CN105279777A - Static PET image reconstruction method based on improved sequential filtering - Google Patents

Abstract

A static PET image reconstruction method based on improved sequential filtering comprises the following steps: 1) a state spatial model of a PET system is built; 2) a concentration reconstruction process based on improved sequential filtering is as follows: 2.1) an initial value and an initial estimate covariance of radioactive concentration distribution are set firstly, so that xhat<0>(0) = xhat(0) and P<0>(0) = P(0); 2.2) sonogram data y(t) are obtained; 2.3) the sonogram data y(t) are divided into r blocks; 2.4) an equation (3) is used to calculate a gain matrix K<i>(t); 2.5) a measurement value y<i>(t) and the gain matrix K<i>(t) are used to calculate a spatial concentration estimate xhat<i>(t) according to a status updating equation (2); 2.6) if i < r, then i = i + 1, and the method jumps to the step 2.4), otherwise, xhat(t) = xhat<r>(t) and P(t) = P<r>(t); and 2.7) if t < N, then t = t + 1, and the method jumps to the step 2.2), otherwise, the algorithm ends and final reconstruction results are obtained. The method ensures the reconstruction effects, reduces the calculation cost, and improves the reconstruction speed.

Description

Static PET image reconstruction method based on improved sequential filtering

Technical Field

The invention relates to the technical field of positron emission tomography, in particular to a static PET image reconstruction method based on improved sequential filtering.

Background

Positron Emission Tomography (PET) technology is a medical imaging technology that tracks and monitors metabolic changes at the molecular level to achieve the purposes of early detection and prevention of diseases. The PET technique, as a functional imaging technique, can reflect the metabolism of a patient's body, thereby enabling earlier detection of a lesion. Therefore, the PET technology is widely applied to the diagnosis of tumors, heart diseases, nervous and mental system diseases, experimental biological imaging, drug screening and development, and has shown great application prospect.

When PET scanning is carried out, firstly an accelerator is used for generating positron-emitting nuclide, then the medicine labeled by the radioisotope nuclide is injected into the human body, and the substances form a certain distribution in various tissues and organs in the human body through blood circulation. Due to the short half-life of the radioisotope and its extreme instability, decay will occur very quickly. During decay, the radioisotope produces a positron which annihilates with a nearby free electron, producing a pair of oppositely directed, equal-energy gamma photons. At this time, photons can be detected by a detector outside the body by using a coincidence technique, and then projection data can be obtained by noise correction. The spatial concentration distribution of the radioactive substances of the human body can be reconstructed by utilizing projection data to carry out inversion solving through some reconstruction methods.

Currently, PET image reconstruction methods can be roughly classified into two categories: analytic methods and iterative statistical methods. The former method is mainly a filtered back projection method, which has high calculation speed and low cost, but cannot well inhibit noise, so that the quality of reconstructed images is not high. Therefore, an iterative statistical method typified by a maximum likelihood method has appeared. The iterative method is based on a statistical model, so that the method has good adaptability to incomplete data, and gradually becomes the focus of PET reconstruction algorithm research. Compared with an analytic method, the reconstructed image obtained by the iterative method is clearer. Accurate modeling of the imaging process of the PET system is critical to influencing the results of the iterative reconstruction. The introduction of the state space strongly promotes the development of the iterative method. The state space method can model the process of PET imaging (giving a mathematical expression of the PET imaging process) and scan PETThe statistical properties of the program and the physiological and structural properties of the organism are correspondingly described, so that the model combined with some existing estimation algorithms can provide better reconstruction effect. The existing state space solving method is mainly based on H₂Filtering (i.e. Kalman filtering), H_∞Filtering, etc. However, the sharpness of the reconstructed image depends on the number of voxels, and the latter is directly related to the calculation amount of the filtering reconstruction algorithm. Therefore, in most cases, the filtering algorithm-based PET image reconstruction method faces very high-dimensional matrix inversion operation, which brings a great operation burden to general filtering algorithm-based PET image reconstruction.

Disclosure of Invention

In order to overcome the defects of higher operation cost and lower reconstruction speed of the conventional PET image reconstruction method, the invention provides a static PET image reconstruction method based on improved sequential filtering. When the system carries out filtering based on the low-dimensional observation, the original high-dimensional matrix which needs inversion is replaced by a plurality of low-dimensional matrixes which are consistent with the dimension of the low-dimensional observation, so that the calculation cost is greatly reduced, the reconstruction speed is improved, and the reconstruction effect which is the same as that of the PET reconstruction method based on the Kalman filtering is provided.

In order to solve the technical problems, the following technical scheme is provided:

a static PET image reconstruction method based on improved sequential filtering comprises the following steps:

1) establishing a state space model of the PET system:

$\left\{\begin{array}{c}x\left(t+1\right)=x\left(t\right)+v\left(t\right)\\ y\left(t\right)=Dx\left(t\right)+e\left(t\right)\end{array}\right.---\left(1\right)$

wherein t represents time; y (t) is an observed value, namely sinogram data obtained after noise correction; d represents the projection matrix of the projection relation between the radioactive concentration in the human body and the PET scanning, and is determined by the inherent characteristic of the PET device; x (t) is the radioactive concentration distribution, i.e. the object to be reconstructed; v (t) is process noise; e (t) is residual measurement noise after data acquisition and correction;

2) a reconstructed image based on improved sequential filtering is obtained according to the following equation:

${\hat{x}}_i\left(t\right)={\hat{x}}_{i-1}\left(t\right)+K_i\left(t\right)\left(y_i\right(t)-D{\hat{x}}_{i-1}(t\left)\right),{\hat{x}}_0\left(t\right)=\hat{x}\left(0\right)---\left(2\right)$

$K_i\left(t\right)=P_i\left(t\right)D_i^T{(D_i{P}_i\left(t\right)D_i^T+R_i)}^{-1}---\left(3\right)$

$P_i\left(t\right)=P_{i-1}\left(t\right)+Q_i-K_i\left(t\right)\left(D_i{P}_i\left(t\right)D_i^T+R_i\right)K_i^T\left(t\right),P_0\left(t\right)=P\left(t\right)---\left(4\right)$

$\hat{x}\left(t\right)={\hat{x}}_r\left(t\right)---\left(5\right)$

P(t)＝P_r(t)(6)

wherein,is a filtered reconstructed value of the spatial concentration,is the initial filtering value of space concentration, y (t) is the corrected measured sinogram data, P (t) is the estimated error covariance matrix of space concentration, P (0) is the estimated error covariance of initial space concentration, divide y (t) into r blocks, y_i(t) is the ith block of y (t), D_i,R_i,Q_iIs corresponding to y_i(t) matrix D, R, partitioning of Q, K_i(t) is a radical corresponding to y_i(t) a filter gain matrix of (t),is based on { y (0), …, y (t-1), y₁(t),…,y_i(t) } spatial concentration filtered reconstruction value, P_i(t) is a number corresponding to { y (0), …, y (t-1), y₁(t),…,y_i(t) } filter error covariance matrix. Iteration from initial valueStarting from P (0), measuring value y (t), and obtaining radioactivity concentration distribution through N iterationsThe reconstruction process is as follows:

2.1) first set the initial value and initial estimated covariance of the radioactivity concentration distributionP₀(0)＝P(0)；

2.2) acquiring sinogram data y (t);

2.3) dividing the sinogram data y (t) into r blocks;

2.4) calculating the gain matrix K using equation (3)_i(t)；

2.5) Using the measurement value y_i(t) and a gain K_i(t) calculating a spatial concentration estimate from the state update equation (2)And deducing the corresponding estimated error covariance matrix P according to (4)_i(t)；

2.6) if i < r, i ═ i +1, jump to step 2.4), otherwise,P(t)＝P_r(t)；

2.7) if t is less than N, t is t +1, jumping to step 2.2), otherwise, the algorithm ends.

The technical conception of the invention is as follows: based on the analysis of the imaging principle of the PET system by using the Kalman filtering method, the problem that the reconstruction speed of the reconstruction method is slow is found in the inversion of a high-dimensional matrix caused by high-dimensional observation, and the inversion of the low-dimensional matrix is only needed to be solved finally by blocking the high-dimensional observation into relatively low-dimensional observation, so that the problems are effectively solved.

According to the technical scheme, the beneficial effects of the invention are mainly shown in that: on the premise of ensuring the same reconstruction effect (as the PET image reconstruction directly based on Kalman filtering), the inversion of a high-dimensional matrix is effectively avoided, and the reconstruction speed is further improved.

Drawings

FIG. 1 is a schematic flow chart of the steps of the PET image reconstruction method of the present invention.

Detailed Description

In order to describe the present invention more specifically, the method for reconstructing the static PET concentration according to the present invention will be described in detail with reference to the accompanying drawings and embodiments.

As shown in fig. 1, a method for reconstructing a static PET image based on improved sequential filtering includes the following steps:

1) according to the PET imaging principle, a state space system is established:

$\left\{\begin{array}{c}x\left(t+1\right)=x\left(t\right)+v\left(t\right)\\ y\left(t\right)=Dx\left(t\right)+e\left(t\right)\end{array}\right.---\left(7\right)$

wherein t represents time; y (t) is an observed value, namely sinogram data obtained after noise correction; d represents the projection matrix of the projection relation between the radioactive concentration in the human body and the PET scanning, is determined by the inherent characteristics of the PET device, and is obtained by adopting a single response line model to carry out approximate calculation in the experiment; x (t) is the radioactive concentration distribution, i.e. the object to be reconstructed; v (t) is process noise; e (t) is residual measurement noise after data acquisition and noise correction; v (t), e (t) are respectively normal Gaussian distribution of diagonal matrixes Q and R according to covariance matrixes;

2) a reconstructed image based on improved sequential filtering is obtained according to the following equation:

${\hat{x}}_i\left(t\right)={\hat{x}}_{i-1}\left(t\right)+K_i\left(t\right)\left(y_i\right(t)-D{\hat{x}}_{i-1}\left(t\right)),{\hat{x}}_0\left(t\right)=\hat{x}\left(0\right)---\left(8\right)$

$K_i\left(t\right)=P_i\left(t\right)D_i^T{(D_i{P}_i\left(t\right)D_i^T+R_i)}^{-1}---\left(9\right)$

$P_i\left(t\right)=P_{i-1}\left(t\right)+Q_i-K_i\left(t\right)\left(D_i{P}_i\left(t\right)D_i^T+R_i\right)K_i^T\left(t\right),P_0\left(t\right)=P\left(t\right)---\left(10\right)$

P(t)＝P_r(t)(12)

wherein,is a filtered reconstructed value of the spatial concentration,the initial value of the spatial density is a filtering initial value, y (t) is measurable corrected sinogram data, P (t) is an estimated error covariance matrix of the spatial density, P (0) is an estimated error covariance of the initial spatial density, a subscript i is 1, …, r, y (t) is divided into r blocks, y (t) is a mean value of the initial spatial density, and the initial spatial density is measured by a computer_i(t) is the ith block of y (t), D_i,R_i,Q_iIs and y_i(t) partitioning of compatible matrices D, R, Q, K_i(t) is a radical corresponding to y_i(t) a filter gain matrix of (t),based on until observation y_i(t) spatial concentration filtered reconstruction value, P_i(t) is the value corresponding to the observation up to the observation y_i(t) filter error covariance matrix. Iteration from initial valueStarting from P (0), measuring value y (t), and obtaining radioactive concentration distribution through multiple iterationsAs shown in fig. 1, the iterative process of image reconstruction based on improved sequential filtering is as follows:

2.1) first of all, the initial values and the initial variances of the radioactive concentration distributions are setP₀(0) The two values are given by empirical values, but without a priori knowledge, the two values can be given byP₀(0) Respectively taking a zero matrix and an identity matrix;

2.2) acquiring sinogram data y (t);

2.3) dividing the sinogram data y (t) into r blocks;

2.4) calculating the gain matrix K using equation (9)_i(t)；

2.5) Using the measurement value y_i(t) and a gain K_i(t) calculating a spatial concentration estimate from the state update equation (8)And deducing the corresponding estimated error covariance matrix P according to (10)_i(t)；

2.6) if i < r, i ═ i +1, jump to step 2.4), otherwise,P(t)＝P_r(t)；

2.7) if t is less than N, and t is t +1, jumping to the step 2.2), otherwise, finishing the algorithm to obtain a final concentration reconstruction result.

Claims (1)

1. A static PET image reconstruction method based on improved sequential filtering is characterized in that: the image reconstruction method comprises the following steps:

1) establishing a state space model of the PET system:

$\left\{\begin{array}{c}x(t+1)=x\left(t\right)+v\left(t\right)\\ y\left(t\right)=Dx\left(t\right)+e\left(t\right)\end{array}\right.---\left(1\right)$

wherein t represents time; y (t) is an observed value, namely sinogram data obtained after noise correction; d represents the projection matrix of the projection relation between the radioactive concentration in the human body and the PET scanning, and is determined by the inherent characteristic of the PET device; x (t) is the radioactive concentration distribution, i.e. the object to be reconstructed; v (t) is process noise; e (t) is residual measurement noise after data acquisition and correction;

2) a reconstructed image based on improved sequential filtering is obtained according to the following equation:

${\hat{x}}_i\left(t\right)={\hat{x}}_{i-1}\left(t\right)+K_i\left(t\right)\left(y_i\right(t)-D{\hat{x}}_{i-1}(t\left)\right),{\hat{x}}_0\left(t\right)=\hat{x}\left(0\right)---\left(2\right)$

$K_i\left(t\right)=P_i\left(t\right)D_i^T{\left(D_i{P}_i\right(t)D_i^T+R_i)}^{-1}---\left(3\right)$

$P_i\left(t\right)=P_{i-1}\left(t\right)+Q_i-K_i\left(t\right)\left(D_i{P}_i\right(t)D_i^T+R_i)K_i^T\left(t\right),P_0\left(t\right)=P\left(t\right)---\left(4\right)$

$\hat{x}\left(t\right)={\hat{x}}_r\left(t\right)---\left(5\right)$

P(t)＝P_r(t)(6)

wherein,is a filtered reconstructed value of the spatial concentration,is the initial filtering value of space concentration, y (t) is the corrected measured sinogram data, P (t) is the estimated error covariance matrix of space concentration, P (0) is the estimated error covariance of initial space concentration, divide y (t) into r blocks, y_i(t) is the ith block of y (t), D_i,R_i,Q_iIs corresponding to y_i(t) matrix D, R, partitioning of Q, K_i(t) is a radical corresponding to y_i(t) a filter gain matrix of (t),is based on { y (0), …, y (t-1), y₁(t),…,y_i(t) } spatial concentration filtered reconstruction value, P_i(t) is a number corresponding to { y (0), …, y (t-1), y₁(t),…,y_i(t) } filter error covariance matrix. Iteration from initial valueStarting from P (0), measuring value y (t), and obtaining radioactivity concentration distribution through N iterationsThe reconstruction process is as follows:

2.1) first set the initial value and initial estimated covariance of the radioactivity concentration distributionP₀(0)＝P(0)；

2.2) acquiring sinogram data y (t);

2.3) dividing the sinogram data y (t) into r blocks;

2.4) calculating the gain matrix K using equation (3)_i(t)；

2.5) Using the measurement value y_i(t) and a gain K_i(t) calculating a spatial concentration estimate from the state update equation (2)And deducing the corresponding estimated error covariance matrix P according to (4)_i(t)；

2.6) if i < r, i ═ i +1, jump to step 2.4), otherwise,P(t)＝P_r(t)；

2.7) if t is less than N, and t is t +1, jumping to step 2.2), otherwise, ending the algorithm and obtaining a final reconstruction result.