Noise is a consistent problem for x-ray transmission images of High-Energy-Density (HED) experiments because it can significantly affect the accuracy of inferring quantitative physical properties from these images. We consider experiments that use x-ray area backlighting to image a thin layer of opaque material within a physics package to observe its hydrodynamic evolution. The spatial variance of the x-ray transmission across the system due to changing opacity serves as an analog for measuring density in this evolving layer. The noise in these images adds nonphysical variations in measured intensity, which can significantly reduce the accuracy of our inferred densities, particularly at small spatial scales. Denoising these images is thus necessary to improve our quantitative analysis, but any denoising method also affects the underlying information in the image. In this paper, we present a method for denoising HED x-ray images via a deep convolutional neural network model with a modified DenseNet architecture. In our denoising framework, we estimate the noise present in the real (data) images of interest and apply the inferred noise distribution to a set of natural images. These synthetic noisy images are then used to train a neural network model to recognize and remove noise of that character. We show that our trained denoiser network significantly reduces the noise in our experimental images while retaining important physical features.

All imaging systems are subject to noise, but the type and extent of the noise vary depending on the system and subject. As such, image denoising and optimal filtering are active areas of research, with many varied approaches depending on the use case and desired sensitivity. In this paper, we are primarily interested in reducing the noise observed on x-ray images from high-energy-density (HED) experiments at the National Ignition Facility (NIF) and similar facilities. Specifically, we focus on x-ray images from the Multi-Shock (MShock) series of experiments1 in which we drive a shock into an initially solid physics package with intense laser beams and image the system to observe its evolution over tens of nanoseconds. The noise in images from these experiments comes from a combination of x-ray source and detector effects and is generally not well-characterized or modeled. The noise also varies between experiments and even between individual shots within an experimental campaign. Given the high level of noise on many HED x-ray images, denoising is necessary for improving quantitative inference of physical properties from these images. In this paper, we adapt neural network (NN) denoising methods, which have found success on optical image denoising to our experimental images.

Neural networks are a versatile and powerful form of machine learning methods that have been used for many varied applications, from language processing and prediction tasks to computer vision applications, such as image segmentation and object classification.2–6 Convolutional neural networks are a type of network that uses weighted spatial convolutions whose weights and biases are learned via training. Convolutional neural networks are commonly applied to image reconstruction tasks, particularly super-resolution7 and denoising.8 The Denoising Convolutional Neural Network (DnCNN) approach presented by Zhang et al.8 showed that convolutional networks with many successive layers are able to regularly outperform other denoisers on certain image reconstruction tasks, when trained properly.

Following these successes, we want to create a neural network denoising method that can be applied to our data images in such a way that we can anticipate which features it will affect. HED images are subject to a variety of noise in both form and amount, and this noise usually falls outside the range of existing NN models, so we must create a new model that is applicable to our data. We adapt our neural network architecture from existing machine learning methods, and we train our denoiser model using an approximation of the NIF Hardened Gated X-ray Detector (HGXD)9 camera noise observed in our experiments. Our denoiser is tailored to the thin-layer MShock experiments but would also apply to many other experiments using these cameras under similar conditions and with similar noise10–15 and, in principle, could be extended to any type of noise.

At the basic level, our neural network denoiser takes as input a noisy image and returns two residual arrays representing additive and multiplicative contributions, which return the denoised image when applied to the input image. The general process for creating the desired denoiser is as follows: (1) estimate the noise distribution of the target images; (2) apply this noise distribution to a set of clean images to generate training and testing image sets; (3) train the network using the noisy and clean image pairs; (4) calculate the average denoising error on the testing set; and (5) apply to the data. We work through these steps for our new denoiser in the remainder of this paper.

This paper is organized as follows: Sec. II describes the MShock experiments and the corresponding images we are interested in denoising and shows our estimate of the noise in this dataset. In Sec. III, we introduce the neural network denoiser, describe the network architecture and training procedure, show the results on the synthetic noise images using different noise models, and compare with other common denoising methods. In Sec. IV, we apply the trained denoising models to some data images and discuss the results. Section V concludes the paper and provides notes for potential future directions of model improvement.

The x-ray images we are concerned with in this paper come from the thin-layer MShock1 series of HED hydrodynamics experiments performed at the NIF to study the RM and RT instabilities. An illustration of the physics package is shown in Fig. 1 and consists of an initially solid shock tube target, which is driven by irradiating gold Hohlraums on both ends of the target with lasers. The shock tube consists of an inner beryllium (Be) tube lining with 2.25 mm inner diameter and 3.05 mm outer diameter in the region of the viewing window, and only 190 µm of Be outside of the window, with a fluorinated plastic casing making up the rest of the outer diameter. The interior of the targets contains a plastic disk of polyamide-imide (PAI) with an interior strip of iodine-doped plastic (CHI) as the opaque tracer layer for x-ray imaging. One side of the PAI/CHI disk is machined to have a large-scale sinusoidal mode, which is either smooth or with an additional amount of machined roughness. The plastic layer is surrounded on both sides by low-density (100 mg/cc) CH foam. The goal of these experiments is to observe the evolution of hydrodynamic instabilities from the perturbed surface via x-ray images of the opacity.

FIG. 1.

Illustration of the NIF MShock thin-layer geometry. The target is made up of a plastic shock tube (cross section shown at center), filled with a low-density CH foam, and a solid plastic disk, onto which a perturbation profile is machined for RM-RT instability study. In the center of the plastic disk is a tracer strip of iodine-doped plastic (CHI) to increase the opacity for measurement. The shock tube is driven with variable laser drives by irradiating the Au Hohlraums at the top and bottom. The backlighter is flat against the back of the target and is driven separately using tiled lasers. The interior of the tube is beryllium, which is more transparent to the x-ray signal, with a plastic casing around the tube at the left and right. Closer to the tracer layer, the plastic casing ends and tube is entirely beryllium, allowing better imaging of the tracer.

FIG. 1.

Illustration of the NIF MShock thin-layer geometry. The target is made up of a plastic shock tube (cross section shown at center), filled with a low-density CH foam, and a solid plastic disk, onto which a perturbation profile is machined for RM-RT instability study. In the center of the plastic disk is a tracer strip of iodine-doped plastic (CHI) to increase the opacity for measurement. The shock tube is driven with variable laser drives by irradiating the Au Hohlraums at the top and bottom. The backlighter is flat against the back of the target and is driven separately using tiled lasers. The interior of the tube is beryllium, which is more transparent to the x-ray signal, with a plastic casing around the tube at the left and right. Closer to the tracer layer, the plastic casing ends and tube is entirely beryllium, allowing better imaging of the tracer.

Close modal

The thin-layer MShock campaign obtained many x-ray images of the system evolution under different initial conditions using the same camera and backlighter configuration. The x-ray source is a Big-Area Backlighter16 (BABL) laser-driven x-ray source, which is mounted flat against the target, ∼5 mm from the center of the shock tube, as shown in Fig. 1. The camera for these experiments is the two-strip NIF HGXD,9 which allows us to take two 200 ps time-integrated images per shot at two different delays. The HGXD is very similar in design to the Gated X-ray Detector (GXD)17,18 cameras, and they share much of the same noise concerns. Both the GXD and HGXD convert incident x rays into electrons at the front of the detector using gold-coated strips. The electron signal is then amplified through a microchannel plate (MCP) array, and the electrons are then converted back into photons at a phosphor layer. These photons travel through the fiber-optic faceplate and to a CCD (GXD) or film (HGXD). These cameras allow us to take high-resolution images with time gating on the order of hundreds of picoseconds, but each step of the HGXD imaging process adds noise and distortion, which affect our ability to quantitatively determine the true transmission through the object being imaged and which alter the inference of material density. The primary sources of noise for many of these experiments, which are common to both cameras, are likely the x-ray conversion at the gold cathode, the electron signal amplification by the MCP, and the electron conversion at the phosphor layer, coupled with the short time gating. Because the GXD and HGXD have many similar components, the noise between the two imagers is roughly comparable.

Over the short time-scales during which we image these systems—usually 200 ps—the variation of photon statistics is noticeable, and Poisson noise is observed on the images. Barring the effect of other physical processes, Poisson noise is directly tied to the illumination of an image, so regions of greater transmission should experience lower relative noise than regions with reduced transmission. The x-ray illumination of the laser-driven backlighter also contributes to the noise distribution on the images. Although the lasers are generally tiled in such a way as to produce a somewhat uniform irradiation pattern,16 nonuniform illumination from the backlighter also affects the level of Poisson noise observed across the image. In addition, the interaction of the lasers with the plasma around the backlighter source can generate a lesser amount of higher-energy x-rays, which produce much higher, localized signal on the detector. The higher-energy x-ray strikes can be considered as a mostly additive noise on top of the main signal. In addition, there is an additive or speckle component seen on the images, with single hits on the detector showing up even in regions of very low transmission. The detector effects also include blurring, as from pinhole apertures, the spread of electrons across an MCP gap, and signal bleeding at the phosphor and the detector itself. Among the two cameras, the GXD typically has a finer resolution than the HGXD but has an additional hexagonal pattern of dark pixels imposed by the extended fiber optics. The HGXD has a larger gap between the MCP and the phosphor, which may increase the blur, and the film grain and scanner can add additional artifacts to the image.

The listed noise contributions can be considered explicitly to create a forward model, as explored by Trosseille et al.,19 but for simplicity, we assume that the structure of the noise can be condensed into a combination of Poisson noise, additive noise, and a Gaussian blur, applied in that order. Ideally, we want a denoising method that preferentially removes noise while retaining important physical properties, but the combination of noise sources on these images makes recovering information from the images difficult. Inferences we make from these noisy images may be highly sensitive to the method of denoising or post-processing. Therefore, we want a denoiser that is able to handle the expected noise, which we address with a neural network model, and to have some understanding of how our chosen denoiser affects the underlying information as well.

Because we have an extensive dataset using the same nominal parameters, we can use the data to estimate the type of noise and power in these images. Figure 2 displays two representative HGXD x-ray images from our dataset, in which the structured high-opacity region in the center of each image is the thin tracer layer. For these images, we know that density fluctuations in the regions of heated and ionized foam on either side of the opaque tracer should be too small to create any significant change in opacity when integrating across the tube. Therefore, we assume that the fluctuations in signal intensity within small patches of the image should primarily correspond to noise.

FIG. 2.

X-ray images from the NIF MShock campaign, with shot designation and HGXD strip number shown. The structured region of low transmission toward the center of the images is the thin tracer layer. The edges of the beryllium window are visible on the left and right, where the transmission suddenly decreases. Spatial fiducial teeth (tantalum) are seen at the top and bottom of the image, with a gold grid fiducial at the top right. The outlined boxes (1a–1d) and (2a–2d) in these images are small regions over which the transmission is expected to be mostly uniform, so fluctuations are assumed to be due to noise. The plots on the right show the normalized intensity distributions of the boxes, with synthetic noise distributions overlaid.

FIG. 2.

X-ray images from the NIF MShock campaign, with shot designation and HGXD strip number shown. The structured region of low transmission toward the center of the images is the thin tracer layer. The edges of the beryllium window are visible on the left and right, where the transmission suddenly decreases. Spatial fiducial teeth (tantalum) are seen at the top and bottom of the image, with a gold grid fiducial at the top right. The outlined boxes (1a–1d) and (2a–2d) in these images are small regions over which the transmission is expected to be mostly uniform, so fluctuations are assumed to be due to noise. The plots on the right show the normalized intensity distributions of the boxes, with synthetic noise distributions overlaid.

Close modal

We outline four regions with boxes on each image of Fig. 2, 1a–1d for shot N180724-001 on top, and 2a–2d for N190923-001 on bottom. The frames we show are 900 × 720 pixels surrounding the data, which are selected from larger images. Boxes a and d on each image are outside of the beryllium window and have lower transmission than boxes b and c, which are inside the window. We use the distribution of signals within each box to estimate the noise on these images. Note that we cannot exactly reproduce the noise from images alone, due to the non-uniqueness associated with any form of noise. Instead, we estimate an approximate range of possible noise as a combination of Poisson and additive white Gaussian noise (AWGN). The noise in these images is not pixel-scale, so a Gaussian blur with σ = 1 must be applied to mimic the spatial extent of the noise.

Within each box (normalized to the maximum of each image) the initial, unblurred, pixel-level noise is estimated as a Poisson distribution about the mean of that box, with an added 5% AWGN. To fit the noise, we create a synthetic noise distribution by first initializing a 2D map of pixels, which are set to the mean of the data box. The Poisson noise is then applied by scaling the map up by the proposed λ (essentially, the maximum λ for a signal level of 1) and applying a Poisson distribution at each pixel. The noisy map is then scaled back down by the chosen λ, which sets the mean back to the initially applied mean and the variance to that mean divided by λ. The 5% AWGN is applied to the rescaled map, and then, the Gaussian blur is applied. The normalized intensity distributions of the boxes are shown in the plots on the right of Fig. 2, with the estimated noise distributions overlaid in dashed lines. The lower-intensity regions of boxes 1a and 1d are estimated to have an effective Poisson λ = 10, and the higher-intensity regions 1b and 1c have λ ≈ 5. For N190923-001 strip 2, the noise in boxes 2a, 2b, and 2d can be approximated with λ = 10, and in box 2c with λ = 20. This range of noise remains mostly consistent for all images with good signal. Interestingly, there seems to be greater noise in the regions of higher intensity, unlike what we would expect from pure Poisson noise. Because the noise is observed to have a different character in different regions of the image, we seek to create a denoiser that applies to the range of estimated noise in our data.

In this section, we describe our neural network denoising model, its architecture, how it is trained, and its performance on a test set of synthetic noisy images. The Denoising Convolutional Neural Network (DnCNN) approach presented by Zhang et al.8 showed that convolutional neural networks, when trained properly, are able to regularly outperform other methods for image reconstruction (i.e., denoising) tasks. In addition, some advances in network architecture have been proposed since the original DnCNN, which we incorporate into our models. In particular, we apply the concatenation of feature maps presented in the DenseNet by Huang et al.6 to our denoising model. Following their results, we apply neural network denoising to the data x-ray images from our experiments, something that has not yet been reported for this type of imaging system. Pre-trained denoising models using DnCNN (and other models) are available, but we cannot directly apply such models and expect good performance because most models (including DnCNN) are only trained on additive Gaussian noise. We create new neural network denoising models with the modified DenseNet network architecture, trained on the range of estimated noise distributions from our data images.

The fundamental blocks of the DnCNN proposed by Zhang et al.8 use 3 × 3 convolutional layers with 64 learned filters per layer, followed by batch normalization,20 which reduces variance among the images simultaneously passed to the network, and a Rectified Linear Unit (ReLU) activation function, which modifies all positive values by a set multiplicative factor and sets negative values to 0. The DnCNN networks of Zhang et al.8 were between 16 and 20 layers deep, depending on the image reconstruction task, and produced good results on blind image denoising for AWGN up to 20%. Our denoising model maintains the general ideas from the DnCNN denoiser of Zhang et al.8 but incorporates the recent work of dense connections by Huang et al.6 The DenseNet architecture of Huang et al.6 uses blocks in which the results from successive convolutional layers are concatenated—called a dense connection or a concatenated skip connection—rather than added together as with additive skip or residual connections,3 which increase the effective receptive field of the network—the length scale at which patterns can be recognized—with fewer layers. DenseNets were originally applied to image classification problems, but by changing the transition layers—which reduce model complexity between blocks—to reduce the number of feature maps while retaining the image size instead of reducing the image dimensionality, this architecture is easily translated to image denoising.

Our proposed denoising network is shown in Fig. 3. The network starts with a 7 × 7 convolutional layer to produce 64 feature maps from the noisy input image, followed by a leaky ReLU activation function21–23 with a negative slope of −0.2, rather than setting negative values to 0. The initial 7 × 7 convolutional size provides a relatively large, but not computationally intensive, receptive field to determine features in the image to start.6,24 These initialization layers are then followed by six dense blocks. Each dense block uses eight successive concatenation skip blocks, with each of these sub-blocks consisting of a 2D batch normalization followed by a leaky ReLU activation function and a 2D 3 × 3 convolutional layer with 16 output channels. The results from each of these blocks are concatenated with the prior input at each step, increasing the size of the input to the next concatenation block by 16 channels. After the concatenation blocks, another series of a 2D batch normalization, leaky ReLU, and 2D convolution make up the transition layer, which reduces the model complexity by reducing the number of channels to 64 + 16i, where i is the index of the dense block, so the number of feature maps increases by 16 after each block. After the final dense block, the feature maps undergo another 2D batch norm and leaky ReLU layer and are reduced to two output maps by a final 3 × 3 convolutional layer. The two output feature maps are envisioned as an additive and multiplicative component of the noise.

FIG. 3.

Illustration of our denoising neural network architecture. The network takes as input a noisy image of any size and generates an additive and multiplicative component, which produces the denoised image when applied to the input noisy image. The architecture of our denoising model is closely based on the DenseNet model of Huang et al.,6 but adapted to denoising. The first layers of the network convert the noisy input into 64 feature maps by a 7 × 7 convolution layer, to which a leaky ReLU activation layer (of negative slope 0.2) is applied. The network then uses six dense blocks (shown at bottom), of which each comprises eight concatenation skip blocks and a transition layer. Each concatenation block applies a 2D batch norm to the input, followed by a leaky ReLU activation function and a 3 × 3 convolution layer with 16 feature maps. The output of each layer is the concatenation of the feature maps from the convolution layer with the input, growing the number of feature maps by 16 at each iteration. After the eight concatenation blocks, the number of feature maps is reduced back to 64 by the transition layer consisting of a 2D batch norm, leaky ReLU, and a 3 × 3 convolution. Following the dense blocks, the final step of the network is to convert the many feature maps into the 2D additive and multiplicative components of the denoising process. The denoised image is generated as Idenoised=Inoisy+outadditive×outmult.

FIG. 3.

Illustration of our denoising neural network architecture. The network takes as input a noisy image of any size and generates an additive and multiplicative component, which produces the denoised image when applied to the input noisy image. The architecture of our denoising model is closely based on the DenseNet model of Huang et al.,6 but adapted to denoising. The first layers of the network convert the noisy input into 64 feature maps by a 7 × 7 convolution layer, to which a leaky ReLU activation layer (of negative slope 0.2) is applied. The network then uses six dense blocks (shown at bottom), of which each comprises eight concatenation skip blocks and a transition layer. Each concatenation block applies a 2D batch norm to the input, followed by a leaky ReLU activation function and a 3 × 3 convolution layer with 16 feature maps. The output of each layer is the concatenation of the feature maps from the convolution layer with the input, growing the number of feature maps by 16 at each iteration. After the eight concatenation blocks, the number of feature maps is reduced back to 64 by the transition layer consisting of a 2D batch norm, leaky ReLU, and a 3 × 3 convolution. Following the dense blocks, the final step of the network is to convert the many feature maps into the 2D additive and multiplicative components of the denoising process. The denoised image is generated as Idenoised=Inoisy+outadditive×outmult.

Close modal

The DnCNN model of Zhang et al.8 outputs a single additive residual of the noise, which is then added to the noisy image to produce the denoised image. In general, computing the residual appears to be superior to trying to output the denoised image directly, as the model enforces that the noise the network recognizes must be relatively minimal. However, we modified the residual approach, so the network output provides two maps, a multiplicative and additive component of the noise, and the final denoised image is computed as Idenoised=Inoisy+outadditive×outmultiplicative. We take this multi-component approach because we assumed that the general noise in our images is a combination of the multiplicative Poisson noise, additive noise, and a Gaussian blur. The multiplicative component should allow the model to better address both the signal-dependent Poisson noise and the Gaussian blur. In practice, this combined approach greatly improves model performance for our noise when compared to using only an additive component. The model outputs are bounded by a Sigmoid activation function and scaled, with the multiplicative output being limited to the range {0, 10} and the additive component to {−0.5, 0.5} rather than being unbounded. These bounds are appropriate because our network is designed to take input normalized between 0 and 1. In addition, the bounded outputs better enforce the restriction that the output should be similar to the input, resulting in faster training and reduced variation in performance during training.

After the architecture, the most important aspect of any neural network model is the training set, because any network can only be as good as the architecture and training set allow. At the most basic level, training consists of giving the network a normalized noisy image as an input, creating the denoised image from the output, and calculating the mean-square error (MSE) loss by comparing the final denoised image against the known clean image. In the most common denoising schemes, it has generally been sufficient to use sets of natural images, which are then corrupted with the desired level of noise. For scientific analysis, it is also common to have a set of simulated data that resemble the real data and are corrupted by whatever processes are expected to physically occur. However, aside from the example we use later for testing purposes, we do not have sufficient simulated data to use as a training set. In addition, training on simulated images of the full experimental system could introduce unwanted bias and reduce generalizability of a denoiser model in that it might cause the model to expect certain features to be present. We find that, for our purposes, it is not necessary to train the network on a simulated image set that resembles the data. Instead, what is more important is that the network is trained on the types of noise we encounter in the data, regardless of the underlying image. As such, our training set is derived from the 200 natural images of the training set in the Berkeley Segmentation Dataset 30025 (BSD300) image database, and the 100 images of the testing set are used for testing model performance. From the training set images, we generate our own training sets of synthetic noisy images each with multiple instances of the desired additive noise, Poisson noise, and Gaussian blur within the range of noise estimated from Fig. 2.

The natural images are further made sufficient for training by splitting each training image into smaller patches and training the model on these patches. By using smaller patches, we are essentially increasing the number of images in the training set and likely improve performance on full images not found in this set by training the model on more small-scale patterns. The patches can be of any size, but we have found the best results with patches of size 50 × 50 pixels and larger. Training on larger patches can improve the denoising performance by giving access to more large-scale patterns, depending on the network depth and width, but we have observed diminishing returns in our models for patches larger than 80 × 80 pixels, particularly when considering the increased memory demands and corresponding increased training time.

For our training set, we generate 100 000 noisy–clean pairs of 80 × 80-pixel patches. The patches are randomly selected from the training set images of the BSD300 image set, with an equal number of patches used for each image. For the models in this work, we select 500 unique patches from each image, so no patch is repeated. In order to ensure that the patches are distributed approximately evenly across each image, we generate the location of each patch using a random Poisson process such that no point is repeated.

Before adding the noise, the patch is first rescaled by randomly subtracting up to 0.15 from the image, cutting off values less than 0, and renormalizing. This first data augmentation step allows these sub-images to more accurately represent the range of values encountered in our data, whose distributions are biased toward lower intensities. We further augment the training set by randomly transforming ∼4% of the patches to a randomly chosen constant value to ensure that the network is exposed to regions of uniform value. Noise is then added to each patch based on the noise estimated from the data, which for our primary model consists of Poisson noise generated by scaling the maximum per-pixel counts of each patch to λ ∈ [5, 50], AWGN between 0% and 10%, and the σ = 1 Gaussian blur. The amount of each noise is randomly selected for each patch, and we use a unique random seed for the amounts of noise and the applied distribution to ensure that there is no overlap or unintended preferential repetition, and for consistency in training. Each noisy–clean pair is saved in memory for reference. An example from the training set is shown in Fig. 4, displaying the original full image and the resulting noisy and clean patches. Our test set is generated similarly, but each image of the BSD300 test set is only used once, at full resolution instead of in patches, and for one specified level of noise which we set as λ = 10 and 5% AWGN.

FIG. 4.

An example of our training set generation, showing the original image, a 50 × 50 pixel patch, and the same patch after adding noise.

FIG. 4.

An example of our training set generation, showing the original image, a 50 × 50 pixel patch, and the same patch after adding noise.

Close modal

During training, our model is given mini-batches of multiple images to denoise simultaneously at each step. Training with mini-batches allows the model to optimize over a wider set of inputs at each step, usually increasing the stability of the training between steps because of the variance between the training patches, although large mini-batches can reduce the generalizability of the model. Mini-batch training tends to both reduce the time required to reach acceptable results and improve convergence during the later stages of training because of the increased stability. We train our models using the Adam26 optimizer (an adaptive optimization scheme) with an initial learning rate of 1 × 10−4 and mini-batches of 50 patches of 80 × 80 pixels at each iteration, which are randomly selected from the set of noisy images of the training set. We train the models for 61 epochs, where one epoch consists of one full cycle over all images in the training set. At epoch 30, we reduce the learning rate by a factor of 10 to more quickly converge. We test our model’s performance by recording the average RMS error over both the training and test sets every epoch. Initializing our model in PyTorch and training with CUDA using an NVIDIA RTX A6000 GPU with 48 GB of memory, it can take up to 18 h to train a model. Training with patches of 50 × 50 pixels reduces the training time to less than 9 h and can still produce good results.

Figure 5 displays plots of the model performance on both the training and testing sets of images as a function of training epoch for three different models trained on different levels of Poisson noise. The top of Fig. 5 shows the root-mean-squared error (RMSE) between the denoised image and the clean image, averaged over the entire image, and the bottom shows the corresponding average peak signal-to-noise ratio (PSNR). PSNR is a common metric in image reconstruction and is log-scale inverse of the RMSE calculated as PSNR = 20 log 10(Imax/RMSE), where Imax is the maximum value of the image. Because of the large number of images in the training set, for all models, we see a rapid decrease in the average RMSE in the first stages of training, before leveling off and approaching a limiting value. The error on the training sets expectedly decreases as the level of noise decreases, but note that the test set has a consistent noise separate from the training set, with Poisson λ = 10 and additive noise of 5%. The model with the highest average PSNR on the test set is the model trained with Poisson noise of λ ∈ [5, 50], reaching 24.93, and we use this as our preferred model. The model trained with λ ∈ [10, 100] is very close, while the model with λ ∈ [20, 100] is noticeably lower. The models that perform best on the test set contain the same λ = 10 Poisson noise in training.

FIG. 5.

Plots of model performance in RMSE (top) and PSNR (bottom) on the training and testing sets during training for noise models with different Poisson λ. Different models are denoted by different colors, with the solid lines representing the test set performance and the dashed lines representing the training set. After 61 training epochs, the best model with λ ∈ [5, 50] produces an average RMSE of 0.0493 on the training set and 0.0567 on the testing set, with the corresponding PSNR of 26.14 and 24.93, respectively.

FIG. 5.

Plots of model performance in RMSE (top) and PSNR (bottom) on the training and testing sets during training for noise models with different Poisson λ. Different models are denoted by different colors, with the solid lines representing the test set performance and the dashed lines representing the training set. After 61 training epochs, the best model with λ ∈ [5, 50] produces an average RMSE of 0.0493 on the training set and 0.0567 on the testing set, with the corresponding PSNR of 26.14 and 24.93, respectively.

Close modal

The results of our trained denoiser model are illustrated in more detail for the castle image of the BSD300 test set in Fig. 6 with two applied noise models, Poisson λ = 10 and λ = 5, which we label noise model 1 and noise model 2, respectively. Our trained denoiser significantly reduces the noise for both noisy images, with denoising performance on model 1 noise reaching a significantly higher PSNR (26.01) than for the higher-noise model 2 noise (24.53), which is to be expected. The additive residuals between the noisy and denoised images at the right show that the model identifies a fairly continuous pattern of noise across the entire image, with shifts due to the changing intensity values on the image. The denoiser also brings out many sharper features as seen around sharp edges in the image, particularly in the enlarged insets, demonstrating its effectiveness at deblurring as well. The consistency of structure in the residuals between different regions of the image suggests that the models are primarily removing noise as opposed to real features. Despite being trained to remove noise of the same type used in these noisy test images, the model cannot fully recover all information, although the fidelity of the denoised images is greatly improved.

FIG. 6.

Demonstrating the denoising performance for the castle test image at the two Poisson noise levels and the corresponding trained neural network models. Both noise models incorporate an AWGN of 5% and a Gaussian blur of σ = 1, with a Poisson noise of λ = 10 for noise model 1 (top) and λ = 5 for noise model 2 (bottom). The peak signal-to-noise ratio and root-mean-square error (RMSE) are shown for the noisy and denoised images, showing that our denoising models significantly increase the fidelity of the image. The additive residuals between the noisy and denoised images are displayed at right, and they display the scale and intensity of the noise inferred by our model.

FIG. 6.

Demonstrating the denoising performance for the castle test image at the two Poisson noise levels and the corresponding trained neural network models. Both noise models incorporate an AWGN of 5% and a Gaussian blur of σ = 1, with a Poisson noise of λ = 10 for noise model 1 (top) and λ = 5 for noise model 2 (bottom). The peak signal-to-noise ratio and root-mean-square error (RMSE) are shown for the noisy and denoised images, showing that our denoising models significantly increase the fidelity of the image. The additive residuals between the noisy and denoised images are displayed at right, and they display the scale and intensity of the noise inferred by our model.

Close modal

Figure 7 shows NN denoising results on synthetic transmission images from simulations of the experiment under similar perturbation conditions and at a similar time as the early time image of Fig. 2. The clean image of Fig. 7(a) shows a lot of structure that is recognizable from the data image, like the shocked opaque layer toward the center moving to the right of the image with visible perturbations to the left. Other important features are the reduced transmission in the shocked region trailing the thin layer on the left and also in the curved shocked region on the right. The noisy image Fig. 7(b) uses the same λ = 10 noise model as the castle image at the top of Fig. 6. Figure 7(c) shows the difference between the noisy image and the clean image. Figures 7(d)7(f) are denoising results using models trained with different levels of Poisson noise as noted, with λ ∈ [20, 100], λ ∈ [10, 100], and λ ∈ [5, 50], respectively, to see the effect of training. The residual images of Figs. 7(g)7(i) illustrate how well the denoisers remove noise. The model used in Figs. 7(d) and 7(g) was trained with the least amount of Poisson noise, and while the noise is significantly reduced, the results suffer from increased hallucinations of fine structure throughout. The denoising models of Figs. 7(e) and 7(f) both include λ = 10 Poisson noise in training as was used on the synthetic image, and they perform equally well on the broad features as seen in the residual images, with only some slight variation in small-scale features. These results give us confidence that our trained denoisers can be effectively applied to real data to improve analysis.

FIG. 7.

Comparison of denoising results on a simulated noisy image. The clean transmission map is shown in (a), simulated under conditions similar to the data image of 9 (a). Noise is added to the image following the 5% AWGN, λ = 10 Poisson noise, and σ = 1 Gaussian blur noise model, and the noisy image is shown in (b), with the difference from the clean image shown in (c). Results of denoising models trained on different ranges of Poisson noise are shown in (d)–(f), with the difference from the clean image shown beneath in (g)–(i). The lower Poisson noise model of (d) leaves some hallucinated features and achieves a lower PSNR compared to the results from (e) and (f), which achieve a better definition of smooth features and are comparable in PSNR.

FIG. 7.

Comparison of denoising results on a simulated noisy image. The clean transmission map is shown in (a), simulated under conditions similar to the data image of 9 (a). Noise is added to the image following the 5% AWGN, λ = 10 Poisson noise, and σ = 1 Gaussian blur noise model, and the noisy image is shown in (b), with the difference from the clean image shown in (c). Results of denoising models trained on different ranges of Poisson noise are shown in (d)–(f), with the difference from the clean image shown beneath in (g)–(i). The lower Poisson noise model of (d) leaves some hallucinated features and achieves a lower PSNR compared to the results from (e) and (f), which achieve a better definition of smooth features and are comparable in PSNR.

Close modal

To show that our new machine learning denoiser is preferable, we compare the results with other methods of image processing commonly used in the field. Figure 8 shows the denoising results from some of these methods. Figures 8(b) and 8(c) apply Gaussian blurs of the listed σ = 5 and σ = 10, respectively, to the noisy image. It is apparent that blurring reduces the noise, as indicated by the increased PSNR, but it also destroys real small-scale features like the edges of the shocked foam and the opaque tracer layer, and reduces the contrast. At σ = 5, the blurred image is still susceptible to longer-range fluctuations, but reducing those fluctuations by further blurring to σ = 10 destroys real features throughout the image, like the perturbations in the tracer layer as seen in the inset. The median filter replaces the value of each pixel with the median of a surrounding neighborhood of pixels, and the 9 × 9 pixel window median filter in Fig. 8(d) produced the best PSNR for this filter type. The median filter reduces the noise but tends to create macro-pixels and leaves similar long-range fluctuations as the smaller Gaussian blur. The Butterworth filter convolves a frequency-dependent gain curve with the image in frequency space. The Butterworth filter in Fig. 8(e) used was of order n = 1 and width f = 37 and produced the best PSNR of this filter type.

FIG. 8.

Comparing results of some common denoising methods with our NN denoiser. (b) and (c) Gaussian blurs of the listed σ, (d) a median filter with a 9 × 9 pixel window, (e) a Butterworth filter of order n = 1 and cutoff f = 37, and (f) our NN model from Fig. 7(e). Our trained neural network denoiser model has the highest PSNR of these methods.

FIG. 8.

Comparing results of some common denoising methods with our NN denoiser. (b) and (c) Gaussian blurs of the listed σ, (d) a median filter with a 9 × 9 pixel window, (e) a Butterworth filter of order n = 1 and cutoff f = 37, and (f) our NN model from Fig. 7(e). Our trained neural network denoiser model has the highest PSNR of these methods.

Close modal

Of all the filters shown in Fig. 8, our neural network denoiser outperforms them all on the noise model combining Poisson noise, additive noise, and a Gaussian blur. Our denoiser significantly reduces the short- and long-range perturbations from the noise while retaining contrast and sharpening real features. In addition, the other filters all require hyperparameters that must be tuned for each image, and the ideal tuning cannot be known for real data images. In contrast, our denoiser is trained to remove a specific type of noise, which we estimated from the data, and is applicable to a range of noise without the need for additional hyperparameters. The hyperparameters of our model are essentially the noise we imparted on the training set. In addition, the noise we impart on this training set is not guaranteed to be an accurate representation of the real noise. If the noise of a system falls outside of what we use in our training set, it is relatively simple to train a new model using a new noise model using the same architecture and methodology we have presented.

Figure 9 shows the results of our trained denoiser model on the data images previously shown in Fig. 2. Note that the trained denoising model is fully self-contained and it does not require any additional hyperparameters in application, as these are determined by the training process. The NN denoiser produces images that retain much of the internal structure of the opaque layer while significantly smoothing the signal in the surrounding foam regions. The blue boxes show how the denoiser performs on the opaque tracer layer, where much of the data analysis is focused. In the N180724-001 image in particular, much of the small-scale noise has been reduced, and the perturbations on the trailing side are much clearer. The noise is also noticeably reduced in the layer of the N190923-001 image, but features appear blurrier. Additional blurring on the bottom image is also apparent when comparing the red boxes that show the gold fiducial grid. The blurring between images can be explained by the fact that the second image is a strip 2 image, taken 4.5 ns after strip 1, allowing much more time for the high-Z materials to absorb x rays from the continuous backlighter drive. In addition, the edges around the fiducial teeth at the top and bottom and around the sides of the beryllium window are sharpened, and the apparent additive noise in the low-intensity regions of the images is significantly reduced. The subtractive residual of the NN denoised image from the noisy data image shows that the estimated noise resembles a continuous 2D Gaussian process throughout, modified only by changes in the local intensity while remaining of the same general character.

FIG. 9.

Results of our trained denoising model on the two HGXD x-ray images from Fig. 2. For both images, the additive residual between the denoised image and the noisy image shows that denoiser has removed a predominantly random field of noise of consistent correlation scale throughout, with a multiplicative dependence on the intensity. The enlarged insets of the high-opacity tracer layer show that many of the real intensity variations have been preserved.

FIG. 9.

Results of our trained denoising model on the two HGXD x-ray images from Fig. 2. For both images, the additive residual between the denoised image and the noisy image shows that denoiser has removed a predominantly random field of noise of consistent correlation scale throughout, with a multiplicative dependence on the intensity. The enlarged insets of the high-opacity tracer layer show that many of the real intensity variations have been preserved.

Close modal

We show the results of three models trained with different levels of Poisson noise on the data images in Fig. 10. Similar to our comparison on synthetic noise in Fig. 7, the differences between the higher noise λ ∈ [10, 100] and λ ∈ [5, 50] models are negligible. However, in the lower noise λ ∈ [20, 100] model, there are significant artifacts throughout. These kinds of artifacts tend to occur when the input image to a model has significantly different noise than what the model was trained on and suggest that our estimate of the noise from the data for the other two models is at least fairly representative of the real noise.

FIG. 10.

Comparison of denoising results on data images using the same model architecture but trained on different levels of Poisson noise. In general, training on lower levels of noise produces increased hallucination of connected structures from the noise. Models trained on higher levels of noise produce smoother images but may introduce additional levels of blur on features that can be seen by eye.

FIG. 10.

Comparison of denoising results on data images using the same model architecture but trained on different levels of Poisson noise. In general, training on lower levels of noise produces increased hallucination of connected structures from the noise. Models trained on higher levels of noise produce smoother images but may introduce additional levels of blur on features that can be seen by eye.

Close modal

In addition to the specific set of MShock hydrodynamics experiments, our denoiser model also works well on x-ray images from other experiments without requiring retraining another model. In Fig. 11, we show the result of our denoising model on an image from the LANL’s shock–shear campaign at the NIF.12,27,28 The shock–shear campaign was very similar to the MShock campaign, being its predecessor, and used a similar camera and backlighter configuration. While there are likely more aspects of the system that need to be considered, the denoiser performs fairly well at removing noise-like features. This result demonstrates the utility of our denoiser and shows that this method can apply well to any type of image so long as the model is trained on a reasonable approximation of the detector noise.

FIG. 11.

Example of the neural network denoiser on an image from NIF shot N150604-002 from the shock–shear campaign that used a similar camera and backlighter configuration.

FIG. 11.

Example of the neural network denoiser on an image from NIF shot N150604-002 from the shock–shear campaign that used a similar camera and backlighter configuration.

Close modal

In this paper, we have presented a new, neural network-based method for denoising x-ray images from HED hydrodynamics and ICF experiments. Focusing on data from the NIF MShock campaign, we estimated the noise in our images to resemble a combination of additive and Poisson noises with a Gaussian blur. We modeled our denoising architecture after previously successful models, albeit with some significant additions of dense connections and splitting the output into additive and multiplicative components of the noise. We trained our denoiser on a large set of noisy training images generated from the BSD300 set of natural images by augmenting clean patches with our estimated range of noise. Our denoiser effectively reduces noise from the training and testing sets of images and also appears to work well on our noisy experimental data, retaining many important physical features. With these results, we should be able to more accurately infer quantitative values from our data, with confidence that the structures being removed by our denoiser are predominantly noise.

Although we focus on a specific approximation of the noise observed on the HGXD images from our experiments in this paper, our method could easily be adapted for GXD CCD images as well, and neural networks can, in principle, be trained for any type of noise. There are some minor downsides to neural network denoising, however—the trained model is not a universal denoiser, and it will primarily attempt to recognize the level and types of noise it was trained on, which is likely to reduce its denoising effectiveness on images that exhibit noise outside of this range. In particular, image denoising with a model trained on lower noise than the input may not remove as much noise as it would otherwise, and on the contrary, using a model trained on higher noise levels than an input image may over-filter it, reducing some of the real information.

In the future, it may be possible to further improve the denoising performance for our HED experiments by training on synthetic images, which incorporate a more physical model of the camera noise. It should also be possible to improve the denoising model by changing the model architecture, but there is no simple or straightforward path for improvement. The simple addition of more blocks or model parameters to the network is not guaranteed to improve denoising performance; instead, there is a complicated and unknown relationship between model size and generalizability. In tests we have run, we find that adding more blocks tends to reduce the generalizability of the model to data, in the form of additional blur to the denoised images. Conversely, with fewer blocks, the network tends to hallucinate more connected features from the noise, similar to reducing the Poisson noise in training as in Fig. 10. Despite the above downsides, neural network models are an attractive option for denoising a wide range of data that could be put to broader use.

Los Alamos National Laboratory is operated by Triad National Security, LLC for the National Nuclear Security Administration of U.S. Department of Energy under Contract No. 89233218CNA000001.

The authors have no conflicts to disclose.

Joseph M. Levesque: Writing – original draft (lead); Writing – review & editing (lead). Elizabeth C. Merritt: Writing – original draft (supporting); Writing – review & editing (supporting). Kirk A. Flippo: Writing – original draft (supporting); Writing – review & editing (supporting). Alexander M. Rasmus: Writing – original draft (supporting); Writing – review & editing (supporting). Forrest W. Doss: Writing – review & editing (supporting).

The code used for training the neural network models and some pre-trained models used in this work will be made available. Data images may be made available upon reasonable request.

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