CN111696166A - FDK (finite Difference K) type preprocessing matrix-based circumferential cone beam CT (computed tomography) fast iterative reconstruction method - Google Patents

FDK (finite Difference K) type preprocessing matrix-based circumferential cone beam CT (computed tomography) fast iterative reconstruction method Download PDF

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CN111696166A
CN111696166A CN202010523774.8A CN202010523774A CN111696166A CN 111696166 A CN111696166 A CN 111696166A CN 202010523774 A CN202010523774 A CN 202010523774A CN 111696166 A CN111696166 A CN 111696166A
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刘华锋
王婷
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Zhejiang University ZJU
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Abstract

The invention discloses a circumferential cone beam CT fast iterative reconstruction algorithm based on an FDK type preprocessing matrix, which effectively improves the quality and speed of image reconstruction by utilizing a CT image reconstruction model under the conditions of low dose and sparse view angle and combining the advantages of high speed of an analytic reconstruction algorithm and good effect of an iterative reconstruction algorithm, accelerates the three-dimensional iterative reconstruction process by introducing the preprocessing matrix based on the FDK algorithm in the process of solving the model, solves the three-dimensional iterative reconstruction process by using an alternative projection approximation algorithm, and iterates repeatedly until the termination condition is met; the experiment comparison with the existing reconstruction algorithm shows that the method can obtain better reconstruction effect.

Description

FDK (finite Difference K) type preprocessing matrix-based circumferential cone beam CT (computed tomography) fast iterative reconstruction method
Technical Field
The invention belongs to the technical field of CT imaging, and particularly relates to a circumferential cone-beam CT fast iterative reconstruction algorithm based on an FDK type preprocessing matrix.
Background
X-ray CT is a nondestructive testing technique and has been widely used in various fields such as medical diagnosis, security inspection, industrial nondestructive testing, and product quality testing. In the field of CT, there are many different scanning methods for measuring projection data, which can be divided into parallel beam scanning, fan beam scanning, cone beam scanning, and the like; the cone beam is the natural shape of the X-ray source, compared with the parallel beam and the fan beam, the data volume which can be obtained by one projection of the cone beam is much larger, so that the cone beam scanning structure is more beneficial to improving the scanning speed and the quality of the reconstructed image, but the corresponding three-dimensional image reconstruction algorithm is much more complex and has a large amount of calculation compared with the two-dimensional image reconstruction algorithm corresponding to the parallel beam and fan beam scanning structure. The detector structure used by the cone beam CT mainly comprises a plane detector and a cylindrical surface detector, wherein detector units of the plane detector and the cylindrical surface detector are arranged at equal intervals in the vertical direction, but the detector units are arranged at equal intervals in the horizontal direction, and the detector units are arranged at equal angles in the horizontal direction; the cone beam CT can be further divided into a circumferential cone beam CT, a helical cone beam CT, and the like according to a scanning trajectory of the light source.
The FBP algorithm based on Radon transformation has wide application in the reconstruction of two-dimensional images corresponding to parallel beam and fan-shaped beam scanning structures, and Feldcamp, Davis, Kress and the like in 1984 propose a cone beam projection reconstruction algorithm suitable for a plane circular scanning structure on the basis, which is called as an FDK algorithm; FDK is an approximate reconstruction algorithm, and the reconstruction principle is to decompose a cone beam into a plurality of inclined fan-shaped beams, then individually filter and back-project each of the inclined fan-shaped beams, and add all the back-projection results to obtain a three-dimensional reconstruction image. The FDK algorithm utilizes two-dimensional filtering and two-dimensional back projection, the mathematical theory is simpler, the calculated amount is smaller, the FDK algorithm is actually used, and when the cone angle is smaller than 10 degrees, the artifact in the reconstructed image is smaller; besides approximate reconstruction algorithms, there are also accurate reconstruction algorithms for cone beam CT, but they tend to be complex in calculation and slow in reconstruction speed.
The analytical reconstruction algorithms such as FDK have high requirements on projection data, so that the performance of the analytical reconstruction algorithms is often poor in the low-dose and sparse view angle CT reconstruction problem, and more obvious artifacts appear. Iterative Reconstruction algorithms, which model the Reconstruction problem as a linear system and then solve it using linear Algebraic methods, such as a series of algorithms and Expectation-Maximization (EM) methods of the Kaczmarz family represented by Algebraic Reconstruction Techniques (ART), are more advantageous in these cases. In order to improve the effect of the iterative reconstruction algorithm, a plurality of researchers add various prior constraints in the reconstruction process, such as various Total Variations (TV) of the image, low-rank characteristics of the image, sparse coding and the like; because a noise model can be fused and various prior information can be utilized, the iterative reconstruction algorithm usually has a better reconstruction effect than an analytic reconstruction algorithm in low-dose and sparse view CT, which is proved by years of research and practice; however, since the iterative reconstruction algorithm usually needs a plurality of iterative operations in the implementation process, the reconstruction speed depends on the design of the algorithm and the operational capability of the equipment.
In recent years, with the rapid development of computer technology, particularly GPU (graphical Processing Unit) acceleration technology, an iterative reconstruction algorithm is receiving more and more extensive attention; aiming at the problem of slow speed, many people research how to accelerate the speed and have achieved many results, wherein one main research direction is to use various acceleration methods to realize rapid convergence of Iterative algorithms, such as Fast Iterative threshold Shrinkage algorithm (FISTA) and Ordered Subset (OS) method; another main research direction is to combine an iterative algorithm with a parallel computing Method to accelerate the algorithm, such as dividing projection data, image pixel points, or a system matrix, and then performing parallel operation on each subset, or decomposing an optimization problem into a plurality of sub-problems to solve, such as an alternating direction Multiplier (ADMM) Method for solving a convex problem.
Disclosure of Invention
In view of the above, the invention provides a circumferential cone-beam CT fast iterative reconstruction algorithm based on an FDK type preprocessing matrix, which covers two situations of low dose and sparse view angle, and the algorithm combines the advantages of an analytic reconstruction algorithm and an iterative reconstruction algorithm, so that a high-quality reconstructed image can be obtained with fewer iteration times.
A fast iterative reconstruction algorithm of a circular cone beam CT based on an FDK type preprocessing matrix comprises the following steps:
(1) acquiring projection data of CT images in different angle directions by using a circumferential cone beam CT system to form a projection data set
Figure BDA0002532999630000031
If projection data of m angular directions are acquired during the measurement, a projection data set is formed
Figure BDA0002532999630000032
Is m × N, N being the number of detector units in a circumferential cone-beam CT system, the projection data set
Figure BDA0002532999630000033
The projection data vector is formed by projection data vectors acquired under the corresponding m angle directions, the dimensionality of the projection data vector is N, and each element value in the vector is projection data measured by a corresponding detector unit;
(2) respectively establishing CT image reconstruction models under the conditions of low dose and sparse view angle;
(3) preprocessing the CT image reconstruction equation under the two conditions, and converting the preprocessed CT image reconstruction equation into a saddle point problem by using a Lagrangian dual method to obtain a corresponding target function;
(4) and selecting a corresponding objective function according to the actual situation, and carrying out optimization solution on the objective function to reconstruct to obtain the CT image.
Further, the CT image reconstruction model in the case of low dose in step (2) is expressed as follows:
Figure BDA0002532999630000034
the CT image reconstruction model under sparse view angle condition is expressed as follows:
Figure BDA0002532999630000035
wherein:
Figure BDA0002532999630000036
is a CT image data set and
Figure BDA0002532999630000037
wherein each element value is a corresponding image in the CT image to be reconstructedThe X-ray absorption coefficient of the pixel point, J is the number of pixel points of the CT image to be reconstructed, T represents transposition, A is a system matrix,
Figure BDA0002532999630000038
for the penalty term used to constrain the objective function, β is given a weighting factor, | | | | represents a 2-norm.
Further, for the CT image reconstruction equation under the low dose condition in the step (3), an additional variable is first introduced
Figure BDA0002532999630000039
Rewriting formula (1) to the following form:
Figure BDA00025329996300000310
wherein: variables of
Figure BDA00025329996300000311
Expressed as a vector of
Figure BDA00025329996300000312
Projection data obtained by performing orthographic projection calculation;
then, introduce the non-singular matrix PconeFormula (3) is further rewritten as follows:
Figure BDA00025329996300000313
lagrange equation corresponding to equation (4)
Figure BDA00025329996300000314
The definition is as follows:
Figure BDA0002532999630000041
wherein:
Figure BDA0002532999630000042
is a Lagrangian vector;
further, equation (4) is converted into the following saddle point problem:
Figure BDA0002532999630000043
finally, through the pair of variables
Figure BDA0002532999630000044
Minimization calculations can be performed to eliminate it from the saddle point problem, resulting in the following objective function:
Figure BDA0002532999630000045
wherein:
Figure BDA0002532999630000046
is composed of
Figure BDA0002532999630000047
And
Figure BDA0002532999630000048
the inner product operation result of (2).
Further, for the CT image reconstruction equation under the sparse view angle condition in step (3), firstly, a non-singular matrix P is introducedconeThe formula (2) is rewritten as follows:
Figure BDA0002532999630000049
lagrange equation corresponding to equation (8)
Figure BDA00025329996300000410
The definition is as follows:
Figure BDA00025329996300000411
further, equation (9) is converted into the saddle point problem, and the corresponding objective function is as follows:
Figure BDA00025329996300000412
wherein:
Figure BDA00025329996300000413
is the lagrange vector.
Further, the non-singular matrix PconeThe expression of (a) is as follows:
Figure BDA00025329996300000414
P＝F-1R(ω)F
wherein: at low dose
Figure BDA00025329996300000415
In sparse view conditions
Figure BDA00025329996300000416
GcIs a diagonal matrix with dimension n, n is the number of detector units in each horizontal row in the circumferential cone-beam CT system, GcEach diagonal element in (a) is a correction term value, F and F, of the projection data measured by each detector unit in the corresponding line in the corresponding angular direction-1The method comprises the following steps of respectively using a one-dimensional Fourier transform operator and an inverse Fourier transform operator, wherein omega is a frequency domain variable obtained by Fourier transform of projection data vectors acquired in a corresponding angle direction, and tau is a given parameter.
Further, if the circular cone-beam CT system employs a planar detector, the diagonal matrix GcThe diagonal element in (A) is
Figure BDA0002532999630000051
If the circular cone beam CT system adopts a cylindrical surface detector, the diagonal matrix GcThe diagonal element in (A) is
Figure BDA0002532999630000052
Wherein: s is a ray of light andthe intersection point position value of the corresponding detector unit in the horizontal direction, h is the intersection point position value of the light and the corresponding detector unit in the vertical direction, D is the distance from the light source to the rotation center, and gamma is the included angle between the light and the central light on the horizontal central plane of the cone beam.
Further, in the step (4), an alternating projection approximation algorithm is adopted to perform optimization solution on the objective function.
Further, the penalty term
Figure BDA0002532999630000053
And adopting a total variation operator.
Further, for a circular cone-beam CT system, the non-singular matrix PconeInstead of acting directly on all projection data, the projection data measured by each row of detector units for each angle is processed separately.
According to the circumferential cone beam CT image reconstruction algorithm, a CT image reconstruction model under the conditions of low dose and sparse view angle is utilized, the advantages of high speed of an analytic reconstruction algorithm and good effect of an iterative reconstruction algorithm are combined, the quality and the speed of image reconstruction are effectively improved, a preprocessing matrix based on an FDK algorithm is introduced in the process of solving the model to accelerate the three-dimensional iterative reconstruction process, the preprocessing matrix is solved by using an alternative projection approximation algorithm, and iteration is repeated until the termination condition is met; the experiment comparison with the existing reconstruction algorithm shows that the method can obtain better reconstruction effect.
Drawings
FIG. 1 is a schematic flow chart of a circumferential cone-beam CT image reconstruction algorithm according to the present invention.
Fig. 2 is a sectional view of a three-dimensional Shepp-Logan head model showing a gray scale range of [1.00, 1.05], wherein (a) and (b) are respectively cross-sectional views of two vertical central axes of the head model in two vertical directions, and (c) and (d) are respectively cross-sectional views of two dotted lines L1 and L2 from top to bottom in (a) in a horizontal direction.
Fig. 3 is a graph showing the root mean square error of the reconstructed image obtained by the three methods in the reconstruction process compared with the true value image as a function of the iteration number, in which the angle range of the projection data is 360 ° and the angle number is 120.
Fig. 4 is a vertical axis cross-sectional view of the reconstruction results obtained by three methods, wherein (a) and (b) are cross-sections in two different directions, respectively, and the reconstructed images in the figure correspond to the results obtained by the reconstruction using the method of the present invention, the ART + TV method and the FISTA method from top to bottom in rows and correspond to the reconstruction results obtained by the 5 th, 10 th, 15 th and 20 th iterations from left to right in columns.
Fig. 5 is a horizontal cross-sectional view of the three methods of reconstruction results closer to the axial plane, in which the reconstructed images correspond to the results of reconstruction by the method of the present invention, the ART + TV method, and the FISTA method from top to bottom in rows, and to the reconstruction results of the 5 th, 10 th, 15 th, and 20 th iterations from left to right in columns.
Fig. 6 is a horizontal cross-sectional view of the three methods of reconstruction results further from the axial plane, wherein the reconstructed images correspond to the results of the reconstruction by the method of the present invention, the ART + TV method and the FISTA method from top to bottom in rows and to the reconstruction results from the 5 th, 10 th, 15 th and 20 th iterations from left to right in columns.
Detailed Description
In order to more specifically describe the present invention, the following detailed description is provided for the technical solution of the present invention with reference to the accompanying drawings and the specific embodiments.
As shown in FIG. 1, the fast iterative reconstruction algorithm for the FDK type preprocessing matrix-based circular cone beam CT of the present invention comprises the following steps:
(1) collecting projection data of CT images measured by the detector in different angle directions to form a projection data set
Figure BDA0002532999630000061
Projection data set
Figure BDA0002532999630000062
Composed of projection data vectors acquired corresponding to all angle directions, and the projection vector dimension is NdetAnd each element value in the vector is the projection data measured by the corresponding detector, NdetIs the number of detectors, if the projection angle is NdegThen the total number of projection data is Ndeg×Ndet。
(2) Respectively establishing a CT image reconstruction model under the conditions of low dose and sparse view angle:
the CT image reconstruction model in the low dose case can be expressed as follows:
Figure BDA0002532999630000063
the CT image reconstruction model for sparse view can be expressed as follows:
Figure BDA0002532999630000064
wherein: vector quantity
Figure BDA0002532999630000071
The CT image reconstruction method is an image array which is expressed discretely, wherein each element value is an X-ray absorption coefficient at a corresponding pixel point in a CT image to be reconstructed; vector quantity
Figure BDA0002532999630000072
Is the corresponding projection data; a is a system matrix;
Figure BDA0002532999630000073
a penalty term for constraining the objective equation, β is a given weight coefficient.
(3) Preprocessing a CT image reconstruction equation under two modes to obtain a corresponding target function, and converting the reconstruction equation into a saddle point problem by utilizing a Lagrange dual:
for the CT image reconstruction model under the condition of low dose, an additional variable is firstly introduced
Figure BDA00025329996300000717
Rewrite equation (1) to the following form:
Figure BDA0002532999630000074
wherein:
Figure BDA0002532999630000075
can be understood as a pair
Figure BDA0002532999630000076
And performing forward projection calculation to obtain projection data.
Then introducing a nonsingular matrix PconeEquation (3) is further rewritten as follows:
Figure BDA0002532999630000077
corresponding Lagrange equation
Figure BDA0002532999630000078
The definition is as follows:
Figure BDA0002532999630000079
wherein:
Figure BDA00025329996300000710
for lagrange vectors, T denotes transpose.
Equation (4) can be translated into the following saddle point problem:
Figure BDA00025329996300000711
finally by relating to variables
Figure BDA00025329996300000712
Minimization calculations can be performed to eliminate it from the saddle point problem, resulting in the following objective function:
Figure BDA00025329996300000713
Figure BDA00025329996300000714
P＝F-1R(ω)F
Figure BDA00025329996300000715
wherein:
Figure BDA00025329996300000716
is a Lagrangian vector; t represents transposition; (-) is an inner product operation; gcA diagonal matrix, wherein the diagonal elements of the diagonal matrix are values of correction terms corresponding to each projection datum; gc 1/2Is GcRoot mean square matrix of (d) due to GcIs a diagonal matrix, so that the square of all diagonal elements can be obtainedc 1/2(ii) a F and F-1Respectively a one-dimensional Fourier transform operator and an inverse Fourier transform operator; omega is a frequency domain variable obtained by Fourier transform of projection data vectors acquired in the corresponding angle direction, m is the number of the angle directions, and tau is a given parameter; for circular cone-beam CT, the matrix PconeInstead of acting directly on all projection data for each angle, the projection data for each angle is processed individually for each detector row.
Diagonal matrix GcThe specific calculation of each diagonal element in (1) is as follows:
when the detector used in the circular cone-beam CT is a flat detector, the projection data is assumed to be Rβ(s, h) β is the angle between the coordinate axis and the line connecting the light source and the rotation center, s is the intersection position of the light and the detector in the horizontal direction, h is the intersection position of the light and the detector in the vertical direction, and the diagonal matrix G is formedcThe specific calculation mode of each diagonal element in the method is
Figure BDA0002532999630000081
D is the distance from the light source to the center of rotation.
When the circumferential cone beam CT makesWhen the detector used is a cylindrical detector, the projection data is assumed to be Rβ(gamma, h) is the included angle between the connecting line of the light source and the rotation center and the coordinate axis, gamma is the included angle between the light ray and the central light ray on the horizontal central plane of the cone beam, h is the intersection point position of the light ray and the detector in the vertical direction, and then the diagonal matrix G is formedcThe specific calculation mode of each diagonal element in the method is
Figure BDA0002532999630000082
For a CT image reconstruction model under the condition of sparse view angle, a nonsingular matrix P is introducedconeThen equation (2) can be rewritten as follows:
Figure BDA0002532999630000083
corresponding Lagrange equation
Figure BDA0002532999630000084
The definition is as follows:
Figure BDA0002532999630000085
equation (8) is converted to the saddle point problem, which corresponds to the objective function as follows:
Figure BDA0002532999630000086
Figure BDA0002532999630000087
P＝F-1R(ω)F
Figure BDA0002532999630000088
(4) the objective functions (7) and (10) are solved using an alternative projection approximation algorithm.
4.1 initialization
Figure BDA0002532999630000089
And sets the values of the parameters: total number of iterations NiterIteration termination threshold, penalty term weight coefficient β, parameter tau and parameter sigma, wherein the total iteration number NiterThe value range of (1) to (50000) and the value range of the iteration termination threshold value of (10)-10-1, the value range of the penalty item weight parameter β is 10-6-1, the parameter τ has a value range of 10-3～103The value range of the parameter sigma is 10-3～103。
4.2 initial iteration count k is 1;
4.3 calculation of
Figure BDA0002532999630000091
When k is 1:
Figure BDA0002532999630000092
when k is not 1:
low dose CT:
Figure BDA0002532999630000093
sparse view angle CT:
Figure BDA0002532999630000094
4.4 pairs of
Figure BDA0002532999630000095
Updating:
Figure BDA0002532999630000096
if the total variation is selected as the penalty item, the above formula can be solved by using a TV denoising algorithm of Chambolle, and iteration is needed for multiple times until an iteration termination condition is met.
4.5 pairs
Figure BDA0002532999630000097
Updating:
low dose CT:
Figure BDA0002532999630000098
sparse view angle CT:
Figure BDA0002532999630000099
4.6 judging the reconstructed image
Figure BDA00025329996300000910
And reconstructing the image
Figure BDA00025329996300000911
Is less than an iteration end threshold or if k is greater than Niter: if yes, executing step 4.7; if not, let k equal to k +1, and execute steps 4.3-4.5.
4.7 terminating the iteration to obtain the final reconstructed image
Figure BDA00025329996300000912
In the following, we verify the practicability and reliability of the method of the present invention by reconstructing the sinogram of the three-dimensional Shepp-Logan head model, the cross-sectional view of which is shown in fig. 2, and since the gray values of most of the structures are very close, the gray range [1.00, 1.05] is selected here to display the image for clear display of the structures.
The projection data (the adopted projection angle range is 360 degrees, the angle degree is 120) are reconstructed by adopting the method, the ART + TV method and the FISTA method, the reconstructed images obtained by the three methods in the reconstruction process are shown in figure 3, compared with the root mean square error of a truth diagram, the root mean square error changes along with the change of the iteration times, the three methods can be seen to gradually converge along with the increase of the iteration times, the reconstruction result root mean square error of the method is smaller, and the convergence speed is higher than that of the other two methods.
The reconstructed image at the maximum iteration number (i.e. 20 times) is selected and displayed, fig. 4 (a) and (b) are respectively vertical central axis cross-sectional views of the reconstructed result in two different directions, fig. 5 is a horizontal cross-section of the reconstructed result closer to the horizontal central axis, and fig. 6 is a horizontal cross-section of the reconstructed result farther from the horizontal central axis. In fig. 4 (a) and (b), fig. 5, and fig. 6, the reconstructed images are reconstructed by the method of the present invention, the ART + TV method, and the FISTA method, respectively, from top to bottom in rows, and the reconstructed results are obtained by the 5 th, 10 th, 15 th, and 20 th iterations, respectively, from left to right in columns; from the results, the method has good reconstruction effect, the reconstruction results of the horizontal cross section which is close to the horizontal central axis or the horizontal cross section which is far from the horizontal central axis are good, the boundary and the internal details of the image are recovered well, obvious artifacts do not exist, in addition, the convergence speed of the method is high, and better reconstruction results can be obtained with fewer iteration times.
The embodiments described above are presented to enable a person having ordinary skill in the art to make and use the invention. It will be readily apparent to those skilled in the art that various modifications to the above-described embodiments may be made, and the generic principles defined herein may be applied to other embodiments without the use of inventive faculty. Therefore, the present invention is not limited to the above embodiments, and those skilled in the art should make improvements and modifications to the present invention based on the disclosure of the present invention within the protection scope of the present invention.

Claims (9)

1. A fast iterative reconstruction algorithm of a circular cone beam CT based on an FDK type preprocessing matrix comprises the following steps:
(1) acquiring projection data of CT images in different angle directions by using a circumferential cone beam CT system to form a projection data set
Figure FDA0002532999620000011
If projection data of m angular directions are acquired during the measurement, a projection data set is formed
Figure FDA0002532999620000012
Is m × N, N being the number of detector units in a circumferential cone-beam CT system, the projection data set
Figure FDA0002532999620000013
The projection data vector is formed by projection data vectors acquired under the corresponding m angle directions, the dimensionality of the projection data vector is N, and each element value in the vector is projection data measured by a corresponding detector unit;
(2) respectively establishing CT image reconstruction models under the conditions of low dose and sparse view angle;
(3) preprocessing the CT image reconstruction equation under the two conditions, and converting the preprocessed CT image reconstruction equation into a saddle point problem by using a Lagrangian dual method to obtain a corresponding target function;
(4) and selecting a corresponding objective function according to the actual situation, and carrying out optimization solution on the objective function to reconstruct to obtain the CT image.
2. The circular cone beam CT fast iterative reconstruction algorithm according to claim 1, characterized in that: the CT image reconstruction model under the condition of low dose in the step (2) is expressed as follows:
Figure FDA0002532999620000014
the CT image reconstruction model under sparse view angle condition is expressed as follows:
Figure FDA0002532999620000015
wherein:
Figure FDA0002532999620000016
is a CT image data set and
Figure FDA0002532999620000017
wherein each element value is the X-ray absorption coefficient of the corresponding pixel point in the CT image to be reconstructed, J is the number of pixel points of the CT image to be reconstructed, T represents transposition, A is a system matrix,
Figure FDA0002532999620000018
for the penalty term used to constrain the objective function, β is given a weighting factor, | | | | represents a 2-norm.
3. The circular cone beam CT fast iterative reconstruction algorithm according to claim 2, characterized in that: for the CT image reconstruction equation under the condition of low dose in the step (3), firstly, an additional variable is introduced
Figure FDA0002532999620000019
Rewriting formula (1) to the following form:
Figure FDA0002532999620000021
wherein: variables of
Figure FDA0002532999620000022
Expressed as a vector of
Figure FDA0002532999620000023
Projection data obtained by performing orthographic projection calculation;
then, introduce the non-singular matrix PcomeFormula (3) is further rewritten as follows:
Figure FDA0002532999620000024
lagrange equation corresponding to equation (4)
Figure FDA0002532999620000025
The definition is as follows:
Figure FDA0002532999620000026
wherein:
Figure FDA0002532999620000027
is a Lagrangian vector;
further, equation (4) is converted into the following saddle point problem:
Figure FDA0002532999620000028
finally, through the pair of variables
Figure FDA0002532999620000029
Minimization calculations can be performed to eliminate it from the saddle point problem, resulting in the following objective function:
Figure FDA00025329996200000210
wherein:
Figure FDA00025329996200000211
is composed of
Figure FDA00025329996200000212
And
Figure FDA00025329996200000213
the inner product operation result of (2).
4. The circular cone beam CT fast iterative reconstruction algorithm according to claim 2, characterized in that: for the CT image reconstruction equation under the condition of sparse view angle in the step (3), firstly, a non-singular matrix P is introducedcomeThe formula (2) is rewritten as follows:
Figure FDA00025329996200000214
lagrange equation corresponding to equation (8)
Figure FDA00025329996200000215
The definition is as follows:
Figure FDA00025329996200000216
further, equation (9) is converted into the saddle point problem, and the corresponding objective function is as follows:
Figure FDA00025329996200000217
wherein:
Figure FDA0002532999620000031
is the lagrange vector.
5. The circumferential cone beam CT fast iterative reconstruction algorithm according to claim 3 or 4, characterized in that: the nonsingular matrix PcomeThe expression of (a) is as follows:
Figure FDA0002532999620000032
P=F-1R(ω)F
wherein: at low dose
Figure FDA0002532999620000033
In sparse view conditions
Figure FDA0002532999620000034
GcIs a diagonal matrix with dimension n, n is the number of detector units in each horizontal row in the circumferential cone-beam CT system, GcEach diagonal element in (a) is a correction term value, F and F, of the projection data measured by each detector unit in the corresponding line in the corresponding angular direction-1Respectively, a one-dimensional Fourier transform operator andand omega is a frequency domain variable obtained by Fourier transformation of the projection data vector acquired in the corresponding angle direction, and tau is a given parameter.
6. The circular cone beam CT fast iterative reconstruction algorithm according to claim 5, characterized in that: if the circular cone-beam CT system uses a planar detector, the diagonal matrix GcThe diagonal element in (A) is
Figure FDA0002532999620000035
Figure FDA0002532999620000036
If the circular cone beam CT system adopts a cylindrical surface detector, the diagonal matrix GcThe diagonal element in (A) is
Figure FDA0002532999620000037
Wherein: s is the intersection point position value of the light and the corresponding detector unit in the horizontal direction, h is the intersection point position value of the light and the corresponding detector unit in the vertical direction, D is the distance from the light source to the rotation center, and gamma is the included angle between the light and the central light on the horizontal central plane of the cone beam.
7. The circular cone beam CT fast iterative reconstruction algorithm according to claim 1, characterized in that: and (4) carrying out optimization solution on the objective function by adopting an alternative projection approximation algorithm.
8. The circular cone beam CT fast iterative reconstruction algorithm according to claim 2, characterized in that: the penalty term
Figure FDA0002532999620000038
And adopting a total variation operator.
9. The circumferential cone beam CT fast iterative reconstruction algorithm according to claim 3 or 4, characterized in that: for a circular cone beam CT systemSystem, said non-singular matrix PcomeInstead of acting directly on all projection data, the projection data measured by each row of detector units for each angle is processed separately.
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