A Self-Supervised Approach to Reconstruction inSparse X-Ray Computed Tomography

Rey Mendoza∗ 1 , Minh Nguyen∗1 , Judith Weng Zhu∗1 ,Vincent Dumont2 , Talita Perciano2 , Juliane Mueller2,3 , Vidya Ganapati1,2
1
Swarthmore College, 2 Lawrence Berkeley National Laboratory,3
National Renewable Energy Laboratory

Abstract
Computed tomography has propelled scientific advances in fields from biology
to materials science. This technology allows for the elucidation of 3-dimensional
internal structure by the attenuation of x-rays through an object at different rotations
relative to the beam. By imaging 2-dimensional projections, a 3-dimensional object
can be reconstructed through a computational algorithm. Imaging at a greaternumber of rotation angles allows for improved reconstruction. However, taking
more measurements increases the x-ray dose and may cause sample damage. Deepneural networks have been used to transform sparse 2-D projection measurements
to a 3-D reconstruction by training on a dataset of known similar objects. However,obtaining high-quality object reconstructions for the training dataset requires highx-ray dose measurements that can destroy or alter the specimen before imaging iscomplete. This becomes a chicken-and-egg problem: high-quality reconstructions
cannot be generated without deep learning, and the deep neural network cannot
be learned without the reconstructions. This work develops and validates a selfsupervised probabilistic deep learning technique, the physics-informed variational
autoencoder, to solve this problem. A dataset consisting solely of sparse projection
measurements from each object is used to jointly reconstruct all objects of theset. This approach has the potential to allow visualization of fragile samples with
x-ray computed tomography. We release our code for reproducing our results at:https://github.com/vganapati/CT_PVAE.

1Introduction & Related Work

1.1

Computed Tomography

The power of x-rays to penetrate through many materials has allowed for advances in scientificunderstanding. In particular, computed tomography has been used in fields such as biology [1],medicine [2], materials science [3], and geoscience [4] to elucidate the internal structure of samples.
Computed tomography works by measuring the attenuation of an x-ray beam through an object atdifferent rotations relative to the beam. From the projection images created by parallel beams ateach rotation, a 3-dimensional object can be reconstructed with a computational algorithm. Imaging
at many rotation angles allows for improved reconstruction, however, fragile samples can only be
imaged under limited x-ray dose before damage and structural changes occur.
1.2

Reconstruction Algorithms

A 3-dimensional object can be reconstructed from projection images through direct algorithms such asfiltered backpropagation or iterative algorithms [5, 6]. In recent years, algorithms using deep learning∗

Equal contribution. Correspondence to vidyag@berkeley.edu.

Machine Learning and the Physical Sciences workshop, NeurIPS 2022.

have come into prominence [5, 7–12]. In these approaches, the parameters of the deep neural network
are found by training on large datasets of known input-output pairs. The computational burdenof training is high and a training dataset is required, which can be generated through high-quality
experimental data and traditional algorithms. However, once trained, a relatively quick forward passthrough the network can generate the reconstruction [5]. Deep learning has also shown the potential
to improve upon conventional methods when there are a limited number of projection images, as aprior is learned from the data and embedded into the neural network parameters [5, 7].
1.3

Deep Learning without Ground Truth

While deep learning methods for computed tomography are promising, these approaches requirea training dataset of reconstructions. High-quality reconstructions need measurements at many
rotation angles, and this high radiation dose may destroy or alter fragile specimens during the dataacquisition process. A chicken-and-egg problem presents itself: the deep learning algorithm needs
reconstructions, and the reconstructions need the deep learning algorithm.
An approach to solve this problem involves using training datasets consisting only of sparse angle
measurements for every object of the set. The intuition is that these sparse measurements taken
together can provide enough information to infer a prior distribution of the objects. One method
is to train generative adversarial networks to create high-quality reconstructions by rewarding reconstructions that have corresponding sparse measurements that lie in the training distribution ofsparse measurements [13–17]. Another method is to train directly with low-dose measurements,
penalizing reconstructions by the discrepancy between simulated low-dose and true low-dose measurements [18–21]. Finally, other works use multiple low-dose measurements of each object intraining [22, 23], though this may be unsuitable for fragile samples that cannot withstand repeatedmeasurement.
1.4

Probabilistic Formulation

In all deep learning methods without ground truth surveyed, the reconstruction is given as a point
estimate. In this work, we aim to find the posterior probability distribution P (O|M ), where O
is the object being reconstructed and M = [M1 , M2 , ..., Mn ] are the measurements, allowing theuncertainty of the prediction to be modeled. We aim to lower the total number of measurements n tominimize the x-ray dose. We assume that we have a set of m objects {O1 , O2 , ..., Om }, sampled fromsome distribution P (O), and we aim to reconstruct all objects in the set. For each of the m objects, weare allowed n measurements. Each set of measurements for an object j, Mj = [Mj1 , Mj2 , ..., Mjn ]is obtained with chosen rotation angles pj = [pj1 , pj2 , ..., pjn ]. We assume that the forward model
physics P (M |O; p) = P (M |O) is known; in computed tomography the forward model is theRadon transform with Poisson noise. For every object O, we aim to find the posterior distribution(O)
P (O|M ) = P (MP|O)P. This work tackles the following problems that arise in finding the posterior:(M )(1) construction of the prior P (O) with no directly observed O, only measurements M on each objectof the set, (2) calculating P (O|M ) in a tractable manner.

2

Methods

2.1

Physics-Informed Variational Autoencoder

We create a framework for posterior estimation that is inspired by the mathematics of the variational
autoencoder [24, 25]. In a variational autoencoder, the goal is to learn how to generate new examples,
sampled from the same underlying probability distribution as a training dataset of objects. To
accomplish this task, a latent random variable z is created that describes the space on a lowerdimensional manifold. A deep neural network defines a function (the “decoder”) from a sample of z
to a probability distribution P (O|z). The parameters of the deep neural network are optimized tomaximize the probability of generating the objects of the training dataset.
In this work, we aim to find the posterior probability distribution P (O|M ), where O is the object
being reconstructed and M are the projection measurements. In our case, we only have a dataset ofnoisy measurements M and no ground truth objects O. However, the forward model, P (M |O), is2

known. Thus, instead of maximizing the probability of generating O, we can maximize the probabilityof generating M , a formulation we call the “physics-informed variational autoencoder.”Figure 1: The developed physics-informed variational autoencoder.
RR
We aim to maximize P (M ) =P (M |O)P (O|z)P (z)dOdz. To compute this integral in acomputationally tractable manner, we can approximate with sampled values. As in [24, 25], formost randomly sampled values of z and O, the probability P (M |O, z) is close to zero, causing poor
scaling of sampled estimates. Similar to variational autoencoders, our framework solves this problem
by estimating the parameters of P (z|M ) by processing the measurements M using a function withtrainable parameters (called the “encoder,” also formulated as a deep neural network). The estimateof P (z|M ) is denoted Q(z|M ). The Kullback–Leibler divergence between the distributions is given
by D[Q(z|M )||P (z|M )] = Ez∼Q [log Q(z|M ) − log P (z|M )]. We also have, by Bayes’ Theorem,
log P (z|M ) = log P (M |z) + log P (z) − log P (M ). Combining the expressions yields:
Zlog P (M )−D[Q(z|M )||P (z|M )] = Ez∼Q
P (M |O)P (O|z)dO −D [log Q(z|M )|| log P (z)] .
The first term on the right side of this expression can be estimated with sampled values. As
Kullback–Leibler divergence is always ≥ 0 and reaches 0 when Q(z|M ) = P (z|M ), maximizing
the right side (defined here as the −loss) during training causes P (M ) to be maximized while forcing
Q(z|M ) towards P (z|M ). In contrast to a conventional variational autoencoder, we do not attempt
to use this formulation to synthesize arbitrary objects O by sampling P (z) directly. This framework
only attempts reconstruction on the training examples themselves. After training, P (O|M ) for
every object can be sampled by first sampling Q(z|M ) then P (O|z); see Fig. 1. Crucially, unlike
most data-driven approaches to reconstruction in computational imaging, the developed framework
assumes that no ground truth dataset of objects O is available.
In this work, the encoder and decoder of Fig. 1 take the same basic shape as a U-Net [26]. The
input to the encoder is an object reconstruction created with a standard algorithm from the sparse setof projections, using the T OMO P Y Python package [27], as well as specification of the projection
angles used. The skip connections to the decoder are parametrized as Gaussian distributions, andsampled to determine the latent variables z. The likelihood distribution P (M |O) is given by theRadon transform with the addition of Poisson noise.

3

Results & Discussion

We prototype and validate the physics-informed variational autoencoder with synthetic datasets.
Code to reproduce our results are at https://github.com/vganapati/CT_PVAE. We utilize 2-D
objects and 1-D projections without loss of generality; a 3-D object in x-ray computed tomography isreconstructed as a series of 2-D axial slices.
3.1

Simple Toy Example

Our first dataset consists of two unique objects. Each object can be
represented by 2 × 2 pixels, and projections can be taken at rotation
angles of 0 and π2 radians; see Fig. 2. The projection of both theobjects at π2 is the same, but the projections at 0 radians are different.
We assume that P (O = O1 ) = P (O = O2 ) = 12 . We sample P (O),and then take a noisy measurement M = [M1 ] at one rotation angle
p (i.e. n = 1 and P (p = 0) = P (p = π2 ) = 12 ). We complete thissampling and measurement procedure m = 1 024 times. From all
the measurements, and no knowledge of the prior P (O), we aim to
Figure 2: Toy dataset consistdetermine the posterior P (O|M ) for each M by training the physicsing of 2 object types.
informed variational autoencoder. Training on an NVIDIA GeForce3

(a) O1 measured at π2 radians.

(b) O1 measured at 0 radians.

Figure 3: Marginal posterior probabilities for each pixel from measurements on the toy dataset;measurements taken on object O1 of the dataset. See Fig. 5 in the Appendix for posteriors on O2 .
GPU takes 47 minutes. Training parameters are available in the released documentation and code.
The true posterior is compared with the posterior from 20 000 samples of the trained physics-informed
variational autoencoder, showing good qualitative agreement; see Fig. 3. The marginal posterior foreach of the four pixels is compared for four different example measurements. With measurements
taken at the rotation angle π2 , there is approximately equal probability of the pixel values from thetwo object types. However, with measurements taken at 0 radians, the posterior is nearly a delta
function at the true pixel value.
3.2

Synthetic Foam Dataset

We generate the second dataset of 1 000 foam images of 128 × 128 pixels through the Python package
XD ESIGN [28], and projections are taken at 180 equally-spaced angles. The image of the projections
is known as the sinogram, and we emulate sparse sinograms by removing 160 of the projection angles
either uniformly or randomly; see Fig. 6 in the Appendix for example images and corresponding
sinograms of the dataset. We jointly reconstruct all images of this dataset by training the physicsinformed variational autoencoder. Training on an NVIDIA Tesla V100 takes 2.2 hours. Training
parameters are available in the released code and documentation.
Fig. 4 depicts example objects from the synthetic foam dataset, reconstructed with the Fourier
grid reconstruction algorithm (gridrec) [29]. Reconstructions were performed with two otherstandard algorithms, the simultaneous algebraic reconstruction technique [27] and the Total Variation
reconstruction technique [30]; however gridrec outperformed these techniques on image quality
metrics of structural similarity (SSIM), mean-squared error (MSE), and peak signal-to-noise (PSNR).
The reconstructions with gridrec on the sparse sinograms are compared with the reconstructions
from the physics-informed variational autoencoder (P-VAE); see Fig. 4. We observe that the P-VAE
improves over gridrec for the sparse sinograms, and there is a slight improvement by randomly
selecting the angles in every example over uniform sampling; see Fig. 7 in the Appendix for results
averaged over the entire 1 000 object dataset and aggregated over 10 independent trials.

(a) Example 1(b) Example 2(c) Example 3

Figure 4: Foam dataset reconstruction results.

4

Conclusions & Limitations

We present a novel data-driven, self-supervised algorithm for image reconstruction in computed
tomography and validate on synthetic datasets. Our method is probabilistic, allowing for quantificationof uncertainty in applications, and self-supervised, requiring no ground truth or reference training
examples. Though initial results are promising, our work is limited in that we only consider highly
4synthetic data. Real experimental data may pose additional problems such as model mismatch thatneed to be considered in the construction of the likelihood P (M |O). In experiments, thousandsof rotation angles may be collected with measurements on the order of 100 GB per object [6]. An
important direction for future work is high memory management techniques for training.

5

Broader Impact

Computed tomography uses x-rays to probe the 3-dimensional internal structure of objects, and is
widely used in many fields such as medicine, biology, and materials science. However, x-rays cancause sample damage and deformation during the measurement process, limiting the use cases ofcomputed tomography. Much effort has been put into sparse computed tomography, where a limited
number of measurements are used to measure and reconstruct an object. This work proposes a novel
algorithm for reconstruction in sparse computed tomography, with the potential positive broader
impact of making visualization of radiation-damage prone objects possible. However, practitioners
should be careful in applying these methods as there are no theoretical guarantees thus far. Applicationof this method could result in reconstruction artifacts that may have a different appearance thanartifacts from conventional methods, evading detection.

Acknowledgments and Disclosure of Funding
This work was supported in part by the U.S. Department of Energy, Office of Science, Office ofWorkforce Development for Teachers and Scientists (WDTS) under the Visiting Faculty Program(VFP) and the AAUW Research Publication Grant in Engineering, Medicine and Science.

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AAppendix(a) O2 measured at π2 radians.

(b) O2 measured at 0 radians.

Figure 5: Marginal posterior probabilities for each pixel from measurements on the toy dataset;measurements taken on object O2 of the dataset.

Figure 6: Object examples generated for the foam image dataset. Complete sinograms are taken
at 180 equally spaced rotation angles. Uniformly sparse sinograms are taken at 20 equally spaced
angles and randomly sampled sparse sinograms are taken at 20 randomly chosen angles; these angles
are re-sampled for each object of the dataset.

9Figure 7: Image quality metrics of structural similarity (SSIM), peak signal-to-noise (PSNR), andmean-squared error (MSE) for reconstruction algorithms applied to the synthetic foam dataset,averaged over the entire dataset for 10 independent optimization trials. Reconstructions from (a)gridrec with the complete sinogram of 180 rotation angles, (b) gridrec with the sparse sinogramof 20 angles, uniformly spaced, (c) gridrec with the sparse sinogram of 20 angles, randomly spaced,with each example having different angles chosen, (d) the physics-informed variational autoencoderwith the sparse sinogram of 20 angles, uniformly spaced, and (e) the physics-informed variational
autoencoder with the sparse sinogram of 20 angles, randomly spaced.

10