CN114187175A - Emission tomography weight matrix determination method for three-dimensional space free projection - Google Patents
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Abstract
The invention relates to an emission tomography weight matrix determination method for three-dimensional space free projection, which overcomes the problems of imperfect model, low efficiency and low precision in the prior art, and has the advantages of simple method and high calculation precision and efficiency. The invention comprises the following steps: establishing a three-dimensional space free projection emission tomography model considering a lens imaging effect and a camera three-dimensional space position; step two: discretizing the projection integral; step three: and determining a weight matrix of the system based on a bilinear interpolation principle.
Description
The technical field is as follows:
the invention belongs to the technical field of optical computational imaging and combustion field three-dimensional measurement, relates to a combustion field emission spectrum chromatography reconstruction method, and particularly relates to an emission chromatography weight matrix determination method for three-dimensional space free projection.
Background art:
imaging, measurement and diagnosis of combustion fields are fundamental to the design of key instruments and meters in the modern aerospace, missile, weapon and energy industries. Among them, three-dimensional imaging and display of the combustion process are one of the important means essential for the research and control of combustion. Emission Tomography (ECT) is a simple, efficient, non-contact combustion field three-dimensional imaging method. The method utilizes an area array CCD camera or an optical fiber array to collect two-dimensional projection images of the emission spectrum intensity of a combustion field in multiple directions, and then combines a Computed Tomography (CT) theory to carry out three-dimensional reconstruction of the combustion field. Due to the fact that the measuring principle and the measuring device are simple, the method becomes a research hotspot of combustion imaging detection technology in recent years, and has wide application prospect.
However, the existing emission tomography technology is based on two-dimensional Radon transform as mathematics, and the implementation process of the emission tomography system using a CCD camera for projection acquisition is limited by two aspects: (1) the Radon transform adopts a parallel projection model, which is no longer true due to the imaging effect of the lens of the CCD camera; (2) the existing parallel projection emission tomography has strict requirements on the position of a camera, an optical window at an ideal position cannot be provided generally due to the limitation of an actual test environment, and a position error inevitably exists during the assembly of a detector. These two constraints severely affect the accuracy of the tomographic reconstruction, and therefore two factors must be considered when modeling the emission tomography system: (1) imaging effect of the lens; (2) the three-dimensional spatial position of the camera.
Among tomographic reconstruction algorithms, algebraic iterative algorithms are the most commonly used reconstruction methods in tomographic techniques. The method comprises the steps of carrying out discretization representation on a chromatographic projection process, representing the contribution of object functions of different discrete grids in a measured object to projection information by using a weight matrix, converting the projection process into a series of linear equation sets, and solving the linear equation sets by using different forms of algebraic iteration methods in a reconstruction process. The calculation accuracy of the weight matrix seriously affects the accuracy and quality of the chromatographic reconstruction. Different projection models determine different weight matrixes, so that the method has important significance for researching a weight matrix determination method considering the two factors. Although a few studies have been made on the problem by using ray tracing and lens imaging theories, the calculation efficiency and the accuracy are low.
The invention content is as follows:
the invention aims to provide an emission tomography weight matrix determination method for three-dimensional space free projection, which overcomes the problems of imperfect model and low efficiency and precision in the prior art, and has the advantages of simple method and high calculation precision and efficiency.
In order to achieve the purpose, the invention adopts the technical scheme that:
an emission tomography weight matrix determination method for three-dimensional space free projection is characterized in that: the method comprises the following steps:
establishing a three-dimensional space free projection emission tomography model considering a lens imaging effect and a camera three-dimensional space position;
step two: discretizing the projection integral;
step three: and determining a weight matrix of the system based on a bilinear interpolation principle.
The first step comprises the following steps:
s1: the lens imaging effect is expressed by that the object point coordinates (x, y, z) in the camera coordinate system and the image point coordinates (x ', y') in the camera imaging plane coordinate system satisfy the relation
Wherein f is the focal length of the camera lens;
s2: the arbitrary three-dimensional spatial position of the camera is represented as a world coordinate system (x)w,yw,zw) Satisfies a relation with a camera coordinate system (x, y, z)
Wherein R is a rotation matrix and T is a translation vector;
s3: the obtained chromatographic projection model is
Wherein f (x, y, z) is the luminous intensity value of the object point in the camera coordinate system, zminAnd zmaxThe upper and lower boundaries of each projection ray in the camera coordinate system intersecting the reconstruction region.
The second step comprises the following steps:
s1: subjecting a three-dimensional object f (x)w,yw,zw) Uniformly dividing the grid into M multiplied by N multiplied by P discrete grids, wherein the grid size is delta g multiplied by delta g, and assuming the value f of each gridi(xw,yw,zw) Is a constant, where i ═ 1,2, … …, mxnxp;
s2: determining the upper limit z of the projection integral according to the camera coordinate values of 8 vertexes of the reconstruction regionmaxAnd a lower limit zmin；
S3: the integral interval is uniformly divided into Q infinitesimal, the objective function of each infinitesimal interval is assumed to be a constant, and the projection model is
Wherein (x ', y') and (x)m,ym,zm) Satisfy the relationship
Determining the position (x) of the integral infinitesimal in the world coordinate system according to the conversion relation between the world coordinate system and the camera coordinate systemmw,ymw,zmw)。
The third step comprises the following steps:
s1: determining the minimum adjacent discrete grid serial number of each infinitesimal;
s2: determining the distance between the micro element and the center of the minimum adjacent discrete grid;
s3: according to the bilinear interpolation principle, determining the physical function value of the infinitesimal by the physical function of the center point of 8 adjacent discrete grids to obtain the weight factor of each discrete grid to the infinitesimal;
s4: and performing integral operation on all the infinitesimals to obtain a weight matrix of each discrete grid for projection.
Compared with the prior art, the invention has the following advantages and effects:
1. the method obtains the tomography model considering the lens imaging effect and the three-dimensional space position of the camera, and provides a theoretical basis for the emission tomography reconstruction of the three-dimensional space free projection; the weight matrix is determined based on the bilinear interpolation principle on the basis of the model, the principle is simple, and the calculation precision and efficiency are high.
2. The invention can enlarge the projection acquisition range of the emission tomography system and improve the reconstruction precision, and has important significance for the practicability of the emission tomography technology.
Description of the drawings:
FIG. 1 is a schematic diagram of a multi-camera emission tomography system coordinate system definition;
FIG. 2 is a schematic view of a camera projection imaging model;
FIG. 3 is a projection imaging result of a cube at different rotation angles;
FIG. 4 shows a hollow sphere and its reconstruction;
FIG. 5 is a synthetic Shepp-Logan model and its reconstructed results.
The specific implementation mode is as follows:
in order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
The invention relates to an emission tomography weight matrix determination method for three-dimensional space free projection, which comprises the steps of firstly, establishing a three-dimensional space free projection emission tomography model on the basis of considering a lens imaging effect and a camera three-dimensional space position; then, discretizing the projection integral on the basis of the projection model; and finally, determining a weight matrix of the chromatography system by utilizing a bilinear interpolation principle.
The invention specifically comprises the following steps:
step one, establishing a three-dimensional space free projection emission tomography model considering lens imaging effect and camera three-dimensional space position
S1: the lens imaging effect is expressed by that the object point coordinates (x, y, z) in the camera coordinate system and the image point coordinates (x ', y') in the camera imaging plane coordinate system satisfy the relation
Where f is the focal length of the camera lens.
S2: the arbitrary three-dimensional spatial position of the camera is represented as a world coordinate system (x)w,yw,zw) Satisfies a relation with a camera coordinate system (x, y, z)
Where R is the rotation matrix and T is the translation vector.
S3: the obtained chromatographic projection model is
Wherein f (x, y, z) is the luminous intensity value of the object point in the camera coordinate system, zminAnd zmaxThe upper and lower boundaries of each projection ray in the camera coordinate system intersecting the reconstruction region.
Step two: discretizing projection integral:
s1: subjecting a three-dimensional object f (x)w,yw,zw) Uniformly dividing the grid into M multiplied by N multiplied by P discrete grids, wherein the grid size is delta g multiplied by delta g, and each grid is assumed to beValue fi(xw,yw,zw) Is a constant where i ═ 1,2, … …, M × N × P.
S2: determining the upper limit z of the projection integral according to the camera coordinate values of 8 vertexes of the reconstruction regionmaxAnd a lower limit zmin；
S3: the integral interval is uniformly divided into Q infinitesimal, the objective function of each infinitesimal interval is assumed to be a constant, and the projection model is
Wherein (x ', y') and (x)m,ym,zm) Satisfy the relationship
Determining the position (x) of the integral infinitesimal in the world coordinate system according to the conversion relation between the world coordinate system and the camera coordinate systemmw,ymw,zmw)。
Step three: determining a weight matrix of a system based on a bilinear interpolation principle
S1: determining the minimum adjacent discrete grid serial number of each infinitesimal;
s2: determining the distance between the micro element and the center of the minimum adjacent discrete grid;
s3: according to the bilinear interpolation principle, determining the physical function value of the infinitesimal by the physical function of the center point of 8 adjacent discrete grids to obtain the weight factor of each discrete grid to the infinitesimal;
s4: and performing integral operation on all the infinitesimals to obtain a weight matrix of each discrete grid for projection.
Example (b):
the invention comprises the following steps:
the method for obtaining the three-dimensional space free projection emission tomography model specifically comprises the following steps:
see FIG. 1 for a definition of a coordinate system of a transmission tomography system with multiple cameras, the system having a fixed world coordinate system (x)w,yw,zw) The luminous intensity of the measured combustion field is denoted as f (x)w,yw,zw). The camera at any position in the world coordinate system is represented by a camera coordinate system (x, y, z), wherein the plane (x, y) is parallel to the CCD target surface, the x axis is the long side direction of the CCD target surface, the y axis is the short side direction of the CCD target surface, and the origin of coordinates is the intersection point of the optical axis of the camera and the lens. And a coordinate system (x ', y') of each camera imaging surface is a CCD target surface, an x 'axis is an opposite direction of an x axis, a y' axis is an opposite direction of a y axis, and a coordinate origin is an intersection point of a lens optical axis and the CCD target surface.
In three-dimensional space, from the world coordinate system (x)w,yw,zw) To any camera coordinate system (x, y, z) can be determined by Euler angles (nutation angle psi, precession angle theta and rotation angle phi) and three translation amounts (Tx,Ty,Tz) Determining, the conversion relationship can be expressed as
Wherein the rotation matrix
Translation vector
For simplicity, formula (1) is represented as
In practical emission tomography systems, the parameter r in the rotation matrix corresponding to each camera1,r2,…,r9And the parameter T in the translation vectorx,Ty,TzCan be accurately determined by a camera calibration method.
Because the imaging lens with proper depth of field can be selected according to the size of the combustion field area to be measured, the camera imaging model does not need to consider the influence caused by the depth of field, and therefore, the model can be equivalent to a pinhole camera model, as shown in fig. 2.
The straight line formed by any image point P' on the imaging plane and the origin O of the camera coordinate system is called a projection ray. The projection light intersects with the detected region, and all object points Pm(m ═ 1,2, … …, n) will be imaged onto the same image point P'. Assuming that the focal length of the camera lens is f, the positions of the object point and the image point satisfy the triangle similarity principle, that is:
the image point acquires the luminous intensity information of
Wherein z isminAnd zmaxF (x, y, z) is the luminous intensity value of all object points on the projection ray for the upper and lower boundaries where each projection ray intersects the reconstruction region.
Discretizing the projection integral specifically comprises:
subjecting a three-dimensional object f (x)w,yw,zw) Uniformly dividing the grid into M multiplied by N multiplied by P discrete grids, wherein the grid size is delta g multiplied by delta g, and assuming the value f of each gridi(xw,yw,zw) Is a constant where i ═ 1,2, … …, M × N × P. If there are X cameras in the tomography system and the number of pixels of each camera is Y, the tomography system has XY projection rays in total. If the intersecting intercept of the jth projection ray and the ith grid is recorded as omegaijObviously, there are only a few i corresponding to ωijIs non-zero. Thus, the pixel intensity I corresponding to the jth projection rayjCan be discretely expressed as:
wherein ω isijThe projection weight factor of the ith grid to the jth ray is referred to as the contribution of the ith grid to the jth projection. All omegaijA weight matrix of the chromatography system is constructed.
The world coordinate system (x) of 8 vertexes of the reconstruction regionwc,ywc,zwc)pWhere p is 1,2, …,8, and the coordinate values of these 8 points are substituted into equation (1), the positions (x) of the 8 vertices in the camera coordinate system can be obtainedc,yc,zc)p. Comparing 8 zcCan obtain an integration interval [ z ]min,zmax]. Discretizing the camera imaging projection model expressed by the formula (4), and integrating the interval [ z ]min,zmax]Evenly divided into Q infinitesimal intervals, i.e. the step number of integration is Q, and the step length is
The objective function of each infinitesimal interval is assumed to be a constant, so
Wherein (x ', y') and (x)m,ym,zm) Satisfy the relationship
The position of the integral infinitesimal in the world coordinate system can be determined according to the formula (1):
thus, it is possible to obtain
Thirdly, calculating a weight matrix of the chromatography system by utilizing a bilinear interpolation principle, which specifically comprises the following steps:
according to the bilinear interpolation principle, the physical function value at any position in the discrete three-dimensional space can be determined by the physical function value of the central point of the adjacent discrete grid. The specific process is as follows:
(1) determining the minimum adjacent discrete grid serial number of each micro element
(2) Determining the distance between the infinitesimal and the center of the minimum adjacent discrete grid as Deltaxm、ΔymAnd Δ zm。
(3) Determining the physical function value of the infinitesimal by the physical functions of the adjacent 8 discrete grid center points
According to the coefficient determined by the distance between the adjacent discrete grids, the weight factor of each discrete grid to the micro element can be obtained.
(4) The projected pixel value for each ray can be written in the form
And performing integral operation on all the infinitesimal elements according to the formula to obtain a weight matrix of each discrete grid pair projection.
Experimental example:
in order to visually reflect the accuracy of the three-dimensional space camera imaging model and the bilinear difference value-based weight matrix calculation method, firstly, the imaging projection of a cube in a three-dimensional space is calculated. The cube is divided into 50 x 50 discrete grids of 0.4mm size, each grid having a value of constant 1. The analog camera has a pixel count of 138 × 138, a pixel size of 3.75 μm, and a lens focal length of 12 mm. When the translation vector is [0mm,0mm,800mm ], the projections at different rotation angles calculated according to the method proposed by the invention are shown in fig. 3. The calculated projection result is completely matched with the theoretical visual projection result, and the correctness of the method is proved.
Using 2 three-dimensional simulation fields F1And F2So as to verify the effectiveness and the accuracy of the three-dimensional space free projection chromatography reconstruction algorithm. Wherein F1The spherical three-dimensional hollow sphere can reflect the integrity of a reconstruction result; f2Is a commonly used synthetic Shepp-Logan model in chromatographic techniques. Both fields are divided into a 50 x 50 square discrete grid with a grid size of 0.4 mm. Assume that there are 12 cameras in the tomography system to acquire projections in different directions, the number of pixels of the camera is 138 × 138, the pixel size is 3.75 μm, and the focal length of the lens is 12 mm. The translation vectors are all [0mm,0mm,800mm ]]The precession angle theta and the rotation angle phi are both 5 DEG, and the nutation angle phi is [ -90 DEG, 90 DEG °]The inner parts are uniformly distributed.
Firstly, the projection imaging of different cameras is calculated by using the method provided by the invention, and then the chromatography reconstruction is carried out by using an ART algorithm. To the hollow round ball F1The reconstruction result of (2) is shown in fig. 4. From the structural point of view, the sphere structure in the F1 reconstruction result is very complete and is in perfect fit with the original model structure, which indicates that the integrity of the reconstruction result is very high; the root mean square error of the reconstruction result is 2.2083 × 10-4And the reconstruction precision is very high. The reconstruction result of the synthesized Shepp-Logan model is shown in FIG. 5, and the root mean square error of the reconstruction result is 7.2078 × 10-5And has very high reconstruction precision.
The 3 simulation results in the experimental example verify the correctness of the camera imaging projection mathematical model in the three-dimensional space and the accuracy of the weight matrix calculation method.
The above description is only for the preferred embodiment of the present invention, and is not intended to limit the scope of the present invention, and all equivalent structural changes made by using the contents of the specification and the drawings of the present invention should be included in the scope of the present invention.
Claims (4)
1. An emission tomography weight matrix determination method for three-dimensional space free projection is characterized in that: the method comprises the following steps:
establishing a three-dimensional space free projection emission tomography model considering a lens imaging effect and a camera three-dimensional space position;
step two: discretizing the projection integral;
step three: and determining a weight matrix of the system based on a bilinear interpolation principle.
2. The method for determining the emission tomography weight matrix for the free projection of the three-dimensional space according to claim 1, wherein: the first step comprises the following steps:
s1: the lens imaging effect is expressed by that the object point coordinates (x, y, z) in the camera coordinate system and the image point coordinates (x ', y') in the camera imaging plane coordinate system satisfy the relation
Wherein f is the focal length of the camera lens;
s2: the arbitrary three-dimensional spatial position of the camera is represented as a world coordinate system (x)w,yw,zw) Satisfies a relation with a camera coordinate system (x, y, z)
Wherein R is a rotation matrix and T is a translation vector;
s3: the obtained chromatographic projection model is
Wherein f (x, y, z) is the luminous intensity value of the object point in the camera coordinate system, zminAnd zmaxThe upper and lower boundaries of each projection ray in the camera coordinate system intersecting the reconstruction region.
3. The method for determining the emission tomography weight matrix for the free projection of the three-dimensional space according to claim 1, wherein: the second step comprises the following steps:
s1: subjecting a three-dimensional object f (x)w,yw,zw) Uniformly dividing the grid into M multiplied by N multiplied by P discrete grids, wherein the grid size is delta g multiplied by delta g, and assuming the value f of each gridi(xw,yw,zw) Is a constant, where i ═ 1,2, … …, mxnxp;
s2: determining the upper limit z of the projection integral according to the camera coordinate values of 8 vertexes of the reconstruction regionmaxAnd a lower limit zmin；
S3: the integral interval is uniformly divided into Q infinitesimal, the objective function of each infinitesimal interval is assumed to be a constant, and the projection model is
Wherein (x ', y') and (x)m,ym,zm) Satisfy the relationship
Determining the position (x) of the integral infinitesimal in the world coordinate system according to the conversion relation between the world coordinate system and the camera coordinate systemmw,ymw,zmw)。
4. The method for determining the emission tomography weight matrix for the free projection of the three-dimensional space according to claim 1, wherein: the third step comprises the following steps:
s1: determining the minimum adjacent discrete grid serial number of each infinitesimal;
s2: determining the distance between the micro element and the center of the minimum adjacent discrete grid;
s3: according to the bilinear interpolation principle, determining the physical function value of the infinitesimal by the physical function of the center point of 8 adjacent discrete grids to obtain the weight factor of each discrete grid to the infinitesimal;
s4: and performing integral operation on all the infinitesimals to obtain a weight matrix of each discrete grid for projection.
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