CN1641700A - Positive electron emitted computerised tomography full-variation weighted image method - Google Patents

Abstract

The invention discloses a method of total variation total weighting imaging of the least square method positron emission computed tomography visualization. The first is to get the projection data, confirm the initial image range, calculate the system probability matrix, and multiply system probability matrix by initial image to get the forward direction projection, multiply the corrected value of the projection data by system probability matrix to get the image corrected value in imaging iterative process. Then is to do total variation of the initial image after discretization, differentiate for every picture elements to get corrected value of the objective function. The last is to calculate to objective function of the rebuilding image, multiply the objective function by initial image to get the image iteration renew; then the image is used as the initial image. To repeat this process until the rebuilt image is convergence. The invention can improve the quality of the image after imaging, and eliminate influence of the noise on imaging.

Description

The full-variation weighted formation method of Positron Emission Computed Tomography

Technical field

The image that the present invention relates to a kind of full-variation weighted least square method becomes construction method, relates in particular to a kind of full-variation weighted formation method of Positron Emission Computed Tomography.

Technical background

(Positron emission tomography PET) is the nucleus medical image technology of current highest level to Positron Emission Computed Tomography, and the PET scanner is the state-of-the-art large-scale medical diagnosis imaging device that medical circle is generally acknowledged.PET is a kind of modern medicine image technology that utilizes positron emitter radioactive nuclide and tagged compound thereof to carry out body local or whole body imaging.The PET imaging process is by injection or sucks radiopharmaceutical, through time-delay after a while, after radiopharmaceutical is sent to the examine organ, begin to scan, during the radioactive isotope decay, it launches a positron, after positron moves through short distance, meet with an electronics and to bury in oblivion, and produce two almost high-energy photons of propagating of reverse direction, if two photons were detected a very short time, then write down an incident, form the data for projection that radioisotope concentration distributes on the tomography by these incidents that detector detects, use these data for projection, can obtain the two dimensional image that radioactive isotope distributes on this tomography according to formation method.The reason that causes the PET image error has a lot, as quick decay, the high count rate of positive electricity subclass medicine intensity cause system's dead time to lose, meet at random, the influence of scattering and absorption of human body decay, statistical noise etc. have also seriously influenced the PET image quality, so the imaging of PET image high precision is to the commercialization of PET with popularize the effect fatefully that plays.Because the incompleteness of PET data for projection and the ill-posedness of method for reconstructing, cause the image border after the reconstruction irregular, making an uproar phenomenon is obvious.

Summary of the invention

The invention provides a kind of quality that can improve image after the imaging, weaken even eliminate the full-variation weighted formation method of noise the Positron Emission Computed Tomography of imaging influence.

The present invention adopts following technical scheme:

A kind of full-variation weighted formation method of Positron Emission Computed Tomography of least square method:

1) obtains data for projection,, determine the initial pictures scope, given initial ash according to the dimensional requirement for the treatment of reconstructed image

2) the degree value is greater than 1, and becomes 1 dimensional vector to calculate 2 dimension image transformations,

3) according to data for projection scale and the size that requires image x, computing system probability matrix P,

3) system's probability matrix and initial pictures x are multiplied each other, obtain forward projection a,

4) each the data for projection y that takes from the Positron Emission Computed Tomography scanner carries out a square calculating, obtains y2, again with this number divided by forward projection a square, obtain the modified value c of data for projection,

c＝y ²/a ²

5) the probability matrix P of system be multiply by the modified value c of data for projection, obtains the modified value d of image in the image imaging iterative process,

6) initial pictures with discretize carries out full variation, again with the full variation initial pictures of this discretize to each picture element differentiate, obtain the corrected value of objective function

7) with image correction value d divided by 1 with the corrected value of β objective function doubly Obtain being used for the objective function of reconstructed image, the target function value that again this is used for reconstructed image multiply by initial pictures, the image after obtaining iteration and upgrading, again with this image as initial pictures, turned back to for the 3rd step, repeat the image convergence of this process after rebuilding.

Compared with prior art, the present invention has following advantage:

The present invention is fused in the weighted least-squares PET formation method as regular terms and with it with full variation, and the quality of image after the raising imaging is eliminated the influence of noise to imaging.

Description of drawings

Fig. 1 is the abdominal cavity template image that is used for testing formation method.

Fig. 2 is the bianry image of abdominal cavity template.

Fig. 3 is with the result after the PET formation method imaging of existing weighted least-squares method.

Fig. 4 is to the result after the PET formation method imaging of the existing weighted least-squares method of usefulness behind the data for projection adding noise of test template.

Fig. 5 is the result after Fig. 4 binaryzation.

Fig. 6 is that wherein β is 0.002 with the result after the imaging of the present invention.

Fig. 7 is that wherein β is 0.005 with the result after the imaging of the present invention.

Fig. 8 is that wherein β is 0.008 with the result after the imaging of the present invention.

Fig. 9 is that wherein β is and 0.01 with the result after the imaging of the present invention.

Figure 10 is that with the result after the imaging of the present invention, wherein β was 0.002 after test template was added noise.

Figure 11 is that with the result after the imaging of the present invention, wherein β was 0.005 after test template was added noise.

Figure 12 is that with the result after the imaging of the present invention, wherein β was 0.008 after test template was added noise.

Figure 13 is that with the result after the imaging of the present invention, wherein β was 0.01 after test template was added noise.

Figure 14 is the result after Figure 10 binaryzation.

Figure 15 is the result after Figure 11 binaryzation.

Figure 16 is the result after Figure 12 binaryzation.

Figure 17 is the result after Figure 13 binaryzation.

Figure 18 does not contain the root-mean-square error analysis result of noise for data for projection.

Figure 19 contains the root-mean-square error analysis result of noise for data for projection.

Figure 20 does not contain the variance analysis result of noise for data for projection.

Figure 21 contains the variance analysis result of noise for data for projection.

Figure 22 is Fig. 3 the 48th alignment diagram.

Figure 23 is Fig. 9 the 48th alignment diagram.

Embodiment

Embodiment 1

A kind of full-variation weighted formation method of Positron Emission Computed Tomography of least square method:

1) obtain data for projection, according to the dimensional requirement for the treatment of reconstructed image, determine the initial pictures scope, given initial gray-scale value is greater than 1, and becomes 1 dimensional vector to calculate 2 dimension image transformations,

2) according to data for projection scale and the size that requires image x, computing system probability matrix P,

3) system's probability matrix and initial pictures x are multiplied each other, obtain forward projection a,

4) each the data for projection y that takes from the Positron Emission Computed Tomography scanner carries out a square calculating, obtains y ², again with this number divided by forward projection a square, obtain the modified value c of data for projection,

c＝y ²/a ²

5) the probability matrix P of system be multiply by the modified value c of data for projection, obtains the modified value d of image in the image imaging iterative process,

6) initial pictures with discretize carries out full variation, again with the full variation initial pictures of this discretize to each picture element differentiate, obtain the corrected value of objective function

7) with image correction value d divided by 1 with the corrected value of β objective function doubly

Obtain being used for the objective function of reconstructed image, the target function value that again this is used for reconstructed image multiply by initial pictures, obtains the image after iteration is upgraded, again with this image as initial pictures, turned back to for the 3rd step, repeat the image convergence of this process after rebuilding

Obtaining of above-mentioned data for projection is to obtain from the Positron Emission Computed Tomography scanner, or carries out thunder when (Radon) conversion, the data for projection that obtains from the simulation modular image.

Embodiment 2

The present invention obtains after existing PET formation method is improved, the theing contents are as follows of specific embodiments:

1. existing weighted least-squares method of estimation

Used imaging model mainly is that the photo emissions process that supposition PET scanner is detected is to obey on the Poisson distribution basis, promptly on the present commercial PET machine

$y_i~\mathrm{Poisson}\left\{\underset{j}{\mathrm{\Σ}}{p}_{\mathrm{ij}}{x}_j\right\}---\left(1\right)$

Y wherein _iRepresent i the photon number that detector detected, 0≤i≤m, m are the detector sum; x _jRepresent the photon number that j pixel place sends, x _j〉=0,0≤j≤n, n are number of picture elements; p _IjThe photon energy of representing to send at j pixel place is by i the detected probability of detector.According to this hypothesis, we set up the PET reconstruction model that a weighted least-squares is estimated.This model decides concrete weights according to the variance of data.This be because the variance quantitative response confidence level of the overall expectation of sample representative, the big more data confidence level of variance is low more, so rational way obviously is to give the less data of variance with bigger weights, the remaining now quantitative relationship that will determine weights and data variance exactly is so that the variance minimum or the precision of estimated value are the highest, by knowledge of statistics, accomplish that this point should make weights be inversely proportional to variance.For the Poisson statistical error, we know that the variance of data equals expectation, so can describe the modeling problem under the weighted least-squares estimation criterion now, that is to say that we can separating as final estimated value with following optimization problem.It can be expressed as

$\left\{\begin{array}{c}\mathrm{\Φ}:\arg \min \left\{{(\mathrm{Px}-y)}^T{W}^{-1}(\mathrm{Px}-y)\right\}\\ s.t.:x\≥0\end{array}\right.---\left(2\right)$

$\mathrm{\Φ}\left(x\right)={\mathrm{\Σ}}_{i=1}^m\frac{{({\left(\mathrm{Px}\right)}_i-y_i)}^2}{{\left(\mathrm{Px}\right)}_i}---\left(3\right)$

Here W is the power diagonal matrix of a m * m, and its i element is (Px) _i:

w _ij＝diag((Px) ₁，(Px) ₂，....，(Px) _m) (4)

The single order local derviation that makes Φ (x) is zero, and according to the Kuhn-Tucker condition, we have:

$\frac{\∂}{{\∂x}_j}\left(\mathrm{\Φ}\left(x\right)\right)=\underset{i=1}{\overset{m}{\mathrm{\Σ}}}(-\frac{{y_i}^2{p}_{\mathrm{ij}}}{{\left(\mathrm{Px}\right)}_i^2}+p_{\mathrm{ij}})=0,x_j>0---\left(5\right)$

$\frac{\∂}{{\∂x}_j}\left(\mathrm{\Φ}\left(x\right)\right)=\underset{i=1}{\overset{m}{\mathrm{\Σ}}}(-\frac{{y_i}^2{p}_{\mathrm{ij}}}{{\left(\mathrm{Px}\right)}_i^2}+p_{\mathrm{ij}})\≥0,x_j=0---\left(6\right)$

Therefore we draw the PET imaging algorithm of the weighted least-squares method of a point of fixity:

$x_j^{(k+1)}=x_j^{\left(k\right)}\underset{i=1}{\overset{m}{\mathrm{\Σ}}}\frac{y_i^2{p}_{\mathrm{ij}}}{{\left(\underset{j=1}{\overset{n}{\mathrm{\Σ}}}{p}_{\mathrm{ij}}{x}_j^{\left(k\right)}\right)}^2},j=\mathrm{1,2},\mathrm{\Λ},n---\left(7\right)$

The picture contrast that this PET formation method obtains is than higher, and reconstructed image quality is also relatively good, but making an uproar phenomenon is more serious, and the pseudo-shadow on the edge is difficult to eliminate.In order to verify the reconstruction effect of this method, we verify with the PET abdominal cavity phantom template of a Computer Simulation.Fig. 1 has shown this abdominal cavity template, the template image size is 96 * 96 PEL matrix, data scale is 139 * 180, i.e. 180 projection angles, 139 parallel projection lines are arranged on each angle, we make the spacing of parallel lines equate with the length of side of image pixel, so that the probability matrix P's of system is definite.Fig. 2 is the bianry image of this phantom template.Fig. 3 is with the result after formula (7) imaging.From Fig. 3, we can see tangible pseudo-shadow on the edge, and image is fuzzyyer, this shows, projection not plus noise is difficult to reconstruct comparatively ideal image with existing method.Fig. 4 is to the result after usefulness formula (7) imaging behind this template plus noise.

Fig. 5 is the result after Fig. 4 binaryzation, and it further specifies with after the existing method imaging, and picture quality is very undesirable.

2. full variation regularization weighted least-squares PET formation method of the present invention

In order to improve picture quality, reduce noise and keep the edge, we have invented the weighted least-squares PET formation method of the full variation of a kind of usefulness as regular terms.The use of full variation is that mainly it effectively can keep the edge not to be destroyed as far as possible denoising the time.The expression formula of full variation is:

$\mathrm{TV}\left(f\right)={\∫}_{\mathrm{\Ω}}|\▿f|\mathrm{dxdy}={\∫}_{\mathrm{\Ω}}\sqrt{f_x^2+f_y^2}\mathrm{dxdy}---\left(8\right)$

Here $f_x=\frac{\∂}{\∂x}f,$ $f_y=\frac{\∂}{\∂y}f.$ Following formula is about i, and the difference expression of j is:

$U_{\mathrm{TV}}=\underset{i,j}{\mathrm{\Σ}}\sqrt{{(f_{i+1,j}-f_{i,j})}^2+{(f_{i,j+1}-f_{i,j})}^2+{\mathrm{\ϵ}}^2}---\left(9\right)$

We find that parameter ε should be smaller or equal to 1% f maximal value.The ε value conference smoothly fall the edge.The partial derivative of formula (9) is:

$\frac{{\∂U}_{\mathrm{TV}}}{{\∂f}_{i,j}}=\frac{f_{i,j}-f_{i-1,j}}{\sqrt{{(f_{i,j}-f_{i-1,j})}^2+{(f_{i-1,j+1}-f_{i-1,j})}^2+{\mathrm{\ϵ}}^2}}$

$+\frac{f_{i,j}-f_{i,j-1}}{\sqrt{{(f_{i+1,j-1}-f_{i,j-1})}^2+{(f_{i,j}-f_{i,j-1})}^2+{\mathrm{\ϵ}}^2}}$

$-\frac{f_{i+1,j}+f_{i,j+1}-2f_{i,j}}{\sqrt{{\left(f_{i+1,j}{-f}_{i,j}\right)}^2+{(f_{i,j+1}-f_{i,j})}^2+{\mathrm{\ϵ}}^2}}---\left(10\right)$

Weighting minimum target function J based on full variation regular terms of the present invention _βReplace the existing weighted least-squares objective function φ (x) shown in the formula (3), the image after the reconstruction By making new objective function J _β(x) minimum provides:

$\stackrel{)}{x}=\arg \underset{x}{\min }\left(J_{\mathrm{\β}}\left(x\right)\right)---\left(11\right)$

The present invention's objective function here is made up of two parts: common weighted least-squares item and full variation regular terms, objective function J of the present invention _β(x) be

J _β(x)＝φ(y，Px)+βU (12)

Here β is a weight factor, and it will influence the effect degree of full variation regular terms in imaging process, and formula (12) is asked the single order local derviation.And formula (10) is brought into formula (12), so at each pixel x _jAsk J _βThe single order local derviation be:

$\frac{{\∂J}_{\mathrm{\β}}\left(x\right)}{{\∂x}_j}=\underset{i}{\mathrm{\Σ}}(\frac{-{y_i}^2{p}_{\mathrm{ij}}}{{{\left(\mathrm{Px}\right)}^2}_i}+p_{\mathrm{ij}})+\mathrm{\β}\frac{{\∂U}_{\mathrm{TV}}}{{\∂x}_j}---\left(13\right)$

Because $\underset{i=1}{\overset{m}{\mathrm{\Σ}}}{p}_{\mathrm{ij}}=1$

According to the Kuhn-Tucher condition, the point of fixity imaging that addresses this problem is iterative to be:

$x_j^{(k+1)}=\frac{x_j^{\left(k\right)}}{(1+\mathrm{\β}\frac{{\∂U}_{\mathrm{TV}}}{{\∂x}_j})}\underset{i=1}{\overset{m}{\mathrm{\Σ}}}\frac{p_{\mathrm{ij}}{y_i}^2}{{\left(\underset{j=1}{\overset{n}{\mathrm{\Σ}}}{p}_{\mathrm{ij}}{x}_j^{\left(k\right)}\right)}^2}---\left(14\right)$

Because the present invention joins full variation regular terms in the common weighted least-squares PET formation method, make the PET precision of images after the imaging obtain bigger raising, removed noise effectively, acting in the noisy imaging of projection of full variation is particularly evident.The effect of parameter beta is to be used for regulating the influence degree of full variation regular terms to algorithm in formula (14), along with the increase of β, the function of regular terms is strengthened, and image is further smoothed, when β was zero, formula (14) became common weighted least-squares formation method.

Fig. 6 and Fig. 7 are that wherein β is respectively 0.002 and 0.005 with the result after the inventive method (formula (14)) imaging.Fig. 8 and Fig. 9 also be with we the invention new algorithm (formula (14)) imaging after the result, wherein β is respectively 0.008 and 0.01.Test thus, finding does not have under the situation of noise at data for projection, and parameter beta is little to the influence of reconstructed image, but the picture quality after rebuilding will be got well more than common weighted least-squares reconstruction method (Fig. 3).

Figure 10 and Figure 11 are that to the imaging results behind the data for projection plus noise, wherein β is respectively 0.002 and 0.005 with our new method (formula (14)) of invention.Figure 12 and Figure 13 also are that to the imaging results behind the data for projection plus noise, wherein β is respectively 0.008 and 0.01 with the present invention (formula (14)).Can find increase along with β by this test, the smoothing effect of image strengthens after the imaging, and the edge has also obtained effective maintenance.Image after the imaging will be got well more than the common weighted least-squares formation method (Fig. 4) that has same noise conditions in projection.

Figure 14 is the result after Figure 10 binaryzation, Figure 15 is the result after Figure 11 binaryzation, Figure 16 is the result after Figure 12 binaryzation, Figure 17 is the result after Figure 13 binaryzation, image after these binaryzations further illustrates the method for the invention can reconstruct high-precision image, and this is because the adding of full variation regular terms has greatly improved the imaging precision of original method.

The present invention is effective especially to the PET image imaging, this is because the principle of PET imaging has caused the resolution of PET image lower, signal noise ratio (snr) of image is lower, making an uproar phenomenon is serious, and this method of the present invention is particularly effective to the low PET backprojection reconstruction of signal to noise ratio (S/N ratio), by Figure 13 and Figure 17 as can be known along with the increase of β, full variation regularization effect strengthens, denoising effect is obvious, the result (Fig. 2) of result behind the image binaryzation (Figure 17) after near the primary template binaryzation, and will get well more than the result of Fig. 4 binaryzation.The binaryzation of Fig. 4 the results are shown in Figure 5.

3. test result of the present invention

The test of the PET formation method of full-variation weighted least square method of the present invention is that 2.4GHz carries out on the 1.00GB at Pentium 4 CPU.In order to test the validity of the inventive method, and the PET formation method of full-variation weighted least square method there is a more clearly understanding, the degree of closeness of testing the image that generates with certain formation method and primary standard image with certain criterion is weighed the quality of this formation method, and we adopt following two kinds of criterions:

Root-mean-square error, promptly

Assess the quality of image after the imaging with the root-mean-square error between the image after test template image and the imaging (root mean square error (RMS)).Root-mean-square error is defined as:

$\mathrm{RMS}={\left[\frac{1}{n}{\mathrm{\Σ}}_{j=1}^n{(x_j^{\mathrm{rec}}-x_j^{\mathrm{org}})}^2\right]}^{\frac{1}{2}}---\left(15\right)$

Here x _j ^OrgAnd x _j ^RecRepresent the template image of an emulation respectively and rebuild after image in the value of this position of pixel j.Good formation method will have less root-mean-square error.Figure 18 has shown that data for projection does not contain the root-mean-square error analysis result of noise, Figure 19 has shown that data for projection contains the root-mean-square error analysis result of noise, these results shown image after neoteric formation method is rebuild more than existing weighted least require method more near the test template data, because the root-mean-square error between image after the imaging of the present invention and the test template image is little.

Deviation, promptly

$\mathrm{Variance}=\frac{1}{n-1}{\mathrm{\Σ}}_{j=1}^n{(x_j^{\mathrm{rec}}-{\stackrel{\‾}{x}}^{\mathrm{org}})}^2---\left(16\right)$

Here x ^OrgRepresentative is used for the average gray value of test template image.Formation method is good more, and deviation is more little.Figure 20 has shown that data for projection does not contain the variance analysis result of noise, and Figure 21 has shown that data for projection contains the variance analysis result of noise.

Figure 22 has shown that data for projection does not contain the 48th row outline line and the approaching degree of primary template image the 48th row outline line of the image after the existing method imaging of usefulness under the noise situations.

Figure 23 shown data for projection do not contain under the noise situations with the new invention method to the data for projection imaging after the 48th row outline line and the approaching degree of primary template image the 48th row outline line of image.This shows the more approaching original test template image of result that the present invention rebuilds.

Claims (3)

1. the full-variation weighted formation method of the Positron Emission Computed Tomography of a least square method is characterized in that adopting the following step:

1) obtain data for projection, according to the dimensional requirement for the treatment of reconstructed image, determine the initial pictures scope, given initial gray-scale value is greater than 1, and becomes 1 dimensional vector to calculate 2 dimension image transformations,

2) according to data for projection scale and the size that requires image x, computing system probability matrix P,

3) system's probability matrix and initial pictures x are multiplied each other, obtain forward projection a,

4) each the data for projection y that takes from the Positron Emission Computed Tomography scanner carries out a square calculating, obtains y ², again with this number divided by forward projection a square, obtain the modified value c of data for projection,

c＝y ²/a ²

5) the probability matrix P of system be multiply by the modified value c of data for projection, obtains the modified value d of image in the image imaging iterative process,

6) initial pictures with discretize carries out full variation, again with the full variation initial pictures of this discretize to each picture element differentiate, obtain the corrected value of objective function

7) with image correction value d divided by 1 with the corrected value of β objective function doubly

Obtain being used for the objective function of reconstructed image, the target function value that again this is used for reconstructed image multiply by initial pictures, the image after obtaining iteration and upgrading, again with this image as initial pictures, turned back to for the 3rd step, repeat the image convergence of this process after rebuilding.

2. the Bayes image method for reconstructing based on implicit activity profile priori according to claim 1 is characterized in that obtaining from the Positron Emission Computed Tomography scanner of data for projection obtain.

3. the Bayes image method for reconstructing based on implicit activity profile priori according to claim 1 is characterized in that obtaining of data for projection is to carry out thunder from the simulation modular image to work as conversion, the data for projection that obtains.

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