Score-based diffusion models for generating liquid argon time projection chamber images

For the first time, we show high-fidelity generation of Liquid Argon Time Projection Chamber (LArTPC-like) data using a generative neural network. This demonstrates that methods developed for natural images do transfer to LArTPC-produced images, which, in contrast to natural images, are globally sparse but locally dense. We present the score-based diffusion method employed. We evaluate the fidelity of the generated images using several quality metrics, including modified measures used to evaluate natural images, comparisons between high-dimensional distributions, and comparisons relevant to LArTPC experiments.

Figure 1

Visualization of the diffusion processes used to train and sample a generative model for LArTPC-like images. The generative process takes noise images (middle columns of the figure) to images that contain particle trajectories similar to what might be seen in a LArTPC. This generative process can be seen as the reverse of the forward process defined by the stochastic differential equation (SDE) on the left, which transforms LArTPC images into noise images. Implementing the reverse process SDE (on the right) from noise to data requires estimating the score function—a quantity related to the underlying probability distribution of the LArTPC images. The score function is learned through training with data. For simplicity, we have defined a backward-time variable $s=T-t$.

Figure 2

Loss per epoch compared against training set and validation set. The loss function measures the mean-squared error between the predicted and true score functions for a training batch of images. The lighter curve in the background shows the loss for one training batch sampled every 100 iterations. The darker curves in the foreground result from smoothing through a running average over 5 epochs.

Figure 3

Samples of randomly selected training (left) and generated (right) images. See more samples in Appendix pp3.

Figure 4

Results from our Turing test where participants were asked to differentiate 50 generated from 50 “true” LArTPC images. The total accuracy is broken down by track (67) and shower (33) images. This corresponds to a binomial distribution where $1\sigma =\pm 5\%$ from random chance (50%). P-values for the total accuracy are boxed above each entry. Only one participant was statistically significant, suggesting our generated images are a close match but still have room for improvement. Note that some participants did not complete the full test; their accuracies have been extrapolated, but their p-values (marked with an asterisk) are less significant.

Figure 5

Visualization of the output of the semantic segmentation network (SSNet) used to label track and shower pixels. The top row shows four LArTPC images from the training set. The bottom row shows the corresponding pixel labels as part of a tracklike trajectory (yellow) or showerlike trajectory (cyan). Tracklike trajectories are those produced by muons, charged pions, and protons. Showerlike trajectories are due to electromagnetic cascades initiated by either an electron or the interaction of a photon with the argon. Background pixels where the pixel value falls below some threshold are shown in purple. The output of SSNet on background pixels is not used in our analysis.

Figure 6

High dimensional goodness-of-fit tests comparing images generated from various epochs against the full training and validation datasets. Explanations of these measures can be found in the text.

Figure 7

SSNet pixel value histograms. (a) shows the number of shower pixels per image, and (b) is the corresponding certainties of these labels. Similarly, (c) is the number of track pixels per image, and (d) is the certainty of the track labels. Across all distributions, we are comparing our generated images at key epochs against our LArTPC validation dataset. We see that epochs 50 and 150 are nearly indistinguishable from each other and the LArTPC validation dataset.

Figure 8

Our SSNet equivalent of Fréchet inception distance (FID) compares generated images for each epoch against the training set (blue) and validation set (orange). As with previous metrics, we see a rapid improvement and then a plateau in performance. All epochs in the plateau are within a narrow range, with epoch 150 being the minimum. For the convenience of those who might wish to compare future work with our results, the values for each epoch are shown in Appendix pp2, Table 4. The minimum is at epoch 150, with all epochs after 50 being very close.

Figure 9

Histogram showing the fraction of track images per length of the track (excluding background noise). The number of track events is shown in the legend. The peak of lengths at 64 results from nearly straight tracks that pass through the entire $64\times 64$ image. 90 is the maximum possible length along the image diagonal.

Figure 10

The distribution of track widths found in images from the validation dataset and images generated from epochs 50 and 150. Only perfectly vertical or horizontal tracks have a width of 1. The majority of the tracks are 1–6 pixels wide. Tracks widths $\ge 28$ have been collected into an overflow bin.

Figure 11

Example tracks from the validation dataset. The track is highlighted in blue, and the excluded background noise is in white. Intensities at a maximum for visualization purposes. Track (a) has some minor curvature. Track (b) is composed of three connected tracks and a few pixels of intersecting background noise.

Figure 12

The total charge deposited in shower event images from epochs 50, 150, and the validation dataset.

Figure 13

Visual check for mode collapse. On the left-most column, the large images are an example of a showerlike (top) and tracklike (bottom) trajectory generated by our model. For each large image, we use the Euclidean distance (L2 norm) to define the five nearest images in the training dataset (top small row) and generated dataset (bottom small row). The distances are labeled above each comparison image. We have excluded the image itself from the generated dataset. More examples can be found in Appendix pp3, including using EMD distance.

Figure 14

Checking for mode collapse by finding the nearest neighbors using Euclidian distance (L2 norm). The five large images are generated events from 50 epochs of training. For each generated image we have found the five closest matches from the training dataset (top row) and generated dataset (bottom row). We have excluded itself (distance zero) in the generated image search. The number above each comparison image is its distance from the comparison image. There are an equal number of images (50,000) in the training and generated datasets. We see that images are distinct and have reasonably large distances, suggesting that we are generating unique images.

Figure 15

Checking for mode collapse by finding the nearest neighbors using Earth (energy) mover’s distance. The five large images are generated events from 50 epochs of training. For each generated image we have found the five closest matches from the training dataset (top row) and generated dataset (bottom row). We have excluded itself (distance zero) in the generated image search. The number above each comparison image is its distance from the comparison image. There are an equal number of images (50,000) in the training and generated datasets. We see that images are distinct and have reasonably large distances, suggesting that we are generating unique images.

Figure 16

Randomly selected images from the training dataset.

Figure 17

Randomly selected images from the epoch 50 generated image dataset.

Figure 18

SSNet pixel value histograms with VQVAE.

Figure 19

Physics-based metrics with the VQ-VAE generated dataset. (a) Track length, (b) Track width, and (c) Shower charge.