).
Fig. 4 (d) is by outlining the amplified image result in part in Fig. 3 (d).
Fig. 5 (a) is that data counts rate is 1 × 107The deviation broken line of the thoracic cavity lower ROI2 every frame Zubal data image result
Figure.
Fig. 5 (b) is that data counts rate is 1 × 107The variance broken line of the thoracic cavity lower ROI2 every frame Zubal data image result
Figure.
Fig. 6 (a) is that data counts rate is 1 × 107It is lower to use ML-EM method to the thoracic cavity Monte Carlo simulation Zubal data
The 14th frame PET image result rebuild.
Fig. 6 (b) is that data counts rate is 1 × 107It is lower to use LRTV method to the thoracic cavity Monte Carlo simulation Zubal data weight
The 14th frame PET image result built.
Fig. 6 (c) is that data counts rate is 1 × 107It is lower to use the method for the present invention to the thoracic cavity Monte Carlo simulation Zubal data
The 14th frame PET image result rebuild.
Fig. 7 (a) is that data counts rate is 1 × 106It is lower to use ML-EM method to the thoracic cavity Monte Carlo simulation Zubal data
The 14th frame PET image result rebuild.
Fig. 7 (b) is that data counts rate is 1 × 106It is lower to use LRTV method to the thoracic cavity Monte Carlo simulation Zubal data weight
The 14th frame PET image result built.
Fig. 7 (c) is that data counts rate is 1 × 106It is lower to use the method for the present invention to the thoracic cavity Monte Carlo simulation Zubal data
The 14th frame PET image result rebuild.
Fig. 8 (a) is the true picture of the 11st frame of Hoffman brain data.
Fig. 8 (b) is that data counts rate is 1 × 107It is lower to use ML-EM method to Monte Carlo simulation Hoffman brain number
According to the 11st frame PET image result of reconstruction.
Fig. 8 (c) is that data counts rate is 1 × 107It is lower to use LRTV method to Monte Carlo simulation Hoffman brain number
According to the 11st frame PET image result of reconstruction.
Fig. 8 (d) is that data counts rate is 1 × 107It is lower to use the method for the present invention to Monte Carlo simulation Hoffman brain
11st frame PET image result of data reconstruction.
Specific embodiment
In order to more specifically describe the present invention, with reference to the accompanying drawing and specific embodiment is to technical solution of the present invention
It is described in detail.
As shown in Figure 1, the present invention is based on the dynamic PET images method for reconstructing that non local full variation and low-rank constrain, including
Following steps:
S1. according to Model Establishment the measurement data matrix Y and sytem matrix D of dynamic PET scan.
The scanning process of dynamic PET is marked off into a certain number of time frames, detector in each time frame as required
The measurement data matrix Y that can go out a dynamic PET according to the sequential build of time of the coincidence counting vector collected；And
The probability that the photon being emitted at each pixel is received by each detector is counted, to obtain sytem matrix D.
S2. dynamic pet imaging model is established.
For dynamic pet imaging, it needs to carry out a series of continuous time frame samplings to the temporal information of tracer.It is right
In each independent time frame, data for projection y, which is represented, meets thing by what each detector was captured during this period of time
The sum of part, i.e. y={ yi, i=1,2 ..., I }, wherein I is the total number of detector；Corresponding radioactive concentration image is recorded
For x={ xj, j=1,2 ..., J }, wherein J is the total number of pixel；Relationship between measurement data y and unknown concentration x are as follows:
Due to the independence that Poisson is assumed, we have the likelihood function of y:
For convenient for solving, we take logarithm to the likelihood function, and minimize the negative logarithm of likelihood function:
For dynamic PET, we can obtain a data matrix in conjunction with all time frame informations.M frame is thrown
Shadow data, vector ymThe m column of data matrix Y are represented,yimRepresent the throwing that m frame is captured by i-th of detector
Shadow data；Likewise,xjmRepresent the radioactive concentration of j-th of pixel of m frame；Our negative pair to each frame
Number likelihood function summation is available:
Wherein: yimRepresent m frame data for projection ymI-th of detector value,It is m frameI-th entry.
S3. low-rank and sparse constraint.
Low-rank and sparse decomposition are carried out to PET image, wherein L ingredient includes becoming the periodical of radioactive intensity in background
Change, S ingredient refers to those tissues heterogeneous with different metabolic rate, establishes following decomposition model based on this:
It is a convex optimization problem by objective function (5) scaling:
Wherein: | | L | |*Represent the nuclear norm of matrix L, i.e. the sum of singular value of L.||S||1Represent the l of S1Norm.
S4. non local full variational methods.
Enable P=[p1,p2,…,pm,…,pM] image sequence of a M frame is represented, whereinM frame image to
Amount.By each columnIt converts to its matrix formWherein U × V=J is convertedNow in spaceIn.We are successively to each frame image matrixInto
The non local full variational methods of row:
Wherein: u, v are the pixel in the space Ω, are natural number, w (u, v) is non-negative symmetrical non local weight letter
Number, GδIt is the Gaussian kernel that standard deviation is δ, h is a filtering parameter.
S5. the LRNLTV of dynamic PET images is rebuild.
Non local full variational methods are respectively acting on to the L and S decomposited, in conjunction with low-rank and sparse constraint before, I
Have following objective function:
S.t. L+S=X
Wherein: λ, μ, αLAnd αSIt is weight coefficient.
S6. based on the optimization algorithm of augmented vector approach.
Auxiliary variable P and Q are introduced, the Augmented Lagrangian Functions of objective function (8) are write out:
Wherein: Z, ZL, ZSFor Lagrange multiplier, β, βL、βSIt is punishment parameter；It is five subproblems by the PROBLEM DECOMPOSITION
It is solved, according to the property of each subproblem, is classified as three classes.
6.1 solve L, S subproblem:
For L subproblem, other variables is enabled to fix, only extracts item relevant to L, and carry out finishing formulation, have:
Formula (10) are solved using singular value threshold method, then have the iteration newer of L are as follows:
Wherein: Aε(Γ)=UBε(s)VT, UsVTIt is the singular value decomposition of Γ, soft contraction algorithm B thereinε(s) are as follows:
Similarly for S subproblem, have after abbreviation formula:
Formula (12) are solved using soft contraction algorithm, then have the iteration newer of S are as follows:
6.2 solve X subproblem:
The negative likelihood function Ψ (w | X) for hidden variable w is write out first, whereinIndicate that meeting for m frame is examined at line i
The transmitting photon from pixel j measured:
Next optimize X using EM algorithm.
E step: taking the conditional expectation of w,And it is inserted into Ψ (w | X)；Then we have alternative functions
φ(X；Xk):
M step: φ (X is calculated；Xk) for xjmDerivative, and enable its be equal to 0；X is found after abbreviationjmSolution be following secondary
Equation root:
Formula (15) is convex function, we are biggish with updating x by takingjm:
s.t. ajm=β,
Wherein: []jmJ-th of entry of representing matrix.
6.3 solve P, Q subproblem:
For P subproblem, other variables is equally allowed to be fixed, only extracts item relevant to L, and carry out finishing formulation, have:
We solve formula (17) using gradient descent method, we write out the Euler-Lagrange of formula (7) first:
For fixed u, have:
We are by each column of L matrixIt converts to its matrix formZLEach column similarly quilt
It converts to its matrix form, wherein U × V=J, and L and ZLEach frame successively handled；Then for each frame of image,
HaveIteration newer:
Wherein,It is the stepping of gradient decline.After all frames are all processed, by matrixIt is converted back to vector
FormL can be retrieved, restores P and Z in the same wayL, then P subproblem is solved.
For Q subproblem, since it has form identical with P subproblem, therefore it is solved with same method；Its
In, Lagrange multiplier updates as usual.
Us below to the thoracic cavity Zubal of Monte Carlo simulation and Hoffman brain template data carry out experiment to
Verify the accuracy of present system reconstructed results.Fig. 2 (a) and Fig. 2 (b) be respectively test the thoracic cavity Zubal used and
The template schematic diagram of Hoffman brain data, different regions is divided into three interested regions, and (wherein ROI4 is background area
Domain).Test running environment are as follows: 8G memory, 3.40GHz, 64 bit manipulation systems, CPU are intel i7-3770；The PET simulated
Scanner models be Hamamatsu SHR-22000, the radionuclide and drug set as18F-FDG, setting sinogram are
64 projection angle data results that 64 beam acquisitions arrive under each angle, the size of sytem matrix D is 4096 ×
4096.In this trial, to 1 × 106、1×107Data for projection under two different counting rates is tested.
By the PET image result of the reconstruction framework of the present invention image result with two kinds of method for reconstructing of ML-EM and LRTV respectively
It is compared, the two is using identical measurement data matrix Y and sytem matrix D with the comparativity of control result.From Fig. 3 (a)~
As can be seen that noise of the image result of reconstruction framework of the present invention in region is significantly less than other two method and weighs in Fig. 3 (d)
The image built, it is more smooth in functional area in the case where guaranteeing edge contrast.Fig. 4 (a)~Fig. 4 (d) is Fig. 3 (a) respectively
The image result of the outlined part amplification of~Fig. 3 (d), hence it is evident that as can be seen that the image result that the present invention is rebuild has more clearly
Clear ROI boundary information improves significantly for the lesion detection tool of early stage.Fig. 5 (a)~Fig. 5 (b) is illustrated
The quantization error of each frame of the ROI2 of the thoracic cavity Zubal data, further illustrates the accuracy of reconstructed results of the present invention.Fig. 6
(a)~Fig. 6 (c) and Fig. 7 (a)~Fig. 7 (c) is that counting rate is 1 × 10 respectively7With 1 × 106Lower three kinds of method for reconstructing are to Zubal
The reconstructed results of the 14th frame of thoracic cavity data, demonstrating reconstructed results of the invention has robustness to noise, and table 1 is that it is further
Quantized result analysis.Fig. 8 (a)~Fig. 8 (d) is the reconstruction experiment carried out on Hoffman brain data, shows the present invention
Reconstructed results to different template datas have robustness.
Table 1
The above-mentioned description to embodiment is for that can understand and apply the invention convenient for those skilled in the art.
Person skilled in the art obviously easily can make various modifications to above-described embodiment, and described herein general
Principle is applied in other embodiments without having to go through creative labor.Therefore, the present invention is not limited to the above embodiments, ability
Field technique personnel announcement according to the present invention, the improvement made for the present invention and modification all should be in protection scope of the present invention
Within.
Claims (6)
1. a kind of dynamic PET images method for reconstructing constrained based on non local full variation and low-rank, is included the following steps:
(1) biological tissue for being injected with radiopharmaceutical agent is detected using detector, when dynamic acquisition obtains corresponding to each
The coincidence counting vector at quarter, and these coincidence counting vectors are combined into coincidence counting matrix Y；
(2) make dynamic PET image combined sequence at PET concentration distribution matrix X, the measurement side PET is established according to PET image-forming principle
Journey；
(3) low-rank constraint is introduced by measuring equation to the PET, the dynamic PET images for obtaining constraining based on low-rank rebuild mould
Type M1；
(4) non local full variational methods are carried out by low-rank part to every frame image and sparse part, is further based on
The dynamic PET images reconstruction model M2 of non local full variational methods；
(5) objective function for combining to obtain dynamic PET reconstruction for dynamic PET images reconstruction model M1 and M2 is as follows:
S.t.L+S=X
Wherein: | | | |*Indicate nuclear norm, | | | |1Indicate 1- norm, L is that low-rank part contains mechanical periodicity in image background
Radioactive concentration, S be sparse part contain the radioactive concentration of the heterogene structure with different metabolic rate, JNLTV(L)
And JNLTV(S) be respectively low-rank part L and sparse part S after non local full variation as a result, λ, μ, αLAnd αSIt is weight system
Number, Ψ (Y | X) are the likelihood function about X and Y；
(6) PET concentration distribution matrix X is obtained after carrying out optimization to above-mentioned objective function, to restore dynamic
PET image sequence.
2. dynamic PET images method for reconstructing according to claim 1, it is characterised in that: the expression of the PET measurement equation
Formula is as follows:
Y=DX+R+S
Wherein: D is sytem matrix, and R and S are respectively the measurement noise matrix for reflecting chance event and scattering events.
3. dynamic PET images method for reconstructing according to claim 1, it is characterised in that: the dynamic PET images rebuild mould
The expression formula of type M1 is as follows:
M1=| | L | |*+λ||S||1+μΨ(Y|X)
S.t.L+S=X.
4. dynamic PET images method for reconstructing according to claim 1, it is characterised in that: the dynamic PET images rebuild mould
The expression formula of type M2 is as follows:
M2=αLJNLTV(L)+αSJNLTV(S)+μΨ(Y|X)
S.t.L+S=X.
5. dynamic PET images method for reconstructing according to claim 2, it is characterised in that: the likelihood function Ψ (Y | X)
Expression formula is as follows:
Wherein: dijFor the i-th row jth column element value, y in sytem matrix DimFor the i-th row m column element value in coincidence counting matrix Y,
xjmFor jth row m column element value, r in PET concentration distribution matrix XimTo measure the i-th row m column element value in noise matrix R,
simFor the i-th row m column element value in measurement noise matrix S, i, j and m be natural number and 1≤i≤I, 1≤j≤J, 1≤m≤
M, I are the dimension of coincidence counting vector, and J is line number, that is, PET image pixel number of PET concentration distribution matrix X, M PET
The columns of concentration distribution matrix X, that is, sampling time length.
6. dynamic PET images method for reconstructing according to claim 1, it is characterised in that: use ADMM in the step (6)
Algorithm carries out optimization to objective function；Wherein, low-rank restricted problem utilizes singular value thresholding algorithm and soft contraction algorithm
It is iterated Optimization Solution, likelihood function problem is iterated Optimization Solution, non local full variational methods problem using EM algorithm
Optimization Solution is iterated using gradient descent method.
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