Method for calculating optimal tilt angle of object for x-ray cl three-dimensional reconstruction and application thereof

Abstract

The application discloses a kind of object optimal tilt angle calculation method and application of X-ray CL three-dimensional reconstruction, and the method comprises:1, the geometric model of three-dimensional reconstruction algorithm of X-ray CL scanning mode is established,2, the geometric parameter relationship of CL scanning mode is determined, the projection address expression is solved,3, projection address relative error formula is calculated according to projection address expression,4, the influence of object tilt angle on projection address relative error is calculated, and gradient descent method is used, the relationship between object thickness and optimal tilt angle is solved, and polynomial fitting method is used, and object thickness and optimal tilt angle relationship formula are fitted.The object tilt angle of the application is optimized in the process of CL three-dimensional reconstruction, can improve the accuracy of CL three-dimensional reconstruction algorithm, and reduce the interlaced phenomenon after reconstruction.

Description

Technical Field

[0001] This invention is applied to the field of 3D reconstruction of flat industrial objects (CL), specifically a method for calculating the optimal tilt angle of an object in the CL 3D reconstruction process and its application. Background Technology

[0002] X-ray linear projection (CL) scanning is widely used in 3D reconstruction of flat objects. CL scanning methods include linear and rotational types. Commonly used 3D reconstruction algorithms are iterative and analytical. An analytical algorithm, such as the filtered back-projection algorithm, projects the detector's data along the original ray path onto the object's coordinate system. An iterative algorithm, such as the ART algorithm, solves for the object's voxel values ​​by establishing a system of equations based on detector data and the projection matrix. Regardless of the algorithm used, projection address errors in CL scanning lead to aliasing in the reconstructed object. Existing research shows that different tilt angles of the object under CL scanning result in varying degrees of aliasing. This aliasing is caused by two factors: firstly, errors in the actual tilt angle due to manufacturing precision and the tilt angle used for 3D reconstruction, which can be corrected using various methods; and secondly, projection address errors during calculation due to the tilt angle itself cause aliasing in the reconstruction results. While theoretical simulations have shown that the tilt angle affects the degree of aliasing in the reconstruction results, methods to reduce the degree of aliasing based on the underlying principles have not yet been developed.

[0003] In the CL 3D reconstruction system, the size of the object tilt angle affects the accuracy of the reconstruction result by influencing the projection address error. Different object thicknesses, i.e., the number of object layers, will result in the highest reconstruction accuracy at a specific tilt angle. Therefore, a method for defining the optimal tilt angle of an object that depends on the number of object layers is needed. Summary of the Invention

[0004] The present invention aims to address the shortcomings of the prior art by providing a method and application for calculating the optimal tilt angle of an object in X-ray CL three-dimensional reconstruction. The goal is to improve the accuracy of the CL three-dimensional reconstruction algorithm and reduce the interlayer aliasing phenomenon after reconstruction by optimizing the tilt angle of the object during the CL three-dimensional reconstruction process.

[0005] To achieve the above-mentioned objectives, the present invention adopts the following technical solution:

[0006] The present invention provides a method for calculating the optimal tilt angle of an object in X-ray CL three-dimensional reconstruction, characterized in that it is applied to the industrial CL scanning three-dimensional reconstruction process of flat objects, and is performed according to the following steps:

[0007] Step 1: Establish a projection coordinate system o with the center of the detector as the origin O, the vertical direction of the detector plane as the X-axis, the horizontal direction as the Y-axis, and the direction perpendicular to the projection plane XOY as the Z-axis; let the center O′ of the flat object and the X-ray source point S both lie on the Z-axis O′ of the projection coordinate system o, and place the flat object at an angle of inclination. Let O′ be the angle between the plane normal on the flat object and the Z-axis of the projected coordinate system o. Establish the object coordinate system o′ with O′ as the origin, the vertical axis of the upper plane of the flat object as the X′ axis, the horizontal axis as the Y′ axis, and the direction perpendicular to the plane X′O′Y′ as the rotation axis Z′. Let D be the distance between the X-ray source point S and the detector, and z′ be the distance from the detector to the flat object. O′ Thus, a three-dimensional reconstruction model using X-ray CL scanning was established;

[0008] Step 2: Determine the geometric parameter relationships of the X-ray CL scanning mode:

[0009] Step 2.1: According to the coordinate system transformation rules, determine the three coordinate transformation matrices R1, R2, R3 between the object coordinate system o′ and the projected coordinate system o; thus, use equation (1) to obtain the coordinates (x, y, z) of the origin coordinates (x′, y′, z′) of o′ in the object coordinate system to the coordinates (x, y, z) in the projected coordinate system o:

[0010]

[0011] In equation (1), θ is the angle of object rotation in CL scanning mode, (x′ O′ ,y′ O′ ,z′ O′ () represents the coordinates of the object's center O′ in the projected coordinate system o;

[0012] Step 2.2: Given the coordinates (x, y) of the ray source in the projected coordinate system o. oS ,y oS ,z oS ), and based on the X-ray passing through the i-th point to be reconstructed (x i ,y i ,z i Establish the linear equation for X-rays;

[0013] Step 2.3: The intersection of the X-ray and the projection plane XOY is the projection address corresponding to the point to be reconstructed. In the projection coordinate system o, set the Z-axis coordinate of the XOY plane to zero and substitute it into the linear equation of the X-ray. Thus, using equation (2), we can obtain the projection address (x) of the X-ray from the X-ray source to the detector after passing through the object in the CL scanning mode. proj ,y proj ):

[0014]

[0015] Step 2.4: Translate the object coordinate system o′ to the projected coordinate system o, thereby utilizing z′ O′ =0, z oS =D yields the simplified projection address formula as shown in equation (3):

[0016]

[0017] Step 3: Calculate the tilt angle The relative error of the projected address under the influence of:

[0018] Step 3.1: Use equation (3) to... Take the derivative to obtain the formula.

[0019] Step 3.2: Use equation (4) to obtain the tilt angle. When there is an error, the projected address (x) in the x-axis and y-axis directions proj ,y proj The formula for the relative error of ) is:

[0020]

[0021] Step 4: Calculate the optimal tilt angle for object reconstruction when the Z′ axis coordinate in the X′O′Y′ plane is zero.

[0022] Step 4.1: Compare the projected addresses (x, y) of the center and edge regions of the object along the x and y axes according to equation (4). proj ,y proj Due to the error in the tilt angle, the center of the object or its edge region that has a greater impact is selected to determine some parameter values ​​in the optimization objective function;

[0023] Step 4.2: Solve for each rotation angle θ of the object along the x-axis and y-axis in the 360-degree projection. i The projection address error value is obtained, and thus the projection address (x) in the x-axis and y-axis directions under 360-degree projection is obtained using equation (5). proj ,y proj The absolute value of the error is the average value, and then the objective function f is established using equation (6):

[0024]

[0025]

[0026] Step 4.3: Select the center or edge region of the object that is significantly affected by the tilt angle, and set the object coordinates (x′, y′, z′), z′ O′Substituting 0 and distance D together into equation (5), we can find the tilt angle that minimizes the objective function f, which is the optimal tilt angle for object reconstruction.

[0027] Step 5: Establish the optimal tilt angle that depends on the object's thickness Computational model:

[0028] Step 5.1: Using z′ as the thickness of the object, and setting the range of values ​​for the object thickness, and following the process in Step 4, obtain the optimal tilt angle corresponding to z′ for different thicknesses within the range of values ​​for the object thickness.

[0029] Step 5.2: Find the optimal tilt angle corresponding to z′ Perform function fitting to obtain z′ and The calculation model is used to obtain the optimal tilt angle for flat objects of different thicknesses.

[0030] The present invention provides an electronic device, including a memory and a processor, wherein the memory is used to store a program that supports the processor in executing the method for calculating the optimal tilt angle of an object, and the processor is configured to execute the program stored in the memory.

[0031] The present invention discloses a computer-readable storage medium on which a computer program is stored, wherein the computer program, when executed by a processor, performs the steps of the method for calculating the optimal tilt angle of an object.

[0032] Compared with existing technologies, the advantages of this invention are:

[0033] 1. This invention provides a formula for calculating the relative error of the projection address under the geometric model of the CL scanning method, and based on the relative error formula, the relationship expression between the relative error and the tilt angle of the object is obtained. This formula can be applied to the geometric correction and error analysis theory of the CL scanning system, thus providing a theoretical basis for reducing the interlayer aliasing phenomenon in the three-dimensional reconstruction of flat objects by reducing the projection address error.

[0034] 2. Existing research indicates that different tilt angles lead to varying degrees of aliasing in CL reconstruction results. This invention, based on the error calculation formula for the tilt angle of the projection address relative to the object, sets an error objective function. By changing the object's tilt angle and thickness, and aiming to minimize the error objective function, the optimal tilt angle for each object thickness is determined. Furthermore, the object thickness is fitted as a function to the optimal tilt angle, providing a reasonable method for selecting the object tilt angle during CL scanning in practical engineering, thereby improving reconstruction accuracy.

[0035] 3. The optimization method used in this invention is also applicable to other geometric parameter values ​​of the CL scanning system, providing an optimization method for studying higher-precision CL three-dimensional reconstruction systems. Attached Figure Description

[0036] Figure 1 This is a geometric model diagram of the CL scanning 3D reconstruction system used in this invention;

[0037] Figure 2 A graph showing the relative error between the edge region and the center region of an object at a specific tilt angle of the object;

[0038] Figure 3 A graph showing the relationship between the relative error of the object in the central layer and the tilt angle;

[0039] Figure 4 The image shows the fitting result of the object's thickness and the optimal tilt angle. Detailed Implementation

[0040] In this embodiment, to address the issue of achieving higher reconstruction accuracy through optimal object tilt angle in CL scanning, a method for calculating the optimal object tilt angle in an X-ray CL 3D reconstruction algorithm is proposed, comprising the following steps:

[0041] Step 1: The X-ray CL 3D reconstruction scanning method for flat objects differs from the traditional CT scanning method. It requires that the object's rotation axis be tilted at an angle relative to the line connecting the X-ray source and the detector center. This is used to obtain information about the object's depth direction, because the geometric model is established as follows: A projection coordinate system is established with the detector's center as the origin O, the detector's vertical direction as the X-axis, its horizontal direction as the Y-axis, and the direction perpendicular to the projection plane XOY as the Z-axis. The center O′ of the flat object and the X-ray source point S are both located on the Z-axis O′ of the projection coordinate system. The flat object is placed at an angle of inclination. Let O′ be the angle between the normal to the upper plane of the flat object and the Z-axis of the projected coordinate system; establish the object coordinate system o′ with O′ as the origin, the vertical axis of the upper plane of the flat object as the X′ axis, the horizontal axis as the Y′ axis, and the direction perpendicular to the plane X′O′Y′ as the rotation axis Z′; let D be the distance between the ray source and the detector, and z′ be the distance from the ray source to the object. O′ This allows for the establishment of a three-dimensional reconstruction model using X-ray CL scanning. The established CL scanning reconstruction system is as follows: Figure 1 As shown.

[0042] Step 2: Determine the geometric parameter relationships of the X-ray CL scanning mode.

[0043] Step 2.1: According to the coordinate system transformation rules, use equation (1) to determine the coordinate transformation matrices R1, R2, R3 between the rotating coordinate system and the projected coordinate system, where R1 is the transformation matrix for rotating the object around the Z′ axis by θ, and R2 is the transformation matrix for the Z′ axis of the rotating coordinate system and the Z axis of the projected coordinate system. The transformation matrix of the included angle, R3 is the translation transformation matrix from the origin O′ of the rotating coordinate system to the origin O of the projected coordinate system;

[0044]

[0045] Based on the transformation matrix in equation (1), the relationship between the coordinates (x′, y′, z′) in the rotated coordinate system and the coordinates (x, y, z) in the projected coordinate system is obtained as shown in equation (2):

[0046]

[0047] Step 2.2: Establish the linear equation of the X-ray source, given the coordinates of the X-ray source (x... oS ,y oS ,z oS ), the ray passes through the point to be reconstructed (x i ,y i ,z i Find the equation of the ray as a straight line;

[0048]

[0049] Since the coordinates z = 0 of the point to be reconstructed reaching the detector plane, then:

[0050]

[0051] Step 2.3: Substitute the above equation into equation (3) to find the values ​​of x and y when z = 0. These values ​​are the projection addresses. Figure 1 The geometric relationship in, where x oS =y oS =x′ O′ =y′ O′ =0, combined with equation (2), we can obtain the projection address expression (5). In actual implementation, since the object's axis of rotation is tilted, it is difficult to measure the distance from the ray source to the center of the object. In order to avoid absolute measurement, the object coordinate system is translated to the projection coordinate system, i.e., z′ O′ =0, z oS =D, and after simplification, we get equation (6), which represents the projection address formula of the X-ray beam from the X-ray source, after passing through the object and attenuating, to the detector in CL scanning mode.

[0052]

[0053]

[0054] Step 3: Formula for calculating the relative error of the projected address under the influence of tilt angle.

[0055] Step 3.1: Simplified projection address formula The derivative is given by:

[0056]

[0057] Step 3.2: Divide the differentiated formula by the projection address formula to obtain equation (8), which gives the result when... When there is an angular error, the formula for the relative error between the projected addresses of the x-axis and y-axis is as follows: When Corner When the error is calculated, the relative error generated by the projected address is... Since the error during mechanical design is a constant, the most significant factor affecting the magnitude of the projected address error is... The coefficient;

[0058]

[0059] Step 4: Calculate the optimal value when z′=0, i.e., when reconstructing the central layer of the object. Angle calculation.

[0060] Step 4.1: Compare the projected addresses of the object's center and edge regions in the x-axis and y-axis directions according to formula (3). The magnitude of the angular error is considered, and the one with the largest impact is selected to determine some parameter values ​​in the optimization objective function. Assuming the maximum values ​​in the x and y directions of the object are 128, 1 degree, 45 degrees, and 80 degrees are taken respectively to calculate the relative error of the projected addresses of the central and edge regions under a 360-degree projection. Figure 2 As shown.

[0061] Figure 2 This indicates that the projection angle changes in both the x-axis and y-axis directions, resulting in greater projection address errors in the edge regions than in the center regions when projecting onto the object. Therefore, in subsequent analyses, the relative error of the projection address of the object's edge regions is used as the basis for establishing the objective function.

[0062] Step 4.2: Solve for the projection address error of each angle in the x-axis and y-axis directions under 360-degree projection, and establish equation (9) based on this. This equation represents the average absolute value of the projection address error in the x-axis and y-axis directions under 360-degree projection. Based on equation (9), establish the optimization objective function equation (10):

[0063]

[0064]

[0065] Step 4.3: Based on the recipient selected in 4.1 Analysis of edge regions with significant angular influence. The scanning step of the angle is 1 degree. The coordinates of the object edge are defined as (128, 128, 0), and the coordinates of the central region are (1, 1, 0). The constant D in the scanning geometry is set to 300, which comes from the parameter values ​​of the actual CL imaging system. Substituting the constant value into equation (9), the objective function and... angular relationships such as Figure 3 As shown.

[0066] Step 4.4: In X-ray CL scanning mode, The angle range is typically 0-90 degrees, so we use a 1-degree increment to search for the formula in step 4.2 using gradient descent to minimize the objective function. The value is the optimal value for the reconstruction of the central layer. The method determines that the optimal tilt angle for the central layer object is 20 degrees.

[0067] Step 5: Establish the optimal system that depends on the object thickness Angle calculation model.

[0068] Step 5.1: Using a step size of 1, to achieve better fitting results, the object thickness value is set to (1 < z′ < 200). This value is greater than the thickness of flat objects in conventional detection. Using the method established in Step 4, the optimal value for each thickness is obtained. horn;

[0069] Step 5.2: Based on the optimal z′ obtained in Step 5.1 Angle, perform function fitting, and determine z′ and optimal Angular relationship, i.e., optimal relationship depending on the thickness of the object. Angle calculation model, comparing various fitting methods, the sine function fitting method has the highest accuracy, and the fitting results are as follows. Figure 4 As shown, the fitting equation is equation (11):

[0070]

[0071] This equation provides a better guide for selecting the tilt angle for flat objects of varying thicknesses in subsequent engineering projects. This avoids significant aliasing in the reconstruction results due to tilt angle issues, thus improving reconstruction accuracy.

[0072] In this embodiment, an electronic device includes a memory and a processor. The memory stores a program that supports the processor in executing the above-described method for calculating the optimal tilt angle of an object. The processor is configured to execute the program stored in the memory.

[0073] In this embodiment, a computer-readable storage medium stores a computer program that, when executed by a processor, performs the steps of the above-described method for calculating the optimal tilt angle of an object.

Claims

1. A method for calculating the optimal tilt angle of an object in X-ray CL three-dimensional reconstruction, characterized in that, This is applied to the industrial CL scanning 3D reconstruction process of flat objects, and is carried out according to the following steps: Step 1: Establish a projection coordinate system o with the center of the detector as the origin O, the vertical direction of the detector plane as the X-axis, the horizontal direction as the Y-axis, and the direction perpendicular to the projection plane XOY as the Z-axis; let the center O′ of the flat object and the X-ray source point S both lie on the Z-axis O′ of the projection coordinate system o, and place the flat object at an angle of inclination. Let O′ be the angle between the plane normal on the flat object and the Z-axis of the projected coordinate system o. Establish the object coordinate system o′ with O′ as the origin, the vertical axis of the upper plane of the flat object as the X′ axis, the horizontal axis as the Y′ axis, and the direction perpendicular to the plane X′O′Y′ as the rotation axis Z′. Let D be the distance between the X-ray source point S and the detector, and z′ be the distance from the detector to the flat object. O′ Thus, a three-dimensional reconstruction model using X-ray CL scanning was established; Step 2: Determine the geometric parameter relationships of the X-ray CL scanning mode: Step 2.1: According to the coordinate system transformation rules, determine the three coordinate transformation matrices R1, R2, R3 between the object coordinate system o′ and the projected coordinate system o; thus, use equation (1) to obtain the coordinates (x, y, z) of the origin coordinates (x′, y′, z′) of o′ in the object coordinate system to the coordinates (x, y, z) in the projected coordinate system o: In equation (1), θ is the angle of object rotation in CL scanning mode, (x′ O′ ,y′ O′ ,z′ O′ () represents the coordinates of the object's center O′ in the projected coordinate system o; Step 2.2: Given the coordinates (x, y) of the ray source in the projected coordinate system o. oS ,y oS ,z oS ), and based on the X-ray passing through the i-th point to be reconstructed (x i ,y i ,z i Establish the linear equation for X-rays; Step 2.3: The intersection of the X-ray and the projection plane XOY is the projection address corresponding to the point to be reconstructed. In the projection coordinate system o, set the Z-axis coordinate of the XOY plane to zero and substitute it into the linear equation of the X-ray. Thus, using equation (2), we can obtain the projection address (x) of the X-ray from the X-ray source to the detector after passing through the object in the CL scanning mode. proj ,y proj ): Step 2.4: Translate the object coordinate system o′ to the projected coordinate system o, thereby utilizing z′ O′ =0, z oS =D yields the simplified projection address formula as shown in equation (3): Step 3: Calculate the tilt angle The relative error of the projected address under the influence of: Step 3.1: Use equation (3) to... Take the derivative to obtain the formula. Step 3.2: Use equation (4) to obtain the tilt angle. When there is an error, the projected address (x) in the x-axis and y-axis directions proj ,y proj The formula for the relative error of ) is: Step 4: Calculate the optimal tilt angle for object reconstruction when the Z′ axis coordinate in the X′O′Y′ plane is zero. Step 4.1: Compare the projected addresses (x, y) of the center and edge regions of the object along the x and y axes according to equation (4). proj ,y proj Due to the error in the tilt angle, the center of the object or its edge region that has a greater impact is selected to determine some parameter values ​​in the optimization objective function; Step 4.2: Solve for each rotation angle θ of the object along the x-axis and y-axis in the 360-degree projection. i The projection address error value is obtained, and thus the projection address (x) in the x-axis and y-axis directions under 360-degree projection is obtained using equation (5). proj ,y proj The absolute value of the error is the average value, and then the objective function f is established using equation (6): Step 4.3: Select the center or edge region of the object that is significantly affected by the tilt angle, and set the object coordinates (x′, y′, z′), z′ O′ Substituting 0 and distance D together into equation (5), we can find the tilt angle that minimizes the objective function f, which is the optimal tilt angle for object reconstruction. Step 5: Establish the optimal tilt angle that depends on the object's thickness Computational model: Step 5.1: Using z′ as the thickness of the object, and setting the range of values ​​for the object thickness, and following the process in Step 4, obtain the optimal tilt angle corresponding to z′ for different thicknesses within the range of values ​​for the object thickness. Step 5.2: Find the optimal tilt angle corresponding to z′ Perform function fitting to obtain z′ and The calculation model is used to obtain the optimal tilt angle for flat objects of different thicknesses.

2. An electronic device, comprising a memory and a processor, characterized in that, The memory is used to store a program that supports the processor in executing the optimal tilt angle calculation method for an object as described in claim 1, and the processor is configured to execute the program stored in the memory.

3. A computer-readable storage medium storing a computer program, characterized in that, When the computer program is run by the processor, it executes the steps of the method for calculating the optimal tilt angle of an object as described in claim 1.

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