CN104392487B - Quick real-time reconstruction method based on dual-panel PET - Google Patents

Abstract

The invention discloses a quick real-time reconstruction method based on dual-panel PET. The method comprises the steps: 1) calculating coupling positioning and coupling degree between reference LOR and pixels of image matrixes crossed with the image matrixes; 2) obtaining the coupling positioning and the coupling degree between all LOR and the pixels by using the symmetry of the dual planes according to the calculated coupling positioning and coupling degree between the crossed pixels of the reference LOR; and 3) constructing a system matrix according to the coupling positioning and the coupling degree between the LOR and the pixels to carry out PET reconstruction. In the quick real-time reconstruction method based on the dual-panel PET, the coupling positioning and the coupling degree between the LOR and the pixels can be calculated according to the symmetry of the dual-panel structure, so that the calculation of the system matrix of the dual-panel PET is simplified, the memory space and the calculating time needed by the system matrix can be saved and the reconstruction process is accelerated.

Description

Quick real-time reconstruction method based on double flat plate PET

Technical field

The present invention relates to nuclear medicine detection imaging field, more particularly to the real-time reconstruction of the PET of a kind of pair of slab construction Method.

Background technology

PET is important Laminographic device in the field of nuclear medicine, be widely used in medical domain diagnosis and Research.The more compact small-sized PET of panel detector structure becomes popular recently, and the image-forming principle of such small-sized PET is extensive with clinic Using human body PET it is essentially identical, be all will to launch the nucleic of positive electron to inject in patient body as specific tracer, show The positive electron of track agent transmitting falls into oblivion a pair of γ photons of generation with the electronics in human body, and this is recorded and stored down to photon by detector Come, tracer can be obtained by distribution situation in patient body by image reconstruction, so as to realize tumour early detection and Positioning.

The performance indications of small-sized PET are more preferable for human body PET, it is possible to obtain more preferable PET image.The kind equipment Detection mode is also more flexible, and the design of panel detector structure is also more diversified, except traditional ring-shaped P ET structure, more fits The double flat plate PET construction being imaged for small animal imaging and body part becomes new study hotspot.Either do animal to grind Study carefully or clinical practice, the PET of double flat plate possesses the structural advantage that can clamp imaging object, and detection mode is also more flexible, The breast cancer remained high especially for incidence and mortality has application well.

The panel detector structure of double flat plate PET is different from traditional annular PET, and its process of reconstruction is also different, due to The sampled data of double plates is incomplete so that analytical algorithm (such as FBP) is realized more difficult.It is main to consider that three-dimensional iterative reconstruction is calculated Method, for the algorithm, obtains the key that sytem matrix is image reconstruction.The preparation method of current system matrix mainly has three Kind：1 analytical Calculation, 2 test measurements, 3 cover snap gauge intends.Wherein experiment measurement and illiteracy snap gauge are intended being tested in advance or mould Intend, obtain sytem matrix, sytem matrix can be stored in advance, for being called when rebuilding, the two methods are being calculated Better than analytical Calculation is intended in precision, but is generally restricted to the situation of detector size fixation, once panel detector structure System changes it is necessary to re-starting simulation or testing measurement.

For double flat plate PET three-dimensional reconstructions, the μ PET that the toy of Chicago University's development is special intend obtaining using snap gauge is covered Sytem matrix is obtained, the method can be very good to reduce the impact that effect of depth brings, and obtain the preferable image of quality, and the method is very Double flat plate symmetry is utilized well, is reduced and is covered the workload that snap gauge is intended.But the method belongs to covers the category that snap gauge is intended, it is impossible to should For the situation that detector pitch changes, PET reconstructions also quickly cannot be in real time carried out.

The content of the invention

For the problem that prior art is present, it is an object of the invention to provide a kind of make full use of the symmetrical of double flat plate Property, the system matrix computation of simplified double flat plate PET, process of reconstruction of accelerating save the quick real-time of the double flat plate PET that calculating is consumed Method for reconstructing.

For achieving the above object, the present invention is, based on the quick real-time reconstruction method of double flat plate PET, to comprise the following steps：

1) the coupling positioning and coupling between the pixel in the described image matrix that calculating benchmark LOR intersects with image array Degree；

2) according to the coupling positioning and degree of coupling between calculated benchmark LOR intersecting pixel, using double flat plate The symmetry coupling positioning and the degree of coupling that obtain between whole LOR and pixel；

3) PET reconstructions are carried out by the coupling positioning between LOR and pixel and degree of coupling constructing system matrix.

Further, in the step 1) also include step before：Double flat plate spacing is determined, to determine and between the double flat plate Away from corresponding image array size.

Further, the step 2) it is specially：According to the coupling positioning between the pixel that calculated benchmark LOR intersects And degree of coupling, the coupling positioning gone out by translation transformation algorithmic derivation between the pixel that the LOR parallel with benchmark LOR intersects And degree of coupling.

Further, the translation transformation algorithm is：

Wherein,It is the vector parallel to xoy planes,Represent crystal to d_uAnd d_lBetween LOR, Represent the pixel of image array；N=Crystal in xoy planes_len/Voxel_len, Crystal_lenThe crystal unit length of side is represented, Voxel_lenRepresent the pixel length of side.

Further, for translation vectorThe LOR parallel with the benchmark LOR under effect, in the x and y direction Span is identical with the benchmark LOR, it is only necessary to the location of pixels translation intersected to the benchmark LORObtain and the benchmark The position of LOR parallel LOR intersecting pixel.

Further, the step 2) it is specially：According to the coupling positioning between the pixel that calculated benchmark LOR intersects And degree of coupling, the coupling that goes out between the pixel that the LOR parallel with benchmark LOR intersect of algorithmic derivation is converted by axial symmetry fixed Position and degree of coupling.

Further, the axial symmetry become scaling method into：

Wherein,Represent crystal to d_uAnd d_lBetween LOR,Represent the pixel of image array；T_xoyCalculate for symmetrical change Son, xoy planes are the intermediate layer in detector space.

The present invention calculates coupling positioning and the coupling obtained between LOR and pixel according to the symmetric relation of double slab constructions Conjunction degree, and then the reconstruction calculating of double flat plate PET is simplified, and save the memory space needed for sytem matrix.

Description of the drawings

Fig. 1 is the flow chart of the method for reconstructing of the double flat plate PET of one embodiment of the invention；

Fig. 2 is the flow chart of the method for reconstructing of the double flat plate PET of another embodiment of the present invention；

The structure that LOR intersects with pixel between the double flat plate of utilization translation transformations of the Fig. 3 (a) shown in one embodiment of the invention Schematic diagram；

LOR intersects with pixel between the double flat plate of utilization axial symmetry conversion of the Fig. 3 (b) shown in another embodiment of the present invention Structural representation；

Detector physical space formed symmetrical planes of the Fig. 3 (c) shown in another embodiment of the present invention；

Fig. 4 (a) and Fig. 4 (b) represent respectively LOR in y-axis direction (crystal array length direction) and x-axis direction (crystal battle array Column width direction) span.

Fig. 5 (a) and Fig. 5 (b) represent the span of all calculative oblique sequence of layer；

The flow chart of the iterative reconstruction algorithm that Fig. 6 is adopted by the method for reconstructing of the double flat plate PET of one embodiment of the invention.

Specific embodiment

As shown in figure 1, the quick real-time reconstruction method based on double flat plate PET of one embodiment of the invention, including it is following Step：

S113：Between pixel in the described image matrix that calculating benchmark LOR intersects with image array coupling positioning and Degree of coupling；

S115：According to coupling positioning and degree of coupling between the pixel that calculated benchmark LOR intersects, using double flat The symmetry of plate obtains coupling positioning and the degree of coupling between whole LOR and pixel；

S117：PET reconstructions are carried out by the coupling positioning between LOR and pixel and degree of coupling constructing system matrix.

Benchmark LOR in step S113 be consider by calculate coupling positioning and the degree of coupling LOR, remaining with span LOR Coupling positioning and the degree of coupling calculate and obtained through translation all on the basis of benchmark LOR, the selection of benchmark LOR can be all same One is arbitrarily selected in the LOR of span, the LOR on the basis of the LOR of detector summit crystal is chosen in this example, such as Fig. 3 b institutes LOR on the basis of showing four LOR all, four crystal of mark are summit crystal, the determination visible image matrix definition of summit crystal And depending on the definition in detector space, the selection of benchmark LOR should mainly consider that facilitating translation calculation is advisable；And according to double flat Spacing between plate determines image array size, the institute intersected with image array using known Point and Line Model calculating benchmark LOR State the coupling positioning and degree of coupling size between the pixel in image array.

In step S115 according in step S113 between the calculated pixel intersected with benchmark LOR coupling positioning and Degree of coupling, gone out by translation transformation algorithmic derivation between the pixel that the LOR parallel with benchmark LOR intersects coupling positioning and Degree of coupling.Wherein, translation transformation algorithm is specially：

Wherein,It is the vector parallel to xoy planes,Represent crystal to d_uAnd d_lBetween LOR, The pixel of image array.We require the N=Crystal in xoy planes_len/Voxel_len, Crystal_lenRepresent crystal unit The length of side, Voxel_lenRepresent the pixel length of side.

If the translation vector between LOR benchmark LOR in parallel isThen counted according to benchmark LOR Calculation obtains the position of the pixel intersected with it and addsThe position of LOR parallel LOR intersecting pixel on the basis of i.e., its The probable value of coupling is sized for the size of the coupling probability of the intersecting pixel of correspondence in benchmark LOR.So without the need for re-starting again Location Calculation and calculated using known Point and Line Model pixel and LOR the degree of coupling probable value, only by the meter of benchmark LOR The position that result just can go out the intersecting pixels of other LOR with direct derivation is calculated, and simply carries out copy tax to coupling probable value Value.The coupling positioning and degree of coupling size between the pixel that all LOR intersect with it can be so obtained, i.e., using double flat plate Symmetry obtain being coupled between all LOR and pixel positioning and degree of coupling, complete whole system matrix computations.

The calculating of degree of coupling intersects the size of probable value by taking existing Point and Line Model as an example, and figure is assumed in Point and Line Model As matrix is ideal cube, LOR is considered the straight line for connecting upper and lower two crystal bar, can be with computing system matrix element The coupling probability of LOR and pixel is obtained using the method for linear interpolation according to the distance between pixel center and adjacent two LOR Size.More simple Point and Line Model is possible to occur the matrix elements of pixel and the LOR for interacting and is entered as 1, and Row interpolation need not be entered, whole system matrix only has 1 or 0 composition.Sytem matrix is constituted using simple 0,1 value in this example, I.e. when the central plane of pixel has LOR through (the LOR quantity that each pixel center plane is averagely passed through should be 1/N, and the value of N is by N =Crystal_len/Voxel_lenDepending on), then probable value size in LOR sytem matrixes corresponding with the pixel (couples journey Degree) it is 1, then corresponding probable value is 0 to the pixel not intersected for the LOR.In fact, the complete sytem matrix for ultimately generating For sparse matrix, the most value of sytem matrix is all 0.

Using the coupling positioning between the calculated benchmark LOR of symmetry of double flat plate intersecting pixel in step S115 And degree of coupling, however it is not limited to accordingly calculated with translation transformation algorithm in above example.Also it is optional in the present embodiment Selecting property coupling positioning and the degree of coupling that scaling method comes between the intersecting pixel of calculating benchmark LOR is become with axial symmetry.Wherein, axle Symmetry transformation algorithm is：

Wherein,Represent crystal to d_uAnd d_lBetween LOR,Represent the pixel of image array；T_xoyCalculate for symmetrical change Son, xoy planes are the intermediate layer in detector space.T_xoyPixel to image arrayAfter effectThen convert ForIllustrate by taking xoy planes as an example in the present embodiment, for yoz planes or xoz plane symmetries become Change in the same manner.Xoy planes are with position of the yoz planes in detector physical space as shown in Figure 3 c.

As shown in Figure 3 b, 4 LOR in figure are 2 for 1, y directions span absolute value for all x directions span absolute values LOR, will complete the span sequence, on the basis of using translation property, can only calculate the coupling of this 4 LOR and picture element matrix Close positioning and couple probability, further consider the symmetry of xoy planes and yoz planes, can be by the calculating of 4 benchmark LOR Amount is simplified again as 1, only calculates LOR₁Coupling positioning and couple probability, you can further amount of calculation is reduced to into 1/ 4。

The calculating of the position of above-mentioned pixel is not limited to by Point and Line Model, it is also possible to using other conventional models, For example, line integral model, area sub-model, three-dimensional angle model etc., specifically suitable according to pixel and LOR width different choices Model carries out sytem matrix structure.Such as pixel is larger, i.e., when N values are less, big pixel crosses over multiple LOR, usable floor area point mould Type is more accurate；As pixel is less, i.e., when N values are larger, there is no the situation across multiple LOR in small pixel, now using solid Angle model is more accurate.When such as using above-mentioned increasingly complex model, the numerical value that sytem matrix is obtained is not simple 0,1, and It is more fine probable value (floating number), now, the probable value of the pixel being coupled therewith that benchmark LOR is obtained is no longer all It is 1, but related pixel more accurately couples probability size with benchmark LOR, for parallel LOR, its corresponding coupling Close probable value size to be obtained with from benchmark LOR copies.

The method for reconstructing of the present embodiment double flat plate PET go for the constant double slab constructions of spacing, i.e. double flat plate it Between spacing it is immutable, image array presets according to the spacing between double flat plate；Can be used for fixing several double The situation of plate spacing, determines after several pairs of plate spacing using this method for sytem matrix is calculated and is previously stored, in weight It is called to further speed up reconstruction speed when building.

The sytem matrix construction method of above double flat plate PET is illustrated below by way of a specific example, with double flat lath body Array be 75 × 50 as a example by, the crystal array of upper and lower flat board is as shown in Figure 3 a, it is illustrated that in only illustrated with 5 × 4 array.Upper strata Arbitrarily crystal bar d_uCrystal bar d all any with lower floor_lBetween produce LOR, for assume in the case of, have (75 × 50)²Kind LOR situations are produced, i.e., each crystal is to d_uAnd d_lBetween LOR quantity N be 14062500.Line up and down in Fig. 3 a is represented Along the y-axis direction (crystal array length direction) span is -1, and along the x-axis direction (crystal array width) span is also -1 Two row LOR, dashed rectangle illustrates the LOR pairings between crystal.In computing system matrix, all LOR of example in Fig. 3 a Situation only needs to calculate once, and follow-up all can be obtained by above-mentioned translation calculation.

It is enumerated above the example calculated using translational symmetry.The example calculated using axial symmetry, then such as Fig. 3 b Shown, span absolute value is that 1, y directions have the oblique sequence of layer of 4 LOR for 2 in x directions, and Fig. 3 b illustrate all of under this span 4 benchmark LOR, after the method for plane symmetry, it is possible to use LOR₁Obtain remaining 3 benchmark LOR coupling positioning and Coupling probability.Above N is further reduced to into 1/4, as 3515625 for 14062500 amounts of calculation.

Data tissue as shown in Fig. 4 (a) and Fig. 4 (b), different from general PET circuluses, deposit by double flat plate gathered data In the problem of lack sampling, therefore data organizational structure, also different from the sinogram patterns of ring-shaped P ET, here adopts special three Angular structure is depositing.As Fig. 4 (a) and Fig. 4 (b) represent respectively LOR in y-axis direction (crystal array length direction) and x-axis side To the span of (crystal array width).Cited all LOR situations are middle real in such as Fig. 4 (a) and Fig. 4 (b) in Fig. 3 a Wire frame is listed, y-axis direction correspondence Fig. 4 (a), x-axis direction correspondence Fig. 4 (b).Fig. 3 a only illustrate one in x directions and y directions respectively Row, the LOR of remaining identical span is not all listed, and the LOR of actual all spans is obtained by Fig. 4 (a) and 4 (b) combination.It is right The situation of x-axis and y-axis span all for -1 in example, it is actual to produce 74 × 49 LOR sequences, but by translational symmetry, only Need to calculate once, remaining situation is similar to be obtained.

After so simplifying, amount of calculation is from original (75 × 50)²(2 × 75-1) × (2 × 50-1) is reduced to, institute is in need The oblique sequence of layer (all LOR of x and y span identicals are defined as into an oblique sequence of layer) for calculating is dotted line in Fig. 5 (a) and 5 (b) Span combination shown in frame is obtained.

As shown in Figure 3 a, when sytem matrix is generated, for specifically a certain calculated a certain LOR is needed₁, calculating obtains The all pixels position and corresponding probable value size intersected with the LOR is obtained, for translation vectorIt is another under effect LOR₂, span in the x and y direction and LOR₁It is identical, it is only necessary to the location of pixels of each probability event position to be translated, is obtained Obtain location of pixels newly.For LOR₂All intersecting location of pixels, can all pass through LOR₁Corresponding intersecting pixel translates one admittedly Permanent amount gained, the constant byDetermine.Illustrate LOR₁With a series of situation of pixel value simple intersections, for simple 0 Or 1 Point and Line Model, then the pixel for drawing and the coupling probability size be somebody's turn to do all are 1, for more complicated model, it is assumed that LOR₁With The coupling probable value of pixel a1 is p, then corresponding LOR₂P is also entered as with the coupling probable value of pixel a2.

LOR₂Corresponding generation interact location of pixels can simply in LOR₁On the basis of related location of pixels Plus the Fixed constant of translation is obtained, LOR₂Its corresponding coupling probable value direct copying LOR₁Corresponding coupling probable value.

Equally, using axial symmetry for benchmark LOR is further simplified as shown in Figure 3 b, LOR₁Intersect with its pixel a1 Probability is p ', after symmetry transformation, LOR₁The coupling probability of ' with corresponding pixel a1 through conversion ' is also p '.

After completion system matrix is built, using known iterative reconstruction algorithm PET reconstructions are carried out.Make in the present embodiment With maximum likelihood method (ML-EM), in each iterative process, need to be carried out using sytem matrix when calculating and project from image array Calculate.Iterative reconstruction algorithm will use obtained system response model when each time root updates initial pictures F, draw just Projections of the beginning image F to system, the projection D ' for then measuring with experiment is contrasted, and the correction coefficient for obtaining is original for correcting Initial pictures F.In each correction iterative process, more accurate image F will be obtained, that is, reconstruct image.

As shown in fig. 6, the iterative reconstruction algorithm particular flow sheet is as follows：

1. an initial pictures F is set；

2., according to the system response model for being obtained, projection Ds of the image F to system is calculated；

3. test the projection D ' for measuring together to be compared；

4. correction coefficient and more new images F are calculated；

5. judge whether that satisfaction is stopped rule, if be unsatisfactory for, re-executes step 2, and the image F after renewal is defeated Enter in step 2；Otherwise, if it is satisfied, then execution step 6, terminates iterative process.

Quick real-time reconstruction method of the present invention based on double flat plate PET, according to the symmetry of double flat plate LOR and picture are simplified The calculating of coupling positioning and degree of coupling, significantly reduces the operand of computing system matrix between element, saves and calculates consumption While, due to only needing to preserve the coupling positioning and degree of coupling size between the pixel that benchmark LOR intersects with it, save The memory space of sytem matrix.

As shown in Fig. 2 for the quick real-time reconstruction method based on double flat plate PET of another embodiment of the present invention, its with Method for reconstructing shown in Fig. 1 is essentially identical, for wherein identical step will not be described here, itself and the reconstruction side shown in Fig. 1 The difference of method is also to include step S111 before step S113：Double flat plate spacing is determined, to determine and between the double flat plate Away from corresponding image array size.Because the spacing between double flat plate influences whether the size of image array, and then can affect The calculating knot of coupling positioning and degree of coupling between the pixel in the described image matrix that benchmark LOR intersects with image array Really, therefore in order to adapt to the change of spacing between double flat plate to the coupling positioning between pixel and the result of calculation of degree of coupling, It was necessary before step S113 to determine double flat plate spacing first, obtains the image moment corresponding with the double flat plate spacing for being obtained Battle array, for example, if double flat plate spacing diminishes, the number of plies of image array also tails off, (general one in the case that pixel size is constant The pixel of individual equipment is the resolution ratio performance according to whole machine determining), the number of the pixel in image array necessarily tails off, The result of calculation of coupling positioning and degree of coupling so between pixel also will necessarily change.If likewise, double flat plate spacing Become big, then the number of plies of image array also becomes many, and in the case that pixel size is constant, the number of the pixel in image array necessarily becomes Many, the result of calculation of coupling positioning and degree of coupling between such pixel also changes therewith.

In the present embodiment, before the coupling positioning and degree of coupling between pixel is calculated, it is first determined double flat plate spacing, The double flat plate PET of dynamic change can be rebuild, so be directed to double flat plate PET gap variables in clinical practice scene Situation, quickly calculates in real time the sytem matrix of double flat plate PET.

The system architecture of the present embodiment is the double flat plate PET system of gap variable, mainly considers to be applied to mammary gland, first shape The small animal imaging of scanning or the scientific research of the local organs such as gland, four limbs.This require double flat plate spacing be it is variable, it is existing Experiment measurement and cover the sytem matrix building method that snap gauge intends and cannot all meet the requirement of the system.The present embodiment has used one Real-time Analytic Calculation Method is planted, without the need for carrying out testing measurement and simulation calculating in advance, it is determined that carrying out weight after double flat plate spacing Build, carry out the analytical Calculation of sytem matrix in process of reconstruction in real time.

The present invention adopts the panel detector structure of double flat plate, detector to be different from general ring-shaped P ET, make for breast imaging Meeting between upper and lower detector panel is carried out with double flat partitioned detector.When different patients is imaged, double flat plate spacing Difference, when being imaged to same patient, carries out static collection imaging after detector panel clamping mammary gland.For small animal imaging During application scenarios, multiple plate spacing, such as 2cm or 5cm, afterwards using this method can be determined according to the different size of mouse or rat The calculating and storage of sytem matrix are carried out, the real-time calling system matrix in image reconstruction.Except to mammary gland scene and toy Application outside, apply also for thyroid gland, four limbs imaging etc., concrete application scene is the present invention is not limited.

More than it is particularly shown and described the illustrative embodiments of the present invention.It should be understood that the invention is not restricted to institute Disclosed embodiment, on the contrary, it is intended to cover comprising various modifications within the scope of the appended claims and equivalent put Change.

Claims (7)

1. the quick real-time reconstruction method based on double flat plate PET, comprises the following steps：

1) the coupling positioning and coupling journey between the pixel in the described image matrix that calculating benchmark LOR intersects with image array Degree；

2) according to the coupling positioning and degree of coupling between the calculated pixel intersected with benchmark LOR, using double flat plate Symmetry obtains coupling positioning and the degree of coupling between whole LOR and pixel；

3) PET reconstructions are carried out by coupling positioning and degree of coupling constructing system matrix between LOR and pixel.

2. the quick real-time reconstruction method of double flat plate PET is based on as claimed in claim 1, it is characterised in that in the step 1) step is also included before：Double flat plate spacing is determined, to determine the image array size corresponding with the double flat plate spacing.

3. the quick real-time reconstruction method of double flat plate PET, the step 2 are based on as claimed in claim 1) it is specially：According to Coupling positioning and degree of coupling between the calculated pixel intersected with benchmark LOR, is gone out by translation transformation algorithmic derivation Coupling positioning and degree of coupling between the pixel that the LOR parallel with benchmark LOR intersects.

4. the quick real-time reconstruction method of double flat plate PET is based on as claimed in claim 3, and the translation transformation algorithm is：

$h({\stackrel{\→}{l}}_{d_u{d}_l};\stackrel{\→}{f})=h({\stackrel{\→}{l}}_{d_u{d}_l}+\stackrel{\→}{m};\stackrel{\→}{f}+N\·\stackrel{\→}{m})$

Wherein,It is the vector parallel to xoy planes,Represent crystal to d_uAnd d_lBetween LOR,Represent figure As the pixel of matrix；N=Crystal in xoy planes_len/Voxel_len, Crystal_lenThe crystal unit length of side is represented, Voxel_lenRepresent the pixel length of side.

5. the quick real-time reconstruction method of double flat plate PET is based on as claimed in claim 4, for translation vectorUnder effect The LOR parallel with benchmark LOR, span in the x and y direction is identical with benchmark LOR, it is only necessary to pair intersect with benchmark LOR Location of pixels is translatedObtain the position of the pixel that the LOR parallel with benchmark LOR intersects.

6. the quick real-time reconstruction method of double flat plate PET is based on as claimed in claim 1, it is characterised in that the step 2) Specially：According to coupling positioning and degree of coupling between the calculated pixel intersected with benchmark LOR, become by axial symmetry Scaling method derives coupling positioning and the degree of coupling between the pixel that the LOR parallel with benchmark LOR intersects.

7. the quick real-time reconstruction method of double flat plate PET is based on as claimed in claim 6, it is characterised in that the axial symmetry Become scaling method into：

$h({\stackrel{\→}{l}}_{d_u{d}_l};\stackrel{\→}{f})=h(T_{xoy}{\stackrel{\→}{l}}_{d_u{d}_l};T_{xoy}\stackrel{\→}{f})$

Wherein,Represent crystal to d_uAnd d_lBetween LOR,Represent the pixel of image array；T_xoyChange operator for symmetrical, Xoy planes are the intermediate layer in detector space.