Random heterogeneous material reconstruction method combining SWD and gradient optimization

Abstract

The invention discloses a random heterogeneous material reconstruction method, which mainly comprises the following steps of: (1) calculating reconstruction series according to a hierarchical reconstruction series calculation formula, and obtaining an image pyramid of a training image; (2) initializing the highest-level reconstructed image into a two-dimensional or three-dimensional noise image according to the reconstruction task; (3) selecting a corresponding template to extract multipoint statistical information of the training image and the reconstructed image, enabling the number of elements of the training image and the reconstructed image to be equal according to an expansion formula, and calculating an SWD distance; and (4) performing step-by-step reconstruction by using the gradient optimization updating reconstruction image until the lowest-level reconstruction is completed, and obtaining a final reconstruction result. The method can be applied to 2D-2D, 2D-3D and 3D-3D reconstruction tasks, reconstruction can be completed when only a single sample serves as a training image, and good flexibility and generalization are achieved.

Description

Technical Field

[0001] The present invention relates to a method for characterizing and reconstructing a microstructure, in particular to a method for reconstructing a random heterogeneous material by combining SWD and gradient optimization, and belongs to the fields of computational materials science and image processing. Background Art

[0002] Random heterogeneous materials (porous rocks, batteries, piezoelectric ceramics, metals, alloys, etc.) have been widely studied and applied in various engineering fields, such as underground storage of carbon dioxide, oil and gas extraction from porous rocks, electrodes of solid oxide fuel cells, diaphragms of lithium-ion batteries, superalloys, semiconductors, etc. Accurately studying and analyzing their macroscopic physical properties (transport properties, electrochemical properties, mechanical properties, etc.) is crucial for improving performance in various systems. It is worth noting that these macroscopic properties of random heterogeneous materials are closely related to their constituent components and pore space structures. Different material components of the solid skeleton have an impact on its electrical conductivity and mechanical properties, while the pore structure, as the site where transport and reactions occur, also plays a key role in transport properties and electrochemical properties. Therefore, it is particularly important to strengthen the understanding and analysis of the microstructure of random heterogeneous materials and establish the relationship between structure and performance.

[0003] Accurately establishing a digital model of the microstructure is the premise and foundation for subsequent numerical analysis, understanding the transport or reaction mechanism, and controlling related processes. Microscopic imaging techniques such as computed tomography (CT) and focused ion beam scanning electron microscopy (FIB-SEM) can construct real three-dimensional structures. However, due to the cost of imaging equipment, the complexity of operation, and the contradiction between the field of view and resolution, the acquisition of widely available three-dimensional structures is limited. Therefore, computational reconstruction methods have emerged in this context, using two-dimensional images or a small number of existing three-dimensional structures to reconstruct target structures with similar statistical characteristics, mechanisms, or properties. Compared with microscopic imaging techniques, computational reconstruction methods have low cost and high efficiency, and can generate three-dimensional structures with specific properties. In addition, computational reconstruction methods play an important role in the construction of multi-length scale and multi-modal data, the study of structure-property relationships based on material data mining, and material design.

[0004] Researching fast, efficient, general, and highly interpretable computational reconstruction methods has always been our goal. However, current computational reconstruction methods either lack high accuracy and efficiency (traditional methods) or rely on datasets and are difficult to handle various types of reconstruction tasks (deep learning methods). Hybrid methods combine the advantages of deep learning methods and traditional methods and have certain potential in achieving these goals, but need to continue to move forward in terms of generalization, small sample datasets, and heterogeneous structure reconstruction. Summary of the Invention

[0005] The object of the present invention is to improve the generality and generalization of the computational reconstruction method, and to achieve the 2D-2D, 2D-3D, and 3D-3D reconstruction tasks of random heterogeneous materials quickly even in the case of a single sample (a single 2D image or a single 3D structure), to provide a new idea for microstructure characterization using the multi-point statistical information of traditional methods, and for microstructure reconstruction using the sliced Wasserstein distance (SWD) metric and gradient optimization of machine learning methods, and a reconstruction method of random heterogeneous materials combining SWD and gradient optimization.

[0006] The present invention realizes the above object through the following technical solutions:

[0007] (1) Design a hierarchical reconstruction level calculation formula. For the training image M, set the template size to T, calculate the reconstruction level Lv, and obtain the image pyramid representation {M (0) , M (1) ,..., M (Lv)} of the training image by successive downsampling, where M (0) is the training image, and M (Lv) is the training image downsampled Lv times;

[0008] (2) Set the target iteration number K at each reconstruction level. Initialize the current iteration number k to 0 and the current reconstruction level lvl to Lv. Initialize the reconstruction image at the highest level according to the reconstruction task as a noise image with a Gaussian distribution or a uniform distribution. For the 2D-2D reconstruction task, initialize it as a two-dimensional noise image, and for the 2D-3D or 3D-3D reconstruction task, initialize it as a three-dimensional noise image, and start the reconstruction;

[0009] (3) At the current reconstruction level lvl, extract multi-point statistical information from the training image M (lvl) and the reconstruction image using the template of size T set. For two-dimensional images, extract using a two-dimensional template, and for three-dimensional images, extract using a two-dimensional template from the three directions of XY, YZ, and ZX;

[0010] (4) Design a multi-point statistical information expansion formula, and use the multi-point statistical information expansion formula to make the number of elements in the multi-point statistical information of the training image M (lvl) and the reconstruction image equal;

[0011] (5) At the current reconstruction level lvl, calculate the sliced Wasserstein distance between the multi-point statistical information of the training image M (lvl) and the reconstruction image , and calculate the gradient of the sliced Wasserstein distance with respect to the reconstruction image through the chain rule of differentiation;

[0012] (6) At the current reconstruction level lvl, update the reconstructed image using the stochastic gradient optimization method based on the gradient calculated in step (5).

[0013] (7) Determine whether the current iteration number k has reached the target iteration number K. If it has, go to step (8). If not, increment the current iteration number k by 1, and loop through steps (3)-(7) until the target iteration number K is reached.

[0014] Determine whether the current reconstruction level lvl is 0. If it is, output the reconstruction result. For the current reconstructed image Reconstruction ends. If not, the current reconstructed image Is upsampled to the input of the next-level reconstructed image The current reconstruction level lvl is decremented by 1, the current iteration number k is set to 0, and loop through steps (3)-(8) until the reconstruction ends.

[0015] In the above step (1), the calculation formula for hierarchical reconstruction is:[[]]

[0016]

[0017] In the formula, T is the template size, r d Is the downsampling factor, with a value greater than 1, l M Is the size of the training image M, and Lv is the number of levels for the final hierarchical reconstruction.

[0018] In the above step (3), the multiple-point statistical information is a multiset composed of all local patterns (or data events) extracted from the image using the set template. For the training image M (or the reconstructed image ), its expression for the multiple-point statistical information is (or ), (or ) is a single data event extracted by the template.

[0019] In the above step (4), the expression for the extended formula of the multiple-point statistical information is:[[]]

[0020]

[0021] In the formula, mp(M) is the multiple-point statistical information of the training image M and Is an element in mp(M), emp(M) is the extended multiple-point statistical information of the training image M, Is the floor operation, mod is the modulo operation, c M And Are the training image M and the reconstructed image The number of elements in the multi-point statistics, i.e., c M =#mp(M),

[0022] In step (5), for 2D-2D, 2D-3D, and 3D-3D reconstruction tasks, the training image M and the reconstructed image The sliced Wasserstein distance expression between the multi-point statistics is as follows:

[0023]

[0024] In the formula, SW p is the sliced Wasserstein distance, is the direction for setting the template to extract the corresponding multi-point statistics, emp(M) is the extended multi-point statistics of the training image M, is the reconstructed image The multi-point statistics of; and the sliced Wasserstein distance SW p The expression is:

[0025]

[0026] In the formula, N is the number of multi-point statistics elements, L is the number of projection directions, is the sorted element position, θ l is the projection direction, and <·> is the inner product operation. Description of the Drawings

[0027] Figure 1 are Training Images 1 and 2 of the 2D-2D Reconstruction Experiment in the Embodiment of the Present Invention;

[0028] Figure 2 are Training Images 3 and 4 of the 2D-3D Reconstruction Experiment in the Embodiment of the Present Invention;

[0029] Figure 3 are Training Images 5 and 6 of the 3D-3D Reconstruction Experiment in the Embodiment of the Present Invention;

[0030] Figure 4 are the two-dimensional template and the two-dimensional templates in three directions adopted in the embodiment of the present invention;

[0031] Figure 5 is the visual comparison diagram of the training image and the reconstruction result of the 2D-2D reconstruction in the embodiment of the present invention;

[0032] Figure 6 is the visual comparison diagram of the training image and the reconstruction result of the 2D-3D reconstruction in the embodiment of the present invention;

[0033] Figure 7It is a visual comparison diagram of the training image and the reconstruction result of 3D-3D reconstruction in the embodiment of the present invention. Detailed implementation manners

[0034] The present invention will be further described in detail with specific embodiments and the accompanying drawings:

[0035] (1) For the training image M, set the template size to T, calculate the reconstruction level Lv according to the hierarchical reconstruction level calculation formula, and obtain the image pyramid representation {M (0) , M (1) ,..., M (Lv)} of the training image by successive downsampling;

[0036] (2) Set the target iteration number K at each reconstruction level, initialize the current iteration number k to 0, initialize the current reconstruction level lvl to Lv, and initialize the reconstruction image at the highest level according to the reconstruction task as a noise image with a Gaussian distribution or a uniform distribution, and start the reconstruction;

[0037] (3) At the current reconstruction level lvl, extract multi-point statistical information from the training image M (lvl) at the current reconstruction level and the reconstruction image using the set template;

[0038] (4) Use the multi-point statistical information expansion formula to make the number of elements in the multi-point statistical information of the training image M (lvl) at the current reconstruction level and the reconstruction image equal;

[0039] (5) At the current reconstruction level lvl, calculate the sliced Wasserstein distance between the multi-point statistical information of the training image M (lvl) at the current reconstruction level and the reconstruction image , and calculate the gradient of the sliced Wasserstein distance with respect to the reconstruction image by the chain rule of differentiation;

[0040] (6) At the current reconstruction level lvl, update the reconstruction image

[0041] using the random gradient optimization method according to the gradient calculated in step (5);

[0042] (7) Determine whether the current iteration number k has reached the target iteration number K. If it has reached, go to step (8). If it has not reached, increment the current iteration number k by 1, and loop through steps (3)-(7) until the target iteration number K is reached; is the current reconstruction image (8) Determine whether the current reconstruction level lvl is 0. If it is, output the reconstruction result as the current reconstruction image and end the reconstruction. If it is not, the current reconstruction image The upsampling serves as the input for the next-level reconstructed image The current reconstruction level lvl is decreased by 1, the current iteration count k is set to 0, and steps (3)-(8) are looped until the reconstruction is completed.

[0043] Example:

[0044] In the present invention, in step (1), to illustrate the generality of the method, training image 1 (sandstone image) and training image 2 (silica material image) were selected as implementation examples for 2D-2D random heterogeneous material reconstruction, as Figure 1 shown; training image 3 (limestone image) and training image 4 (solid fuel cell electrode material) were selected as implementation examples for 2D-3D random heterogeneous material reconstruction, as Figure 2 shown; training image 5 (carbonate rock image) and training image 6 (graphite electrode material) were selected as implementation examples for 3D-3D random heterogeneous material reconstruction, as Figure 3 shown. Among them, training images 1-6 are all of size 128×128, the template size is uniformly set to 7, the downsampling factor is set to 1.25, and the reconstruction levels of training images 1-6 are calculated to be 6 according to the hierarchical reconstruction level calculation formula. Pyramid images of training images 1-6 are obtained by successive downsampling respectively.

[0045] In step (2), the target iteration count K at each reconstruction level is set to 200. For training images 1 and 2, the highest-level reconstructed image is initialized as a 32×32 two-dimensional Gaussian noise image; for training images 3-6, the highest-level reconstructed image is initialized as a 32×32×32 three-dimensional Gaussian noise image.

[0046] In step (3), both the two-dimensional template and the two-dimensional templates in three directions used to extract the multi-point statistical information are of size 7, as Figure 4 shown.

[0047] In step (4), for training images 1 and 2, 5 and 6, the number of elements of the multi-point statistical information at each reconstruction level is equal. After applying the multi-point statistical information expansion formula, the number of elements of the multi-point statistical information of the training images remains unchanged; for training images 3 and 4, the reconstructed images at each level are three-dimensional images of sizes 32 3 、41 3 、52 3 、65 3 、82 3 、102 3 、128 3 respectively, and the training images at each level are of sizes 32 2 、41 2 、52 2 、65 2 、822 , 102 2 , 128 2 For two-dimensional images of size 、, after applying the multi-point statistical information expansion formula, the number of elements of the multi-point statistical information of each level of training images is expanded to 32, 41, 52, 65, 82, 102, and 128 times the original respectively.

[0048] In the said step (5), the number of projection directions of the sliced Wasserstein distance is set to 49, which is the square of the template size, and the projection directions are random projection directions on the unit sphere.

[0049] In the said step (6), the stochastic gradient optimization method adopts the ADAM optimization method.

[0050] In the said step (7), steps (3)-(6) need to be looped 200 times at each reconstruction level.

[0051] In the said step (8), the reconstruction level decreases step by step from 6 until it reaches 0; the size of the reconstructed image also changes step by step from to , , , , , . 3 to 3 , 3 , 3 , 3 , 3 , 3 .

[0052] Specifically, in order to fully illustrate the effectiveness of the method of the present invention, the present invention has conducted reconstruction experiments on 6 kinds of random heterogeneous materials, including 2 groups of 2D-2D experiments, 2 groups of 2D-3D experiments, and 2 groups of 3D-3D experiments. Each group has carried out 10 reconstructions and obtained 10 reconstruction results. For the first three results of each group, we have made a visual comparison, as Figures 5-7 shown.

[0053] In terms of visual evaluation, this method has achieved pore morphology characteristics and spatial structures consistent with the training images, verifying the effectiveness of the method of the present invention. Compared with other reconstruction methods, the method of the present invention can be applied to different random heterogeneous materials, and is applicable to different reconstruction tasks. It can also achieve reconstruction with only one training image, achieving improvements in flexibility, generalization, and small-sample reconstruction performance.

[0054] The above embodiments are only preferred embodiments of the present invention, and do not limit the technical solutions of the present invention. Any technical solutions that can be implemented on the basis of the above embodiments without creative labor shall be regarded as falling within the scope of the patent rights of the present invention.

Claims

1. A random heterogeneous material reconstruction method combining SWD and gradient optimization, characterized in that: It also includes the following steps: (1) Design the calculation formula for the hierarchical reconstruction level. For the training image M, set the template size as T, calculate the reconstruction level Lv, and obtain the image pyramid representation of the training image by successive downsampling {M (0) , M (1) ,..., M (Lv)}, where M (0) is the training image, and M (Lv) is the training image downsampled Lv times; (2) Set the target number of iterations \(K\) for each reconstruction level. Initialize the current iteration number \(k\) to 0 and the current reconstruction level \(lvl\) to \(Lv\). Initialize the reconstructed image of the highest level according to the reconstruction task. It is a noise image with Gaussian distribution or uniform distribution. Initialize it as a two-dimensional noise image for 2D-2D reconstruction tasks and as a three-dimensional noise image for 2D-3D or 3D-3D reconstruction tasks. Start the reconstruction. (3) At the current reconstruction level lvl, multi-point statistical information is extracted from the training image M of the current reconstruction level using a template of size T set (lvl) and the reconstructed image For a two-dimensional image, the two-dimensional template is used for extraction, and for a three-dimensional image, the two-dimensional template is used to extract from three directions: XY, YZ, and ZX; (4) Design a multi-point statistical information expansion formula, and use the multi-point statistical information expansion formula to make the number of elements in the multi-point statistical information of the training image M (lvl) and the reconstructed image equal in the multi-point statistical information of both; (5) Calculate the training image M at the current reconstruction level lvl (lvl) and the reconstructed image Calculate the sliced Wasserstein distance between the multi-point statistics of both, and calculate the gradient of the sliced Wasserstein distance with respect to the reconstructed image using the chain rule of differentiation; (6) At the current reconstruction level lvl, the reconstructed image is updated using the stochastic gradient optimization method based on the gradient calculated in step (5). (7) Determine whether the current iteration number k reaches the target iteration number K. If it reaches, go to step (8); if not, increment the current iteration number k by 1, and loop through steps (3)-(7) until the target iteration number K is reached; (8) Determine whether the current reconstruction level lvl is 0. If so, output the reconstruction result. For the current reconstructed image Reconstruction ends. If not, the current reconstructed image Is upsampled to the next-level reconstructed image As the input, the current reconstruction level lvl is decremented by 1, the current iteration count k is set to 0, and steps (3)-(8) are looped until the reconstruction ends.

2. The random material reconstruction method according to claim 1, wherein: In the said step (1), the calculation formula for hierarchical reconstruction is: where T is the template size, r d is the downsampling factor, which takes a value greater than 1, and l M is the size of the training image M, and Lv is the number of levels for the final hierarchical reconstruction.

3. The random material reconstruction method according to claim 1, wherein: In the said step (4), the expression of the multi-point statistical information expansion formula is: where mp(M) is the multi-point statistics of the training image M and is an element in mp(M), emp(M) is the extended multi-point statistics of the training image M, is the floor operation, mod is the modulo operation, c M and are the number of elements in the multi-point statistics of the training image M and the reconstructed image respectively, that is, c M =#mp(M), 4. The random material reconstruction method according to claim 1, wherein: In the step (5), for 2D-2D, 2D-3D, and 3D-3D reconstruction tasks, the sliced Wasserstein distance expression between the multi-point statistical information of the training image M and the reconstructed image is as follows: where SW p is the sliced Wasserstein distance, is the direction for setting the template to extract the corresponding multiple-point statistical information, emp(M) is the multiple-point statistical information after the expansion of the training image M, is the reconstructed image of the multiple-point statistical information.

Images