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Abstract
Digital inline holography has gained extensive application in the optical diagnosis of solid propellant combustion. However, this method confronts several challenges. Firstly, the calculation time required for reconstruction and depth of field extension is excessively long. Secondly, the excessive smoke, airflow, and flame during combustion cause significant interference and poor reconstruction quality, which reduces the accuracy of particle identification. To address these issues, we have developed a holographic image reconstruction technique for aluminum particle combustion based on the Attention Mechanism, U-net, and Diffusion models. This approach enables end-to-end reconstruction of aluminum particle combustion holographic images, while effectively circumventing the interference of airflow combustion and flame.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Aluminum powder, widely utilized in modern solid composite propellants, not only discharges considerable energy during combustion but also mitigates combustion fluctuations [1]. Typically, its mass ratio in the propellant ranges between 10-20% [2]. Yet, as the propellant combusts, aluminum particles experience transformations like melting and clustering, resulting in a diverse particle spatial layout. Such variations profoundly influence particle damping efficacy [1]. In extreme scenarios, this dispersed combustion can intensify combustion instability. Thus, examining the dynamics of clustered particles within the propellant is crucial.
In order to study and characterize the spatial distribution of particles during propellant combustion, digital in-line holography has been widely used in the combustion diagnosis of solid propellants [3]. Digital in-line holography can capture particle distribution within a vast depth of field simultaneously, marking a significant reduction in effort compared to conventional optical experiments. Yet, given the intricate combustion process of the propellant under specific pressures, the presence of soot, flames, and airflow often hampers imaging. Occasionally, these elements even obstruct laser light propagation. Accurately reconstructing the holographic image and discerning particle size and spatial distribution remains challenging [4]. Additionally, raw digital holographic images typically require reconstruction and depth-of-field expansion. Currently, regardless of the reconstruction method employed, the image must be initially propagated to various depth positions. Subsequent processing at each depth, using wavelet transform or similar techniques, tends to be computationally sluggish [5]. Therefore, it is necessary to utilize deep learning to improve the reconstruction and depth-of-field extension of aluminum burnt holographic images.
Deep learning has been applied to the reconstruction of digital holography [6]. Tomoyoshi et al. realized the reconstruction of pure particle images at different positions [7]. End-to-end reconstruction of magnitude and phase information of off-axis holographic images achieved by Ren et al. [8]. Zhang et al. trained a modified U-Net on the off-axis DHM dataset generated under different defocus, and then tested it on simulated and real data [9]. Ma et al. successfully performed image segmentation and phase-aberration compensation for off axis digital holographic images using a U-net [10]. In current research, most of them are aimed at the reconstruction of transparent cell objects, or holographic images with relatively simple backgrounds, and the ability to reconstruct images with complex backgrounds and strong interference is relatively weak. Wu et al. successfully proposed a deep learning-based method to detect tiny burning particles which can be as small as 4 pixels in the reconstructed hologram to compensate the traditional threshold methods [11].
Most previous research on digital holographic reconstruction has relied on the comparatively simple U-net structure [12–14]. However, when reconstructing complex image features, such as those in the digital holographic images of propellant combustion in this study, the U-net structure may not be sufficient to extract satisfactory features [15]. While Generative Adversarial Networks (GANs) have been used for such tasks [16,17]. Diffusion models have gained greater attention in recent years, with many researchers demonstrating their superiority to GANs in various fields [18]. Yet, there is currently no research that has applied diffusion models to digital holographic reconstruction.
In this research, we introduce an innovative technique for reconstructing holographic images of aluminum particle combustion, harnessing the pix2pix neural network and denoising diffusion models (DDMs). Our approach facilitates a direct end-to-end depth of field extension from raw holographic images, adeptly navigating the challenges posed by smoke and airflow during combustion. Unlike traditional wavelet-based methods requiring manual parameter tweaking [19], our method capitalizes on deep learning's potential to sidestep these subjective constraints. We deploy two primary network structures: diffusion models and pix2pix, both refined with an attention mechanism. Diffusion models, a neural network variety, employ Markov chains to systematically eliminate predictive noise over time, proficiently salvaging data from Gaussian disturbances. Leveraging simulated holography for our dataset creation [20], objects were diffracted at random intervals using a digital holographic simulation. The ensuing images were enhanced with environmental noise and genuine propellant combustion visuals. These augmented images then served as our neural network's training foundation, with the original images acting as the model's training benchmark. Conclusively, we applied our techniques to reconstruct holograms of aluminum particle combustion and subsequently analyzed particle size statistically. The derived statistical insights validate our methodology's adeptness at reconstructing holographic imagery of aluminum-infused propellant combustion.
2. Methods
2.1 Experiment device and wavelet-based depth-of-field extension
For the digital inline holographic experimental system of aluminum-containing composite propellants, please refer to the previous papers of our laboratory [21]. Figure 1 shows the digital holographic image of the combustion of aluminum-containing propellant, along with its reconstructed and auto-focused image. It is apparent that the burning of the propellant under specific pressures results in the formation of a dense smoke consisting of alumina and incompletely burned carbon, which poses significant challenges to particle identification. Our autofocusing method was adopted from [5], which can be explained by Fig. 2. The reconstructed images were decomposed into the detailed subimages in the high-frequency bands and the approximation subimage in the low-frequency bands with wavelet transform. The depth-of-field extended image can be obtained with all the particles focalized using the maximum selection scheme. Then the 3D position of the particle can be accurately determined in the high-frequency subimages.
Fig. 1. a) Raw digital holographic image of the aluminum-containing propellant burning. b) Digital holographic image after reconstruction and auto-focusing.
2.2 Neural networks
Pix2pix neural network is used for holographic reconstruction in our research. Pix2pix is an improved algorithm of the GAN model [22]. The pix2pix network model is established through a generator (G) and a discriminator (D) [23]. In pix2pix, the image translation task consists of an input data x and a random noise z, where z can be understood as the starting point of the iteration, and finally generates y, which is $G:{\{ }x,z{\} } \to y$. Some related studies use Conditional Generative Adversarial Networks (CGAN) [24], which using conditions to control the output [25]. We know that GAN achieves the purpose of optimizing two models at the same time through the mutual game between the generator G and the discriminator D. Different from the traditional Gan, the input of the D in Pix2pix is the generated image and the original image, and the input of the G is random noise and the original image.
As introduced in [26], we add a regular term to the loss function to improve the quality of the generated image. In order to make the image clearer, the pix2pix network uses the L1 norm, such as formula (1). Finally, we get the loss function of pix2pix (2):
Here z uses random noise, which will not have a significant impact on the result [27].
The generator is developed on the basis of the U-net network [28]. In many similar studies before [7,20], such an encoder-decoder network was used. The advantage of U-net is that the feature map obtained by the deeper the network layer has a larger field of view [29].
In this study, various features of digital holograms play a significant role, including both shallow features and deep features. To effectively eliminate the irrelevant parts in the holographic images, such as the background, and emphasize the prominent features of the particles that require reconstruction, the U-shaped structure skip connection structure (feature stitching) proves to be more beneficial. To achieve this goal, both the U-net utilized in the pix2pix network and the diffusion model incorporate attention mechanisms. In the deepest layer of the U-net, attention blocks are inserted, as we show in Fig. S1 in Supplement 1.
The discriminator uses classic PatchGan [30], whose patch size is 1, because the experimental images have different sizes and may not all have 1024 × 1024 resolution. Our deep learning is based on paddlepaddle Deep learning platform.
Compared to the direct training of pix2pix from random noise to real results, the diffusion model decomposes the process of mapping random noise to real results into a progressive process [31]. If the real result is $q(x)$, we can define a forward diffusion process $q({x_t}|{x_{t - 1}})$, which consists of T steps, each of which adds a small amount of Gaussian noise to the current distribution. It is easy to derive that this forward process is a Markov process. The original holographic image x is input together with other conditions to guide the forward and reverse process. The algorithm is shown in Algorithm 1, and the target function of the forward process is:
In the current phase, ${\varepsilon _t}$ denotes the random Gaussian noise introduced. Given t as the prevailing time step and ${\gamma _t}$ is the intensity of the noise added in the current step. as the noise's intensity, throughout the learning evolution, it becomes feasible to discern both the mean and variance of the Gaussian distribution. Such understanding fosters a methodical denoising and generative progression from ${y_T}$ to ${y_0}$. Algorithm 1 encapsulates the forward training dynamics, whereas Algorithm 2 outlines the retrospective sampling procedure, where ${y_T}$ signifies the output at the time juncture t. while ${y_0}$ is defined as Gaussian noise with a mean of 0 and a variance of 0.5, at is the intensity of noise reduction at time t, and the noise-adding strategy will be discussed in detail later. Figure 3 illustrates the forward and backward processes, where x represents the guidance condition, as the original hologram, used to guide the entire diffusion process.
2.3 Build dataset
Deep learning paradigms hinge upon voluminous datasets for optimal efficacy. Yet, when the training corpus consists of images reconstituted via antecedent depth-of-field extension techniques, the emergent deep learning schema may not invariably transcend its predecessor in the realm of image reconstruction fidelity. Such an outcome runs counter to our investigational aspirations, wherein we endeavor to elevate image reconstruction prowess via the deep learning conduit.
To circumvent this challenge, we curated a bespoke dataset through the simulation of holographic projections. Within this exploration, various objects were disseminated to arbitrary locales leveraging angular spectral transformation diffraction. Genuine stochastic values dictated the transformative distance, spanning between 0 to 10 cm. Parameters crucial for ascertaining this distance, such as wavelength and pixel dimension, were extrapolated from the experimental apparatus: a camera pixel dimension of 10 × 10µm, a lens magnification of 1x, and a 532 nm wavelength laser. The above dataset generation process is shown in Fig. 4. The venerable angular spectral transformation was employed to emulate the holographic projection, with the resultant display portrayed in Fig. 5.
The objects to be transformed were circles, ovals, and FASHION-MNIST objects, respectively. Different objects were chosen to enhance the model's generalization, as previous studies showed that many particles were not round and had irregular shapes [3]. Since aggregated particles in propellant combustion are opaque, all objects were filled with solid colors to simulate the powder phase of the propellant burning field, and diffraction occurred around these parts. All images were 1024 × 1024 pixels, and the number of circles, ovals, and FASHION-MNIST objects was in a 3:1:1 ratio. As shown in Fig. 5(a) is the untransformed image and serves as the ground truth of the dataset, while (b) is the transformed image and is used as the train data. Many other studies based on generation models tend to focus on image restoration and interference removal [32], while our research focuses more on image translation. This is because digital holography involves the diffraction of parallel light on the edge of the object, and we obtain the depth and edge information of the object through these diffraction fringes [33]. The goal of generating the model is to obtain accurate spatial information.
Moreover, to negate the noise and airflow resultant from the combustion, background perturbations mirroring genuine combustion are integrated post-holographic transformation. Our approach amalgamates authentic combustion noise directly with the training image. Even when the propellant's combustion surface recedes from view, a profusion of smoke and flames persists until the propellant's full combustion. Our inquiries predominantly target agglomerative particles. Given that such images currently bear limited scholarly import, we infuse them into training images as ambient noise. This strategical inclusion ensures our model's adeptness at identifying and obliterating these disturbances in subsequent training iterations. This segment of the software harnesses OpenCV, deploying 20,000 image pairs for training, wherein a subset of 1,000 arbitrary images is earmarked as test datasets.
To reconstruct the propellant's burning surface, we integrated the extracted and transformed holographic image of the burning surface into our dataset. We employed mathematical morphology to delineate the combustion surface. Varying the size of the morphological operator affects its capability to filter out smaller particles. With an operator size of 50 × 50 pixels, all elements other than the burning surface were effectively eliminated, as depicted in Fig. S3 of Supplement 1. We adopted a methodology akin to previous text, projecting the hologram of the burning surface to arbitrary positions before enhancing half of our dataset images with these visuals. And the final image pare can be seen in Fig. 6. You can use our dataset in Dataset 1 (Ref. [34]).
2.4 Train and infer
We introduce several modifications that enable denoising diffusion to work well on high resolutions. One of the modifications in diffusion models is the noise schedule. Former researchers have found that the noise distribution has strong effect on the results [35]. The most common schedules are the cosine schedule, which under the variance ${\sigma _t}/{\alpha _t} = tan ({\pi t\textrm{ / }2} )$, while $no\textrm{i}s\textrm{e} = 0({t = 1{\; }\& \& {\; }t = 0} )$. Here ${\sigma _t}$, ${\alpha _t} \in \{{0,1} \}$ are scheduled weights of the noise. They are hyperparameters that determine how much signal is destroyed at a timestep $t \in \{{0,1000} \}$. It can be easily proved that image adding noise with lower resolution are more prone to distortion. This is problematic because the generative denoising process only has a small-time window to decide on the global structure of the image. As the cosine noise schedule were originally proposed on CIFAR10 with resolution of 32${\times} $32 and ImageNet of 64${\times} $64. However, for high resolutions not enough noise is added. So, cosine schedule is not suitable for our results. Commonly, diffusion models use network architectures that use downsampling to operate on lower resolution feature maps, in our case with average pooling. Here we have average pool $z_t^{1024 \times 1024}$ with resolution of 1024${\times} $1024 is being pooled. The new pixel is $z_t^{512 \times 512} = ({z_t^{(1 )} + z_t^{(2 )} + z_t^{(3 )} + z_t^{(4 )}} )\textrm{ / }4$. Here the 1, 2, 3, 4 denote the pixels in a 2${\times} $2 square which is being pooled. Letting $x_t^{512 \times 512}$ denote the first pixel of the input average pooled image, we can find that $z_t^{512 \times 512}\sim N({{a_t}x_t^{512 \times 512},{\sigma_t}} )$ . The lower resolution pixel only has half the amount of noise. As equivalent to fewer diffusion time is spent on the lower resolution situation.
It can be inferred that ${a_t}{\; }and{\; }{\sigma _t}$ in this lower resolution can be two times higher. We use S as the intensity of noise, and so $s_t^{512 \times 512} = s_t^{1024 \times 1024} \cdot {2^2}$, or in general:
At the reference resolution we define the noise schedule $s_t^{64 \times 64} = 1/tan {({\pi t\textrm{ / }2} )^2}$ and with full resolution d × d:The square of the ratio of resolution is added as the coefficient to enlarge the noise. In log-space, this implied a simple shift of $2 \cdot log ({64/d} )$. Under a variance preserving process, the diffusion parameters can be calculated as ${a_t}^2 = sigmoid({log({S_t})} )\; and\; {\sigma _t}^2 = sigmoid({ - log({S_t}} ))$.
Another modification in diffusion models is the loss function. The loss in diffusion models is between generated noise and real noise. Lsimple function is commonly used in such tasks [24]. Some relevant researches [36] have found that L1 loss has higher SSIM while implies less diversity. L2 loss has lower SSIM scores among their output samples than those trained with the L1 loss, indicating greater sample diversity. In our task, the precision of reconstruction is the only thing we need to consider, so L1 loss is appropriate in our experiment.
We utilized pixel-level metrics such as PSNR and SSIM since such metrics are reliable measures of sample quality for difficult tasks that don’t require hallucination, observe that PSNR and SSIM correlate well with Image restoration. We also trained some other models to compare the result including eHolonet [37], U-net and pix2pix (without attention block).
As shown in Fig. 7, it is the reconstruction result of the diffusion model of one image in the dataset, and the SSIM between it and ground truth is 0.91. It can be seen that the trained network can successfully remove background noise and reconstruct all objects with different Z distances onto one plane. Table 1 shows the average value of PSNR and SSIM of 2,000 images which was not included in the training set of all the models we trained. It can be seen that our network has good performance. PSNR is around 25 db, SSIM is around 0.95, although the PSNR and SSIM of image generation algorithms are lower than some other scenes, our research focuses on the restoration of each object, and our method can well reconstruct the morphology of the object. We deploy the deep learning model in an experimental computer environment (AMD 5800S), and the predicted speed of atten-pix2pix can reach 30 hz, which can shorten the previous reconstruction method which costs several days to a few minutes. We use an improved-diffusion [31] method to speed up the sampling step of ddm, of which the speed of construct can exceed 1 hz. Figure 6(c) is the image reconstructed by wavelet-based algorithm. After denoising and binarization, the error between it and the original image is 28.9%. In order to compare the specific differences between the two algorithms, we reconstructed the 5,000 images using the previous algorithm, which took 45 hours. The size of the window function was 15 × 15 pixels. The reconstruction process run on the Xeon E5 processor for parallel computation.
Table 1. Quantitative evaluation of different networks
Table 1 shows that the algorithms in this research can restore the intensity of the original image well. The diffusion model has higher reconstruction accuracy compared to the atten-pix2pix model at the same resolution, but its reconstruction speed is relatively slower. The eHolonet only suit for the reconstruction of a single project. Also, a simple U-net is not enough. Because the wavelet-based algorithm only has the functions of reconstruction and depth of field extension, without any denoising ability, the image without interference is used for comparison between algorithms. In addition, in depth of field extension, the method of wavelet decomposition and calculation of regional variance maximum will change the original light intensity for a large area of connected region, resulting in image distortion. In the image of propellant combustion, this situation normally occurs in the solid area. In order to compare the reconstruction ability of the two algorithms, we also need to eliminate this error artificially, so we use the image of the object with only diffraction for reconstruction, without any noise combined with flame or smoke, to compare the algorithms. It can be seen that the method in this paper has strong abilities of denoising and overcoming twin images from Table 1. But some of the details of the reconstruction are missing in Fig. 7. Therefore, it is necessary to build more abundant large-scale data sets for training and improve the robustness of the model.
3. Experiment result
In this paper, learning based algorithm is compared with the traditional wavelet reconstruction. Subsequently, the refined neural network is deployed to reconstruct genuine holographic images of aluminum-propellant combustion. Figure 8(a) is the original holographic image of a propellant agglomeration particle. Figure 8(b) is the image reconstructed by angular spectral transform and then extended by wavelet transform. The particle in the green box in the following figure is outside the reconstruction range of angular spectral reconstruction. Figure 8(c) is the image generated by the ddm in this paper.
Fig. 8. Combustion image of aluminum-containing propellant reconstructed using wavelet-based method and our model a) original holographic image, b) wavelet-based method. c) Holographic image reconstructed by our model.
Compared to traditional methods, the deep learning approach excels in reconstructing propellant agglomeration particles. This method provides images with sharp, distinct edges and without the extraneous noise that previous algorithms introduced due to local variance discrepancies. Crucially, it effectively nullifies interferences like flames, greatly enhancing particle data extraction and identification. From the provided illustrations, it's evident that traditional algorithms, influenced by preset step sizes and reconstruction ranges, fail to reconstruct some particles, leaving them in their diffraction form. This occurs when particles are significantly distant from the focal point. Additionally, as the depth of field expands, varying local selection ranges post-wavelet decomposition introduce substantial interference, an issue adeptly addressed by our deep learning approach. We've set our reconstruction range between -5 to 5 cm. Expanding this range would inevitably escalate both time and reconstruction error. The given particle's deviation in reconstructed distance is 16 mm, as shown in Fig. 9, and incorporating this within the traditional algorithm would inflate reconstruction time and diminish the extended depth of field's signal-to-noise ratio. More digital holographic images of propellant combustion reconstructed by this model are shown in Dataset 1 (Ref. [34]). Next, we will analyze the granularity distribution of the reconstructed results.
Fig. 9. Comparison of the results of single particle reconstruction by traditional reconstruction method and our model.
The main ability of this neural network is to reconstruct a large number of digital holographic images of aggregated particles in different Z positions. The network is used to reconstruct the digital holographic image directly to extract the particle size. The distribution of particle size is usually expressed by two statistical methods, Quantity probability density function ${N_{PDF}}$(Number PDF) And the volume probability density function ${V_{PDF}}$(Volume PDF). The expressions for both are as follows:
Images are taken at 4600 HZ. Many particles show up in several images in a row. To avoid counting these particles more than once, we recorded particle size every 20 images. The propellant's aluminum particles are 50µm in size. We measured particle size at different pressures: 0.1 MPa, 0.2 MPa, and 0.24 MPa. Combustion images at these pressures have different background noises. Higher pressures mean more smoke and airflow. The particle size of propellant groups shows a three-peak pattern, detailed in Table 2. As pressure goes up, the peaks get smaller. Most particles are near the first peak, which are mostly single aluminum particles. At high pressures, bigger unburnt particles are also seen. There's also a faint fourth peak in the results at normal pressure, but it's very small, only about 1‰ of the total. This fourth peak gets even smaller with higher pressures.
Table 2. Peaks and characteristic particle size for propellants
In general, the particle size distribution tends to decrease with the increase of combustion pressure. In order to characterize the particle size more clearly, the characteristic particle size was used for analysis. The characteristic particle size can be expressed as follows:
${D_{10}}$ is the number of particles the average particle size, ${D_{32}}$ is the surface area average particle size, ${D_{43}}$ is the volume mean particle size, and the mass mean diameter (MMD) represents the particle size of the average volume.The particle size distribution of the propellant containing aluminum reconstructed by the method in this paper is consistent with the size of the condensed combustion products (CCPs), which can be verified by each other [39]. Therefore, the method presented in this research can be well applied to the reconstruction of digital inline holographic images of solid propellant aluminum combustion particles.
4. Conclusion
This paper introduces an innovative deep learning-based end-to-end reconstruction technique tailored for holograms of aluminum composite propellant combustion. The approach adeptly discerns aluminum agglomerates amidst significant holographic interference, gleaning particle data from the combustion imagery. It offers precise extraction and reconstruction of particle details, suitable across an extensive depth spectrum.
This study analyzes digital holographic images of propellant combustion, and extracts interference and combustion surface images from the images as part of the dataset. A large number of random objects are used for holographic simulation to construct the dataset. These simulated objects and combustion surfaces are projected onto different positions and real propellant combustion interference is added to the dataset. Diffusion models and Pix2pix neural networks are used for training, and the trained models are used to reconstruct real digital holograms of propellant combustion. This method effectively eliminates background interference from propellant combustion under certain pressures, relatively accurately reconstructs holographic images, and can reconstruct agglomerated particles on a large scale. The strategy for adding noise and loss functions in the training process of the diffusion model is also studied. The reconstruction accuracy is significantly improved compared to previous methods. Although the diffusion model has higher reconstruction accuracy at the same resolution, the GAN has a faster reconstruction speed. In our practical applications, it is worthwhile to trade a slight degree of accuracy for extremely high reconstruction speed.
This method greatly improves reconstruction efficiency, eliminates interference caused by combustion smoke, and enhances the accuracy of particle identification. It is highly beneficial for research on aluminum-containing propellant combustion based on digital holography. However, the reconstruction of details for small objects is still not ideal, and some more complex particle edges are approximated as curves after reconstruction. In the future, the plan is to continue improving the richness of the dataset and exploring the complexity of the generator network to obtain more accurate features.
Funding
National Natural Science Foundation (U2241250).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are available in Supplement 1, including datasets and experiment results. Our model is available in Supplement 1, you can try it. Data are also available in Dataset 1 (Ref. [34]).
Supplemental document
See Supplement 1 for supporting content.
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