CN105832358A - Imaging method based on system calibration for rotating double-panel PET system - Google Patents

Abstract

The invention discloses an imaging method based on system calibration for a rotating double-panel PET system. The imaging method comprises the following steps: acquiring a plurality of sets of single photon data through the rotating double-panel PET system, carrying out pretreatment on the data to obtain a forward projection data set needed for reconstruction, and carrying out reconstruction by adopting an iterative reconstruction algorithm OSEM, thus obtaining a plurality of sets of reconstruction data; calculating the offset vector between registration point coordinates between two adjacent angle reconstruction results, thus obtaining an offset vector set of one group of image domain, and further obtaining the geometrical offset vector of the system; and dividing to obtain subsets by taking the angle as the unit and by utilizing the calculated offset vector set of the image domain and the projection data set, and carrying out reconstruction sequentially on projection data. With the adoption of the imaging method, the accuracy of the reconstruction result is guaranteed, and the resolution of a panel PET reconstruction image is improved; the calculation amount of the system for responding to the array is reduced, and the reconstruction efficiency is improved; by utilizing the offset vector of the image domain, the calculation for the geometrical offset vector of the system is indirectly completed.

Description

A kind of formation method of rotation double flat plate PET system based on system calibration

Technical field

The invention belongs to medicine imaging technique field, particularly relate to a kind of rotation double flat plate PET based on system calibration The formation method of system.

Background technology

PET (PositronEmission Tomography, PET) full name is Positron emission computed tomography, is The image technology that the field of nuclear medicine is advanced.The metabolism utilizing PET system detector can obtain at body is distributed, and is that a kind of function shows As technology, especially the early stage context of detection in tumor occupy very important status.Medical pet detector uses static ring mostly Shape structure, although can obtain the sampled data of full angular, but the complexity of system is high, cost is high, and having of system is certain Closure, be not easy to doctor or the laboratory technician technological guidance to sample objects simultaneously.In order to reduce cost, increase detector Flexibility ratio, in recent years, researcher proposes flat board pet detector, and such as Chien-Min Kao et al. is at " A High- Sensitivity Small-Animal PET Scanner:Development and Initial Performance Measurements,IEEE Transactions onNuclear Science,vol.56,no.5,pp.2678-88,2009” Article proposes the double end flat board pet detector of a kind of static state, simple in construction, it is possible to obtain satisfied imaging results, but Being to be poor in the image resolution ratio being perpendicular to detecting plate direction, tracing it to its cause is that static flat board PET cannot obtain completely The data for projection of angle；In order to solve flat-panel systems angle disappearance problem, researcher is had to propose flat board PET rotary system, logical Cross the single photon data gathering multiple angle, to reach the purpose that systemic resolution is improved, such as, JuneEt al. At " Efficient methodologies for system matrix modelling in iterative image reconstruction for rotating high-resolution PET,Physics inMedicine AndBiology, vol.55, no.7,1833-61,2010 " use four flat board pet detector, detecting head two is the most right, due to Detecting plate is less, and plate spacing is relatively big, and system exists the biggest disappearance angle, and it is solved this problem in that by rotation process；Additionally Chunhui Zhang etc. are at " 90 ° of-rotating dual-head small of Performance evaluation of a Animal PET system, Physics in Medicine and Biology, vol.60, no.15, pp.5873-90, 2015 " use two pieces of large-area flat board PET detecting heads as the major part of system in, be perpendicular to detecting plate side to overcome To the problem of resolution rate variance, system is carried out the rotation of 90 degree by it, it is possible to obtain the sampled data of full angle；But both the above System, in the case of being all based on emulation, the rotary course being not related to mechanical system can produce position deviation problem, at its data Reason process is not suitable for real system.

The calibration of rotary system and the reconstruction of multiple angle-datas that causes due to system geometric error in prior art Result cannot be completely superposed.

Summary of the invention

It is an object of the invention to provide the formation method of a kind of rotation double flat plate PET system based on system calibration, purport At the calibration solving rotary system in prior art and the reconstruction knot of multiple angle-datas caused due to system geometric error The problem that fruit cannot be completely superposed.

The present invention is achieved in that the formation method of a kind of rotation double flat plate PET system based on system calibration, described The formation method of rotation double flat plate PET system based on system calibration is collected by rotation double flat plate PET system and organizes monochromatic light more Subdata, and after data prediction, obtain the forward projection data set needed for rebuilding；Recycling iterative reconstruction algorithm OSEM rebuilds, and obtains many groups and rebuilds data；Data set, in addition to initial angle reconstructed results, other angles is rebuild based on this Reconstruction data rotate to an angle with inverse system direction of rotation；Body is imitated, by adjacent by means of custom-designed registration point The calculating of the offset vector between registration point coordinate between two angle reconstruction results, obtains the offset vector of one group of image area Collection, and then the geometrical offset vector of the system of acquisition；Utilize the image area set of offset vectors and projection data set calculated, with angle For unit dividing subset, successively data for projection is rebuild, each subset rebuild terminate after, reconstructed results is rotated and After offset correction, the initial value rebuild as next subset data joins in process of reconstruction, completes during rebuilding Correction to system.

Further, the formation method of described rotation double flat plate PET system based on system calibration specifically includes following steps:

Step one, uses and rotates double flat plate PET system data acquisition and pretreatment, and rotating double flat plate PET system is by double Flat board pet detector, data fit processing system and control system composition, wherein pet detector is used for gammaphoton ultrahigh speed The capture of scintillation pulse and analysis, data fit processing system is for meeting process, by each detector to collection data The single photon event collected, according to the energy provided and the temporal information of each photon, processes, and obtains needed for rebuilding Data for projection i.e. sinogram data；Control system completes the rotary step to system, the anglec of rotation, gathers setting of duration Fixed；

Step 2, single angle-data is rebuild

Using iterative reconstruction algorithm OSEM to rebuild data, formula is as follows:

$f_j^{k+1,\mathrm{deg}}=\frac{f_j^{k,\mathrm{deg}}}{\underset{i\⋐S_l}{\Σ}{p}_{ij}}\underset{i=1}{\overset{M}{\Σ}}\frac{g_i}{\underset{j\⋐S_l}{\Σ}{p}_{ij}{f}_j^{k,\mathrm{deg}}}{p}_{ij};$

Wherein,For the jth voxel under different angles deg in the image intensity of kth time iteration, retouch for convenience Stating, deg is as follows in definition:

Symbol S_lBe l (l=1,2 ..., L) individual subset, L be divide total number of subsets, M is total number of voxels, g_iFor forward direction Data for projection；p_ijIt is the element of system response matrix P, represents the probability that voxel j is detected by corresponding line i；

Step 3, the acquisition of system geometrical offset vector, o represents central region, o⁰It is the central region under 0 °, definition system System geometrical offset vector isAfter entering 90 ° of rotations, central region moves to o¹And defineDefinition picture Offset vector between element isVectorWithBetween relationship description be:

$\stackrel{\→}{\mathrm{\η}}=\stackrel{\→}{\mathrm{\μ}}\×R;$

Wherein,

MatrixIt is the rotation parameter relevant to anglec of rotation α, in this instance α=90 °, from above two formulas

Calculate vectorRelation between the two is:

$\stackrel{\→}{v}=\stackrel{\→}{\mathrm{\μ}}-\stackrel{\→}{\mathrm{\μ}}\×R=(\mathrm{\δ}X-\mathrm{\δ}Y,\mathrm{\δ}X+\mathrm{\δ}Y);$

Step 4, data of multiple angles is rebuild；

Step 5, three-dimensional reconstruction result shows, the reconstructed results obtaining step 4 carries out three dimensional display.

Further, described data acquisition specifically includes with pretreatment:

First, the rotation double flat plate PET system object to being placed on object stage is utilized to carry out the number no less than 2 angles According to collection, each angle stops the identical time, obtains the reception condition of the γ photon of single detector under each angle；Pass through Double flat plate PET system is rotated by control system at equal intervals, and detector often rotates one group of single photon data of an angle acquisition And carry out real-time data storage, rotate double flat plate PET system and rotate 180 ° around rotary shaft；

Secondly, in units of angle, according to gathering the time that data provide, energy, positional information, carry out single angle Event matches, rejects random noise and shot noise successively, obtains forward projection data set, i.e. sinogram data g⁰,g¹。

Further, described vectorCalculating process as follows:

The first step, utilizes OSEM algorithm to rebuild forward projection data g⁰,g¹, obtain two single angle reconstruction results

Second step, by carrying out rotation transformation to image coordinateObtain postrotational result

3rd step, calculates 13 registration point centre coordinate under two angles respectively.

Further, described 3rd step specifically includes:

Step one, fromIn choose the section that be perpendicular to rotary shaft relevant to registration point；

Step 2, extracts the border of each section and calculates the centre coordinate of registration pointWherein i= 1 ..., num, num are number a little, num=13；

Step 3, the mean deviation amount of the centre coordinate in calculation procedure two:

$(\mathrm{\δ}x,\mathrm{\δ}y)=\left(\frac{{\mathrm{\Σx}}_i^0-x_i^1}{num},\frac{{\mathrm{\Σy}}_i^0-y_i^1}{num}\right);$

Step 4, then vector

Further, described data of multiple angles is rebuild and is specifically included:

The first step, initializes f=f^k=1, k=0；

Second step, initializesOSEM algorithm is utilized to rebuild forward projection data g⁰, obtain reconstructed results

3rd step, willCoordinate utilize formulaCarry out rotating translation transformation, obtain result Wherein,Represent the coordinate of pixel j, i.e.

4th step, utilizesAs the initial value of initial value OSEM algorithm, rebuild forward projection data g¹, obtain and rebuild knot Really

5th step, willCoordinate utilize formulaCarry out rotating translation transformation, obtain result Wherein R' is the transposed matrix of matrix R；

6th step, k=k+1；

7th step, jumps to second step, until terminating iteration after getting suitable reconstructed results.

The formation method of the rotation double flat plate PET system based on system calibration that the present invention provides, it is possible to realize biology Being accurately positioned and super-resolution reconstruction of internal radionuclide, can be used for Rotating Plates PET system imaging field；Propose one Plant the formation method of rotation double flat plate PET system based on system calibration, there is the strongest practicality, retain flat-panel systems spirit While activity structure, it is achieved that to complex biological body being accurately positioned and high-resolution weight the metabolism status of radionuclide Build.

The present invention compared with prior art has the advantage that

The first, the present invention utilizes designed registration point to imitate body, calculate the geometrical offset rotating double flat plate PET system to Amount, simply and easily while the correction of completion system, it is ensured that the accuracy of reconstructed results, for being perpendicular to detector side To resolution improve about 3 times, significantly improve flat board PET rebuild image quality.

The second, multi-angle reconstruction model of the present invention, the system response matrix needed for reconstruction and static system Identical, it is not necessary to reset line of response, the amount of calculation rotating double flat plate PET system matrix at least reduces 4 times, improves reconstruction Efficiency.

3rd, the present invention is without being directly corrected system, utilizes the offset vector of image area, indirectly completes being The calculating of system geometrical offset vector.

Accompanying drawing explanation

Fig. 1 is the formation method flow process of the rotation double flat plate PET system based on system calibration that the embodiment of the present invention provides Figure.

Fig. 2 is the system geometrical deviation description figure that the embodiment of the present invention provides.

Fig. 3 is that the registration point used that the embodiment of the present invention provides imitates body schematic diagram.

Fig. 4 is the Comparative result figure of the embodiment of the present invention.

Detailed description of the invention

In order to make the purpose of the present invention, technical scheme and advantage clearer, below in conjunction with embodiment, to the present invention It is further elaborated.Should be appreciated that specific embodiment described herein, only in order to explain the present invention, is not used to Limit the present invention.

Below in conjunction with the accompanying drawings the application principle of the present invention is explained in detail.

As it is shown in figure 1, the formation method of the rotation double flat plate PET system based on system calibration of the embodiment of the present invention includes Following steps:

S101: collect many group single photon data by rotating double flat PET system, and after data prediction, obtain Taking the forward projection data set needed for reconstruction, recycling iterative reconstruction algorithm OSEM rebuilds, and obtains many groups and rebuilds data；Base Rebuilding data set in this, in addition to initial angle reconstructed results, the reconstruction data of other angles rotate one with inverse system direction of rotation Determine angle；

S102: imitate body by means of custom-designed registration point, by the registration between adjacent two angle reconstruction results The calculating of the offset vector between point coordinates, obtains the set of offset vectors of one group of image area, and then the geometrical offset of the system of acquisition Vector；

S103: utilize the image area set of offset vectors and projection data set calculated, with angle for unit dividing subset, depend on Secondary rebuilding data for projection, each subset is rebuild after terminating, and rotates reconstructed results and offset correction, as next The initial value of individual subset data reconstruction joins in process of reconstruction, to realize completing system compensation in process of reconstruction simultaneously.

Below in conjunction with the accompanying drawings the application principle of the present invention is further described.

Step 1, data acquisition and pretreatment

First, the rotation double flat plate PET system object to being placed on object stage is utilized to carry out no less than 2 angles (this Example uses 2 angles) data acquisition, each angle stops identical time, obtains the γ of single detector under each angle Photon accept situation；The method of data acquisition of the present invention is: revolved double flat plate PET system at equal intervals by control system Turning, detector often rotates one group of single photon data of an angle acquisition and carries out real-time data storage, and system is revolved around rotary shaft Turn 180 °.

Secondly, in units of angle, according to gathering the time that data provide, energy, positional information, carry out single angle Event matches, rejects random noise and shot noise successively, obtains forward projection data set, i.e. sinogram data g⁰,g¹。

Further, rotating double flat plate PET system is by double flat plate pet detector, data fit processing system and control System forms.Wherein pet detector is used for capture and analysis, the data fit processing system of gammaphoton ultrahigh speed scintillation pulse For collection data are met process, obtain the forward projection data needed for rebuilding, i.e. sinogram data, control system Mainly complete the rotary step to system, the anglec of rotation, gather the setting of duration, it is achieved the control to system operates.

Step 2, single angle-data is rebuild

Using iterative reconstruction algorithm OSEM to rebuild data, formula is as follows:

$f_j^{k+1,\mathrm{deg}}=\frac{f_j^{k,\mathrm{deg}}}{\underset{i\⋐S_i}{\Σ}{p}_{ij}}\underset{i=1}{\overset{M}{\Σ}}\frac{g_i}{\underset{j\⋐S_i}{\Σ}{p}_{ij}{f}_j^{k,\mathrm{deg}}}{p}_{ij}$

Wherein,For the jth voxel under different angles deg in the image intensity of kth time iteration, retouch for convenience Stating, deg is as follows in definition:

This its dependent variable being defined on the present invention is still suitable for.Symbol S_lBe l (l=1,2 ..., L) individual subset, L is The total number of subsets divided, M is total number of voxels, g_iFor forward projection data.p_ijIt is the element of system response matrix P, represents voxel j The probability detected by corresponding line i.The calculating of system response matrix P is at article " Performance evaluation of a 90 ° of-rotating dual-head small animal PET system, Physics in Medicine and Biology, vol.60, no.15, pp.5873-90,2015 " in be described later in detail, no longer its calculating is described.

Step 3, the acquisition of system geometrical offset vector

Inconsistent due to system center of rotation and central region, result in multi-angle and rebuilds image and cannot be carried out overlapping, By Fig. 2, this problem is described.As shown in Fig. 2 (a), o represents central region, o⁰It is the central region under 0 °, defines system geometry Offset vector isAfter entering 90 ° of rotations, central region moves to o¹And defineTherefore result in double angle Degree (0 ° and 90 °) rebuilds the skew between the pixel of image, and as shown in Fig. 2 (b), the offset vector between definition pixel isTo AmountWithBetween relation can be described as:

$\stackrel{\→}{\mathrm{\η}}=\stackrel{\‾}{\mathrm{\μ}}\×R,;$

Wherein,

MatrixIt is the rotation parameter relevant to anglec of rotation α, in this instance α=90 °, from above two formulas

In real work, it is difficult to directly calculate system geometrical offset vectorBut can calculate very easily VectorRelation between the two is:

$\stackrel{\→}{v}=\stackrel{\→}{\mathrm{\μ}}-\stackrel{\→}{\mathrm{\μ}}\×R=(\mathrm{\δ}X-\mathrm{\δ}Y,\mathrm{\δ}X+\mathrm{\δ}Y);$

VectorCalculating process as described below:

OSEM algorithm 3a) is utilized to rebuild forward projection data g⁰,g¹, obtain two single angle reconstruction results

3b) by image coordinate is carried out rotation transformationObtain postrotational result

3c) calculate 13 registration point (as shown in Figure 3) centre coordinate under two angles respectively；

3c1) fromIn choose the section that be perpendicular to rotary shaft relevant to registration point；

3c2) extract the border of each section and calculate the centre coordinate of registration pointWherein i= 1 ..., num, num are number a little, num=13 in this example；

3c3) calculation procedure 3c2) in the mean deviation amount of centre coordinate:

$(\mathrm{\δ}x,\mathrm{\δ}y)=\left(\frac{\Σx_i^0-x_i^1}{num},\frac{\Σy_i^0-y_i^1}{num}\right);$

3c4) then vectorial

Step 4, data of multiple angles is rebuild

4a) initialize f=f^k=1, k=0；

4b) initializeOSEM algorithm is utilized to rebuild forward projection data g⁰, obtain reconstructed results

4c) willCoordinate utilize formulaCarry out rotating translation transformation, obtain resultWherein,Represent the coordinate of pixel j, i.e.

4d) utilizeAs the initial value of initial value OSEM algorithm, rebuild forward projection data g¹, obtain reconstructed results

4e) willCoordinate utilize formulaCarry out rotating translation transformation, obtain resultWherein R' is the transposed matrix of matrix R；

4f) make k=k+1；

4g) jump to step 4b), until terminating iteration after getting suitable reconstructed results.

Step 5, three-dimensional reconstruction result shows, the reconstructed results obtaining step 4 carries out three dimensional display.

3 registration point used in the present invention is imitated body it is further described below in conjunction with the accompanying drawings:

Accompanying drawing 3 is that the registration point used in the present invention imitates body, is made up of 3 parts, every part size is 80 × 30 × 10mm³, wherein Part I is made up of 5 holes, and other two parts are made up of 4 holes, and the position in its hole is respectively at two diagonal angles 3 parts are overlapped during experiment by line, and central axial portion, use as an entirety.

The reconstructed results of 4 couples of present invention is further described below in conjunction with the accompanying drawings.

Accompanying drawing 4 is the reconstructed results schematic diagram of the embodiment of the present invention.Wherein, the system anglec of rotation is 90 °, the projection of collection The number of data is 2 groups.

Accompanying drawing 4 is to have carried out, by the step of the present invention, the reconstructed results that obtains after system compensation, and wherein, draw above be singly Angle reconstruction result, figure below is double angle reconstruction results.

The reconstruction structure of the reconstruction effect accompanying drawing 4 of the present invention is compared, it can be seen that the single angle in reconstructed results Reconstructed results also exists pulled out condition in one direction, and image stretch is corrected by multi-angle reconstructed results, explanation The shortcoming that instant invention overcomes prior art with needs the easiest identification characteristic point restricted to data for projection, easily and efficiently Complete the rectification that flat board PET reconstructed results is differentiated rate variance.

The foregoing is only presently preferred embodiments of the present invention, not in order to limit the present invention, all essences in the present invention Any amendment, equivalent and the improvement etc. made within god and principle, should be included within the scope of the present invention.

Claims (6)

1. the formation method of a rotation double flat plate PET system based on system calibration, it is characterised in that described based on system school The formation method of accurate rotation double flat plate PET system collects many group single photon data, and warp by rotating double flat plate PET system After crossing data prediction, obtain the forward projection data set needed for rebuilding；Recycling iterative reconstruction algorithm OSEM rebuilds, Obtain many groups and rebuild data；Rebuilding data set based on this, in addition to initial angle reconstructed results, the reconstruction data of other angles are with inverse System direction of rotation rotates to an angle；By to the skew between the registration point coordinate between adjacent two angle reconstruction results The calculating of vector, obtains the set of offset vectors of one group of image area, and then the geometrical offset vector of the system of acquisition；Utilize the figure calculated Image field set of offset vectors and projection data set, with angle for unit dividing subset, rebuild data for projection successively, each Subset is rebuild after terminating, and rotates reconstructed results and after offset correction, the initial value rebuild as next subset data Join in process of reconstruction, during rebuilding, complete the correction to system.

2. the formation method of rotation double flat plate PET system based on system calibration as claimed in claim 1, it is characterised in that The formation method of described rotation double flat plate PET system based on system calibration specifically includes following steps:

Step one, uses and rotates double flat plate PET system data acquisition and pretreatment, and rotating double flat plate PET system is by double flat plate Pet detector, data fit processing system and control system composition, wherein pet detector flashes for gammaphoton ultrahigh speed The capture of pulse and analysis, data fit processing system is for meeting process to collection data, before obtaining needed for rebuilding To data for projection, i.e. sinogram data, control system completes the rotary step to system, the anglec of rotation, gathers setting of duration Fixed；

Step 2, single angle-data is rebuild

Using iterative reconstruction algorithm OSEM to rebuild data, formula is as follows:

$f_j^{k+1,\mathrm{deg}}=\frac{f_j^{k,\mathrm{deg}}}{\underset{i\⋐S_l}{\Σ}{p}_{ij}}\underset{i=1}{\overset{M}{\Σ}}\frac{g_i}{\underset{j\⋐S_l}{\Σ}{p}_{ij}{f}_j^{k,\mathrm{deg}}}{p}_{ij};$

Wherein,For the jth voxel under different angles deg in the image intensity of kth time iteration, describe for convenience, fixed Deg is as follows for justice:

Symbol S_lBe l (l=1,2 ..., L) individual subset, L be divide total number of subsets, M is total number of voxels, g_iFor forward projection Data；p_ijIt is the element of system response matrix P, represents the probability that voxel j is detected by corresponding line i；

Step 3, the acquisition of system geometrical offset vector, o represents central region, o⁰Being the central region under 0 °, definition system is several What offset vector isAfter entering 90 ° of rotations, central region moves to o¹And defineDefinition pixel it Between offset vector beVectorWithBetween relationship description be:

$\stackrel{\→}{\mathrm{\η}}=\stackrel{\→}{\mathrm{\μ}}\×R,;$

Wherein,

MatrixIt is the rotation parameter relevant to anglec of rotation α, in this instance α=90 °, from above two formulas

Calculate vectorRelation between the two is:

$\stackrel{\→}{v}=\stackrel{\→}{\mathrm{\μ}}-\stackrel{\→}{\mathrm{\μ}}\×R=(\mathrm{\δ}X-\mathrm{\δ}Y,\mathrm{\δ}X+\mathrm{\δ}Y);$

Step 4, data of multiple angles is rebuild；

Step 5, three-dimensional reconstruction result shows, the reconstructed results obtaining step 4 carries out three dimensional display.

3. the formation method of rotation double flat plate PET system based on system calibration as claimed in claim 2, it is characterised in that Described data acquisition specifically includes with pretreatment:

First, the rotation double flat plate PET system object to being placed on object stage is utilized to carry out the data acquisition no less than 2 angles Collection, each angle stops identical time, obtain single detector under each angle γ photon accept situation；By controlling Double flat plate PET system is rotated by system at equal intervals, and detector often rotates one group of single photon data of an angle acquisition and goes forward side by side The data storage that row is real-time, system rotates 180 ° around rotary shaft；

Secondly, in units of angle, according to gathering the time that data provide, energy, positional information, carry out the event of single angle Meet, reject random noise and shot noise successively, obtain forward projection data set, i.e. sinogram data g⁰,g¹。

4. the formation method of rotation double flat plate PET system based on system calibration as claimed in claim 2, it is characterised in that Described vectorCalculating process as follows:

The first step, utilizes OSEM algorithm to rebuild forward projection data g⁰,g¹, obtain two single angle reconstruction results

Second step, by carrying out rotation transformation to image coordinateObtain postrotational result

3rd step, calculates 13 registration point centre coordinate under two angles respectively.

5. the formation method of rotation double flat plate PET system based on system calibration as claimed in claim 4, it is characterised in that Described 3rd step specifically includes:

Step one, fromIn choose the section that be perpendicular to rotary shaft relevant to registration point；

Step 2, extracts the border of each section and calculates the centre coordinate of registration pointWherein i=1 ..., Num, num are number a little, num=13；

Step 3, the mean deviation amount of the centre coordinate in calculation procedure two:

$(\mathrm{\δ}x,\mathrm{\δ}y)=\left(\frac{{\mathrm{\Σx}}_i^0-x_i^1}{num},\frac{{\mathrm{\Σy}}_i^0-y_i^1}{num}\right);$

Step 4, then vector

6. the formation method of rotation double flat plate PET system based on system calibration as claimed in claim 2, it is characterised in that Described data of multiple angles is rebuild and is specifically included:

The first step, initializes f=f^k=1, k=0；

Second step, initializesOSEM algorithm is utilized to rebuild forward projection data g⁰, obtain reconstructed results

3rd step, willCoordinate utilize formulaCarry out rotating translation transformation, obtain resultWherein,Represent the coordinate of pixel j, i.e.

4th step, utilizesAs the initial value of initial value OSEM algorithm, rebuild forward projection data g¹, obtain reconstructed results

5th step, willCoordinate utilize formulaCarry out rotating translation transformation, obtain resultWherein R' Transposed matrix for matrix R；

6th step, k=k+1；

7th step, jumps to second step, until terminating iteration after getting suitable reconstructed results.